anders_h feed - LessWrong 2.0 Reader anders_h’s posts and comments on the Effective Altruism Forum en-us Counterfactual outcome state transition parameters https://www.lesswrong.com/posts/K3d93AfFE5owfpkx4/counterfactual-outcome-state-transition-parameters <p>Today, my paper <a href="https://www.degruyter.com/view/j/em.ahead-of-print/em-2016-0014/em-2016-0014.xml?format=INT">&quot;The choice of effect measure for binary outcomes: Introducing counterfactual outcome state transition parameters&quot; </a>has been published in the journal <em>Epidemiologic Methods</em>. The version of record is behind a paywall until December 2019, but the final author manuscript is available as a preprint at <a href="https://arxiv.org/abs/1610.00069">arXiv</a>. </p><p>This paper is the first publication about an ambitious idea which, if accepted by the statistical community, could have significant impact on how randomized trials are reported. Two other <a href="https://arxiv.org/pdf/1610.00068.pdf">manuscripts</a> from the same project are available as working papers on arXiv. This blog post is intended as a high-level overview of the idea, to explain why I think this work is important. </p><p><em>Q: What problem are you trying to solve?</em></p><p>Randomized controlled trials are often conducted in populations that differ substantially from the clinical populations in which the results will be used to guide clinical decision making. My goal is to clarify the conditions that must be met in order for the randomized trial to be informative about what will happen if the drug is given to a target population which differs from the population that was studied.</p><p>As a first step, one could attempt to construct a subgroup of the participants in the randomized trial, such that the subgroup is sufficiently similar to the patients you are interested in, in terms of some observed baseline covariates. However, this leaves open the question of how one can determine what baseline covariates need to be accounted for. </p><p>In order to determine this, it would be necessary to provide <em>a priori </em>biological facts which would lead to the effect in one population being equal to the effect in another population. For example, if we somehow knew that the effect of a drug is entirely determined by some gene whose prevalence differs between two countries, it is possible that when we compare people in Country A who have the gene with people in Country B who also have the gene, and compare people in Country A who don&#x27;t have the gene with people in Country B who don&#x27;t have the gene, the effect is equal between the relevant groups. Using an extension of this approach, we can try to look for a set of baseline covariates such that the effect can be expected to be approximately equal between two populations once we make the comparisons within levels of the covariates. </p><p>Unfortunately, things are more complicated than this. Specifically, we need to be more precise about what we mean by the word &quot;effect&quot;. When investigators measure effects, they have several options available to them: They can use multiplicative parameters (such as the risk ratio and the odds ratio), additive parameters (such as the risk difference), or several other alternatives that have fallen out of fashion (such as the arcsine difference). If the baseline risks differ between two populations (for example, between men and women), then at most one of these parameters can be equal between the two groups. Therefore, a biological model that ensures equality of the risk ratio cannot also ensure equality of the risk difference. <em>The logic that determines whether a set of covariates is sufficient in order to get effect equality, is therefore necessarily dependent on how we choose to measure the effect</em>. </p><p>Making things even worse, the commonly used risk ratio is not symmetric to the coding of the outcome variable: Generalizations based on the ratio of probability of death, will give different predictions from generalizations based on the ratio of probability survival.. In other words, when using a risk ratio model, your conclusions are not invariant to an arbitrary decision that was made when the person who constructed the dataset decided whether to encode the outcome variable as (death=1, survival=0) or as (survival=1, death=0).</p><p>The information that doctors (and the public) extract from randomized trials is often in the form of a summary measure based on a multiplicative parameter. For example, a study will often report that a particular drug &quot;doubled&quot; the effect of a particular side effect, and this then becomes the measure of effect that the clinicians will use in order to inform their decision making. Moreover, the standard methodology for meta-analysis is essentially a weighted average of the multiplicative parameter from each study. Any conclusion that is drawn from these studies would have been different if investigators had chosen a different effect parameter, or a different coding scheme for the outcome variable. These analytic choices are rarely justified by any kind of argument, and instead rely on a convention to always use the risk ratio based on the probability of death. No convincing rationale for this convention exists. </p><p>My goal is to provide a general framework that allows an investigator to reason from biological facts about what set of covariates are sufficient to condition on, in order for the effect in one population to be equal to the effect in another, in terms of a specified measure of effect. While the necessary biological conditions can at best be considered approximations of the underlying data generating mechanism, clarifying the precise nature of these conditions will be useful to assist reasoning about how much uncertainty there is about whether the results will generalize to other population.</p><p><em>Q: What are the existing solutions to this problem, and why do you think you can improve on them?</em></p><p>Recently, much attention has been given to a solution by Judea Pearl and Elias Bareinboim, based on an extension of causal directed acyclic graphs. Pearl and Bareinboim&#x27;s approach is mathematically valid and elegant. However, the conditions that must be met in order for these graphs to be a reasonable approximation of the data generating mechanism, are much more restrictive than most trialists are comfortable with.</p><p>Here, I am going to skip a lot of details about these selection diagrams, and instead focus on the specific aspect that I find problematic. These selection diagrams abandon measures of effect completely, and instead consider the counterfactual distribution of the outcome under the intervention separately from the counterfactual distribution of the outcome under the control condition. This resolves a lot of the problems associated with effect measures, but it also fails to make use of information that is contained in how these two counterfactuals relate to each other.</p><p>Consider for example an experiment to determine the effect of homeopathy on heart disease. Suppose this experiment is conducted in men, and determines that there is no effect. If we use selection diagrams to reason about whether these conclusions also hold in women, we will have to construct a causal graph that contains every cause of heart disease whose distribution differs between men and women, measure these variables and control for them. Most likely, this will not be possible, and we will conclude that we are unable to make any prediction for what will happen if women take homeopathic treatments. The approach simply does not allow us to try to extrapolate the effect size (even when it is null), since it cannot make use of information about how what happened under treatment relates to what happens under the control condition. The selection diagram approach therefore leaves key information on the table: In my view the information that is left out is exactly those pieces of information that could most reliably be used to make generalizations about causal effects.</p><p>A closely related point is that the Bareinboim-Pearl approach leads to a conclusion that meta-analysis can be conducted separately in the active arm and the control arm. Most meta-analysts would consider this idea crazy, since it arguably abandons randomization (which is an objective fact about how the data was generated) in favor of unverifiable and questionable assumptions encoded in the graph, essentially claiming that all causes of the outcome have been measured.</p><p></p><p><em>Q: What are counterfactual outcome state transition parameters?</em></p><p>Our goal is to construct a measure of effect that allows us to capture the relationship between what happens if treated, to what happens if untreated. We want to do this in a way that avoids the mathematical problems with standard measures of effect, and such that magnitude of the parameters has a biological interpretation. If we succeed in doing this, we will be able to determine what covariates to control for on the basis of asking what biological properties are associated with the magnitude of the parameters. </p><p>Counterfactual outcome state transition parameters are effect measures that quantify the probability of &quot;switching&quot; outcome state if we move between counterfactual worlds. We define one parameter which measures the probability that the drug kills the patient, conditional on being someone who would have survived without the drug, and another parameter which measures the probability that the drug saves the patient, conditional on being someone who would have died without the drug.</p><p>Importantly, these parameters are not identified from the data, except under strong monotonicity conditions. For example, if we believe that the drug helps some people, harms other people and has no effect on a third group, there is no monotonicity and the method cannot be used. However, it is sometimes the case that the drug only operates in one direction. For example, for most drugs, it is very unlikely that the drug prevents someone from getting an allergic reaction to it. Therefore, its effect on allergic reactions is monotonic.</p><p>If the effect of treatment is monotonic, one of the COST parameters is equal to 0 or 1, and the other parameter is identified as the risk ratio. If this is a treatment that reduces incidence, the COST parameter associated with a protective effect is equal to the standard risk ratio based on the probability of death. If on the other hand the treatment increases incidence, the COST parameter associated with a harmful effect is identified as the recoded risk ratio based on the probability of survival. Therefore, if we determine which risk ratio to use on the basis of the COST model, the risk ratio is constrained between 0 and 1. </p><p><em>Q: Is this idea new?</em></p><p>The underlying intuition behind this idea is not new. For example, Mindel C. Sheps published a remarkable paper in the <a href="http://www.nejm.org/doi/full/10.1056/NEJM195812182592505">New England Journal of Medicine in 1958</a>, in which she works from the same intuition and reaches essentially the same conclusions. Sheps&#x27; classic paper has more than 100 citations in the statistical literature, but her recommendations have not been adapted to any detectable extent in applied statistical literature. <a href="http://onlinelibrary.wiley.com/doi/10.1002/sim.1188/abstract">Jon Deeks</a> provided empirical evidence for the idea of using the standard risk ratio for protective treatments, and recoded risk ratio for harmful effects, in Statistics in Medicine in 2012. </p><p>What is new to this paper, is that we formalize the intuition Sheps was working from in terms of a formal counterfactual causal model, which is used as a bridge between the background biological knowledge and the choice of effect measure. Formalizing the problem in this way allows us to clarify the scope and limits of the approach, and points the direction to how these ideas can be used to inform future developments in meta-analysis.</p><p></p> anders_h K3d93AfFE5owfpkx4 2018-07-27T21:13:12.014Z Comment by Anders_H on The New Riddle of Induction: Neutral and Relative Perspectives on Color https://www.lesswrong.com/posts/SuDqsE3DgHnDiutyM/the-new-riddle-of-induction-neutral-and-relative#6H5RjupJGnhozCP4j <p>In my view, &quot;the problem of induction&quot; is just a bunch of philosophers obsessing over the fact that induction is not deduction, and that you therefore cannot predict the future with logical certainty. This is true, but not very interesting. We should instead spend our energy thinking about how to make better predictions, and how we can evaluate how much confidence to have in our predictions. I agree with you that the fields you mention have made immense progress on that. </p><p>I am not convinced that computer programs are immune to Goodmans point. AI agents have ontologies, and their predictions will depend on that ontology. Two agents with different ontologies but the same data can reach different conclusions, and unless they have access to their source code, it is not obvious that they will be able to figure out which one is right.</p><p>Consider two humans who are both writing computer functions. Both the &quot;green&quot; and the &quot;grue&quot; programmer will believe that their perspective is the neutral one, and therefore write a simple program that takes light wavelength as input and outputs a constant color predicate. The difference is that one of them will be surprised after time t, when suddenly the computer starts outputting different colors from their programmers experienced qualia. At that stage, we know which one of the programmers was wrong, but the point is that it might not be possible to predict this in advance.</p> anders_h 6H5RjupJGnhozCP4j 2017-12-02T18:56:46.422Z Comment by Anders_H on The New Riddle of Induction: Neutral and Relative Perspectives on Color https://www.lesswrong.com/posts/SuDqsE3DgHnDiutyM/the-new-riddle-of-induction-neutral-and-relative#bNsCr3cukTf9L6YyN <p>I am not sure I fully understand this comment, or why you believe my argument is circular. It is possible that you are right, but I would very much appreciate a more thorough explanation. </p><p>In particular, I am not &quot;concluding&quot; that humans were produced by an evolutionary process; but rather using it as background knowledge. Moreover, this statement seems uncontroversial enough that I can bring it in as a premise without having to argue for it.</p><p>Since &quot;humans were produced by an evolutionary process&quot; is a premise and not a conclusion, I don&#x27;t understand what you mean by circular reasoning.</p> anders_h bNsCr3cukTf9L6YyN 2017-12-02T17:20:29.841Z The New Riddle of Induction: Neutral and Relative Perspectives on Color https://www.lesswrong.com/posts/SuDqsE3DgHnDiutyM/the-new-riddle-of-induction-neutral-and-relative <p>Nelson Goodman&#x27;s &quot;New Riddle of Induction&quot; has previously been discussed on Less Wrong at <a href="http://lesswrong.com/lw/8fq/bayes">http://lesswrong.com/lw/8fq/bayes </a>and<a href="http://lesswrong.com/lw/mbr/grue"> http://lesswrong.com/lw/mbr/grue</a><em> . </em>Briefly, this riddle shows that any attempt to make future predictions by generalizing from past observations, may depend on arbitrary aspects of the reasoner&#x27;s language. This is illustrated in the context of the proposed time-dependent color predicates &quot;grue&quot; and &quot;bleen&quot;.</p><p>In this article, I propose that the resolution to this apparent paradox lies in the recognition that a neutral perspective exists, and that while an agent cannot know with certainty whether <em>their </em>perspective is neutral, they can assign significantly higher credence to their perspective being neutral because evolution had no reason to introduce an arbitrary time-dependent term in their color detection algorithm. I am unsure whether the argument is novel, but as far as I can tell, this particular solution is not discussed in any of the previous literature that I have consulted. </p><p>(Note: This article was originally written as coursework, and therefore contains an extended summary of Goodman&#x27;s original paper and of previous discussion in the academic literature. Readers who are familiar with the previous literature are encouraged to skip to the section &quot;The Grue Sleeping Beauty Problem&quot;)</p><p></p><p><strong>The Problem of Induction and Its Dissolution</strong></p><p>In a series of lectures published as the 1954 book &quot;Fact, Fiction and Forecast&quot;, Nelson Goodman argued that Hume&#x27;s traditional problem of induction has been &quot;dissolved&quot;, and instead described a different problem that confounds our effort to infer general facts from limited observations. Before I discuss this new problem - which Goodman termed &quot;The New Riddle of Induction&quot; - I briefly discuss what Goodman means when he asserts that the original problem of induction has been &quot;dissolved&quot;.</p><p>In Goodman&#x27;s view, it is true that there can be no necessary relationship between past and future observations, and it is therefore futile to expect logical certainty about any prediction. In his view, this is all there is to the problem of induction: If what you want from an inductive procedure is a logical guarantee about your prediction, then the problem of induction illustrates why you cannot have it, and it is therefore futile to spend philosophical energy worrying about knowledge or certainty that we know we can never have.</p><p>Goodman therefore argues that the <em>real</em> problem of induction is rather how to distinguish strong inferences from weak ones, in the sense that strong inferences are those which a reasonable person would accept, even in the absence of logical guarantees. In other words, he is interested in describing rules for a system of inference, in order to formalize the intuitions that determine whether we consider any particular prediction to be a reasonable inference from the observed data. </p><p>He models his approach to induction on a narrative about how the rules of deduction were (or continue to be?) developed. In his view, this occurs through a feedback loop or &quot;virtuous cycle&quot; between the inferential rules and their conclusions, such that the rules are revised if they lead to inferences we are unwilling to accept as logically valid, and such that our beliefs about the validity of an inference are revised if we are unwilling to revise the rules. Any attempt at deductive inference can then be judged by its adherence to the rules; and if a situation arises where our intuition about any particular inferential validity conflicts with the rules, we would have to adjust one of the two. In Goodman&#x27;s view, a similar approach should, and is, used to continuously improve on an set of rules that capture human beliefs about what generalizations tend to produce good predictions, and what generalizations tend to fail.</p><p>However, as we will see, even if we do not seek the kind of certainty that is ruled out by the lack of logical guarantees and necessary connections, and follow Goodman to focus our attention on distinguishing &quot;good&quot; generalizations from bad ones, there are important obstacles that arise from the fact that our inferential mechanisms are unavoidably shaped by our language, in the sense that the predictions depend on seemingly arbitrary features of how the basic predicates are constructed. </p><p><strong>The New Riddle of Induction: Goodman&#x27;s Argument</strong></p><p>Goodman&#x27;s goal is to describe the rules that determine whether inferring <em>s2</em> from <em>s1</em> is inductively &quot;valid&quot;, in the sense that a reasonable person would consider the truth of <em>s2</em> to be &quot;confirmed&quot; by the truth of <em>s1</em> even without the support of an argument that establishes <em>s1</em>--&gt;s<em>2</em> by deductive logic. In particular, we are interested in situations where <em>s1</em> and <em>s2</em> are very similar statements, where each statement makes some claim about a property that applies to some set of elements/objects, such that the property is equal between <em>s1</em> and <em>s2</em>, but such that <em>s2</em> refers to a larger, more general set of objects (i.e. where the set of elements referred to by<em> s2</em> is determined by relaxing/expanding the set of elements that are referred to by <em>s1</em>). Goodman adopts Hempel&#x27;s view that induction is described by a relation <em>R</em> over these statements, such that <em>s1Rs2</em> holds if and only if <em>s2</em> is &quot;confirmed&quot; by <em>s1</em>. The goal is then to describe the conditions that <em>R</em> must meet in order for a reasonable person to conclude that <em>s1Rs2</em></p><p>In order to demonstrate that this is more difficult than it sounds, Goodman presents the &quot;New Riddle Of Induction&quot;. In this thought experiment, he posits that we have observed several fine stones, and have noted that all the emeralds we have seen so far have been green. Our goal is to determine whether we would be justified in believing that all emeralds are green, or at least in predicting that the next emerald we see will be green. Let us suppose we have seen enough green emeralds to be confident that the next emerald will also be green; and we have checked that the procedure we followed to reach this conclusion meets all the rules of our inductive framework.</p><p>However, we have a friend who speaks a different language, which does not have a word for &quot;green&quot;. Instead, they have a word for &quot;Grue&quot;, which translates as &quot;green before time t, blue after time t&quot;. This language also has a word &quot;Bleen&quot;, which translates as &quot;blue before time t, green after time t&quot;.</p><p>Since our friend saw the same emeralds as us, and since they were all observed before time t, he has correctly observed that all the emeralds he has seen so far have been grue. Moreover, since he is following the same inductive rules as us, he is predicting that the next emerald, to be observed after time t, will also be grue. Since we are following the same inductive rules, his prediction appears to be based on equally strong inference as our prediction. However, at most one of us can be correct. This raises an important challenge to the very idea of reasoning about whether an attempted inference is valid: Any set of rules, even when adhered to strictly, can lead to different predictions depending on how the basic predicates are defined. In fact, this argument can be generalized to show that for any possible future predictions, there exists a &quot;language&quot; such that inductive reasoning based on that language will result in the desired prediction. </p><p>Goodman goes on to argue that the basic problem is one of distinguishing &quot;lawlike&quot; statements from &quot;accidental&quot; statements, where lawlike statements are those that relate to classes of objects such that either all members have the property in question, or such that none have the property. It can then be expected that lawlike predicates are projectable, in the sense that induction based on lawlike predicates lead to valid extrapolation. However, this move does not really resolve the issue: Essentially, it just shifts the discussion down one step, to whether statements are lawlike, which it is not possible to know unless one has information that would only be available after successful inductive inference. </p><p><strong>Can The Riddle Be Adequately Solved?</strong></p><p>An adequate solution to Goodman&#x27;s riddle would require a principled way to distinguish &quot;projectable&quot; predicates from non-projectable ones - a way to &quot;cut reality at the joints&quot; and establish natural, lawlike predicates for use in inductive reasoning.</p><p>Rudolf Carnap suggests that one can distinguish lawlike statements from accidental statements, because lawlike statements are &quot;purely qualitative and general&quot;, in the sense that they do not restrict the spatial or temporal position of the objects that the predicates refer to. Goodman considers and rejects this argument, because from the perspective of someone who thinks in terms of predicates such as grue, grue is time-stable and green is the time-dependent predicate. Therefore, temporality itself can only be defined relative to a given perspective, and there seems to be no obvious way to give priority to one over the other. </p><p>One potential line of argument in favor of giving priority to green over grue has been suggested by several writers, who make the observation that the time &quot;t&quot; is seemingly chosen arbitrarily, and that there are infinitely many versions of &quot;grue(t)<em>&quot;, </em>one for each possible time t, with no reason to prefer one over the other. This argument also suffers from the problem that a &quot;grueist&quot; person can take the same approach, and establish that there are infinitely many possible versions of &quot;green(t)&quot; at which time green transitions from being grue to being bleen.</p><p>In fact, this is a common theme in most arguments attempting to resolve the problem: They can generally be restated from the grueist perspective, and used to reach the opposite conclusion. Goodman therefore holds that it is impossible to distinguish lawlike statements from accidental statements on grounds of their generality. The best we can do is ask whether the statements are based on predicates that are &quot;entrenched&quot;, in the sense that have in the past proved successful at producing accurate predictions. Such a track record then provides some support for a tentative belief that the predicate has caught onto some aspect of reality that is governed by lawlike phenomena. </p><p><strong>The Grue Sleeping Beauty Problem</strong></p><p>Swinburne (1968) sees an asymmetry between the predicates green and grue, which arises from the fact that an individual can judge whether an object has the property &quot;green&quot; even if they do not know what time it is, and argues that this asymmetry can be exploited to give priority to green over grue. I find this argument persuasive but incomplete, and will therefore discuss it in detail, in a slightly altered form.</p><p>Consider the following thought experiment, which is not due to Swinburne but which I believe illustrates his argument: An evil turtle has abducted Grue Sleeping Beauty, the princess of the Kingdom of Grue, in order to perform experiments on her. Specifically, he intends to give her sleeping pill, and then flip a coin to randomly decide whether to wake her before or after time <em>t</em>. In the room, there will be a green emerald. Bowser plans to ask what color the emerald is, without informing her about what time it is.</p><p>One possibility is that Grue Sleeping Beauty gets it right: She will experience the emerald as being grue if she is woken before time <em>t</em>, and as bleen if she is woken after time <em>t</em>. However, this seems unlikely: It requires that the unspecified psychical phenomenon that produces color, interacts with her qualia in a time dependent manner even when she cannot know what time it is. The other options are that she gets it wrong - which seems like a big hit against the idea of &quot;grue&quot; - or that she experiences no qualia at all, which seems unlikely, since all our experience tells us that non-colorblind humans can in general identify the colors of objects.</p><p>Goodman might argue that this parable begs the question, by implicitly assuming that the diamond remains green from the experimenter&#x27;s perspective. One imagines his response might be to reverse the thought experiment, and instead tell a story about a grueist student of philosophy who makes up an elaborate tale about abducting Green Sleeping Beauty. Therefore, this line of reasoning is incomplete, for the same reason as most other attempted solutions to the New Riddle of Induction. However, I believe the two versions of the parable have different implications, such that a reasonable person would assign much higher credence to the implications of the first version. To explain this, in the next section I provide a &quot;patch&quot; to Swinburne&#x27;s argument</p><p><strong>Time Independence: Neutral and Relative Perspectives</strong></p><p>Let us assume there is an underlying regular, lawlike phenomenon (in this case the wavelength of light reflected by emeralds), and that agents implement an algorithm which takes this phenomenon as input, and outputs a subjectively experienced color. A classifier algorithm is said to be time-independent if, for any particular wavelength, it outputs the same subjectively perceived color, regardless of time. From the perspective of an agent that implements any particular classifier algorithm, other classifier algorithms will appear relatively time-dependent if reference to time is needed to translate between their outputs.</p><p>I argue that there exists a classifier algorithm that is time-independent even from a neutral perspective: It is the one that simply ignores input on time. Therefore, a color classifier algorithm is time-independent by default, unless it actively takes time into account. Moreover, if two color classifier algorithms result in predicates that are time-dependent relative to each other, then at least one of the algorithms must contain a term that takes time into account, either directly or through contextual clues. Of course, this may be subconscious and hidden from the meta-level cognition of the agent implementing it, and the agent therefore has no direct way of knowing whether he is implementing an algorithm that is time-independent from a neutral perspective. </p><p>Both versions of the Grue/Green Sleeping Beauty thought experiment implicitly assume that the investigator is implementing a classification algorithm that is time-independent even from a neutral perspective. Since the neutral perspective is unique, at least one of them must be wrong. Finding out which one (if any) is right, is an empirical question, but unanswerable before time t, so any attempt to grant the necessary empirical observations would amount to begging the question. </p><p>But now consider an evolutionary environment where agents must choose whether to eat delicious blue berries, or poisonous green berries. There is significant selective pressure to find find an algorithm that outputs the same subjective qualia, for any given wavelength. There is no reason at all that nature should put a term for time in this algorithm: To do so, would be to add needless complexity and add an arbitrary term that makes reference to the time <em>t</em>, which has no contextual meaning in the setting in which the algorithm is operating and in which it is being optimized. </p><p>This points us to the central difference between the grue and green predicates: The fact that we, as real humans shaped by evolution, are implementing a particular classification algorithm is highly relevant evidence for it not containing arbitrary complications, relative to a hypothetical algorithm that, as far as we know, does not exist in any agent found in nature. The actual human algorithm is therefore more likely to ignore input on time, and more likely to be time-independent from the neutral perspective.</p><p>Note that even in the absence of a theory of optics, it is reasonable to argue that subjective experience of colors arises from some physical regularity interacting with our sensory system, and that this sensory system would be much simpler if it ignores time. Thus, the argument still holds even if the underlying physical phenomenon is poorly understood.</p><p><strong>Conclusions</strong></p><p>Nelson Goodman presents a riddle which illustrates how our predictions for future observations are, to some extent, functions of the seemingly arbitrary categorization scheme used in our mental representation of reality. A common theme between many of the suggested solutions to this riddle, is that they attempt to find an asymmetry between languages that use the predicate grue, and languages that use the predicate green. Such an asymmetry would have to be immune to predictive changes that arise when rephrasing the problem in the other language; it seems likely that no such asymmetry can be found on purely logical or semantic grounds.</p><p>Instead, I argue that one can bring in additional background beliefs in support of the conclusion that the reference frame implemented by humans is &quot;neutral&quot;. In particular, the human color classifier algorithm was chosen by evolution, which had no reason to include a term for time. This licenses me to give significant higher credence to the belief that my representation scheme is neutral, and that hypothetical other classification algorithms that result in constructs such as grue are time-dependent even from a neutral perspective. In some sense, this line of reasoning could be interpreted as an extension of Goodman&#x27;s original argument about entrenchment, but allows the entrenchment to have occurred in evolutionary history. </p><p>Despite the fact that a solution seems possible, the riddle is still important, since an agent can only compensate for the uncertainty in his predictions that results from the reference frame of his predicates, if he has a clear understanding of the problems highlighted by Goodman. </p> anders_h SuDqsE3DgHnDiutyM 2017-12-02T16:15:08.912Z Comment by Anders_H on Odds ratios and conditional risk ratios https://www.lesswrong.com/posts/fYY9yAvpwAzjqHjGX/odds-ratios-and-conditional-risk-ratios#B4jAMyZdbm9Wbnz3Z <p>Update: The editors of the Journal of Clinical Epidemiology have now rejected my second letter to the editor, and thus helped prove Eliezer's point about four layers of conversation.</p> anders_h B4jAMyZdbm9Wbnz3Z 2017-02-02T15:03:19.943Z Comment by Anders_H on Odds ratios and conditional risk ratios https://www.lesswrong.com/posts/fYY9yAvpwAzjqHjGX/odds-ratios-and-conditional-risk-ratios#EnguE7AsHo3wvoaXv <blockquote> <p>Why do you think two senior biostats guys would disagree with you if it was obviously wrong? I have worked with enough academics to know that they are far far from infallible, but curious on your analysis of this question.</p> </blockquote> <p>Good question. I think a lot of this is due to a cultural difference between those of us who have been trained in the modern counterfactual causal framework, and an old generation of methodologists who felt the old framework worked well enough for them and never bothered to learn about counterfactuals.</p> anders_h EnguE7AsHo3wvoaXv 2017-01-25T06:02:48.605Z Comment by Anders_H on Odds ratios and conditional risk ratios https://www.lesswrong.com/posts/fYY9yAvpwAzjqHjGX/odds-ratios-and-conditional-risk-ratios#zDEJ3W7Gv9dWnFtut <p>I wrote this on my personal blog; I was reluctant to post this to Less Wrong since it is not obviously relevant to the core interests of LW users. However, I concluded that some of you may find it interesting as an example of how the academic publishing system is broken. It is relevant to Eliezer's recent Facebook comments about building an intellectual edifice.</p> anders_h zDEJ3W7Gv9dWnFtut 2017-01-25T03:55:43.955Z Odds ratios and conditional risk ratios https://www.lesswrong.com/posts/fYY9yAvpwAzjqHjGX/odds-ratios-and-conditional-risk-ratios anders_h fYY9yAvpwAzjqHjGX 2017-01-25T03:55:04.420Z Comment by Anders_H on [deleted post] https://www.lesswrong.com/posts/y74CHRNvG9iyh8MDF/odds-ratios-and-conditional-risk-ratios-0#WRHghoGgmAEaEdto9 <p>I wrote this on my personal blog; I was reluctant to post this to Less Wrong since it is not obviously relevant to the core interests of LW users. However, I concluded that some of you may find it interesting as an example of how the academic publishing system is broken. It is relevant to Eliezer's recent Facebook comments about building an intellectual edifice. </p> anders_h WRHghoGgmAEaEdto9 2017-01-25T03:48:12.884Z Comment by Anders_H on Is Caviar a Risk Factor For Being a Millionaire? https://www.lesswrong.com/posts/eSosndJmKT3KmDRt8/is-caviar-a-risk-factor-for-being-a-millionaire#edjLCL6etdZPT47uE <p>VortexLeague: Can you be a little more specific about what kind of help you need? </p> <p>A very short, general introduction to Less Wrong is available at <a href="http://lesswrong.com/about/">http://lesswrong.com/about/</a> </p> <p>Essentially, Less Wrong is a reddit-type forum for discussing how we can make our beliefs more accurate.</p> anders_h edjLCL6etdZPT47uE 2017-01-25T02:32:53.938Z Comment by Anders_H on Choosing prediction over explanation in psychology: Lessons from machine learning https://www.lesswrong.com/posts/d7LJ2BasCrBjRuKNN/choosing-prediction-over-explanation-in-psychology-lessons#fiCj4YTLssCGBCjmY <p>Thank you for the link, that is a very good presentation and it is good to see that ML people are thinking about these things. </p> <p>There certainly are ML algorithms that are designed to make the second kind of predictions, but generally they only work if you have a correct causal model</p> <p>It is possible that there are some ML algorithms that try to discover the causal model from the data. For example, /u/IlyaShpitser works on these kinds of methods. However, these methods only work to the extent that they are able to discover the correct causal model, so it seems disingenious to claim that we can ignore causality and focus on &quot;prediction&quot;.</p> anders_h fiCj4YTLssCGBCjmY 2017-01-18T13:58:16.346Z Comment by Anders_H on Choosing prediction over explanation in psychology: Lessons from machine learning https://www.lesswrong.com/posts/d7LJ2BasCrBjRuKNN/choosing-prediction-over-explanation-in-psychology-lessons#ubgdNBZNpePLFqKqN <p>I skimmed this paper and plan to read it in more detail tomorrow. My first thought is that it is fundamentally confused. I believe the confusion comes from the fact that the word &quot;prediction&quot; is used with two separate meanings: Are you interested in predicting Y given an observed value of X (Pr[Y | X=x]), or are you interested in predicting Y given an intervention on X (i.e. Pr[Y|do(X=x)]). </p> <p>The first of these may be useful for certain purposes. but If you intend to use the research for decision making and optimization (i.e. you want to intervene to set the value of X , in order to optimize Y), then you really need the second type of predictive ability, in which case you need to extract causal information from the data. This is only possible if you have a randomized trial, or if you have a correct causal model. </p> <p>You can use the word &quot;prediction&quot; to refer to the second type of research objective, but this is <em>not</em> the kind of prediction that machine learning algorithms are designed to do. </p> <p>In the conclusions, the authors write:</p> <blockquote> <p>&quot;By contrast, a minority of statisticians (and most machine learning researchers) belong to the “algorithmic modeling culture,” in which the data are assumed to be the result of some unknown and possibly unknowable process, and the primary goal is to find an algorithm that results in the same outputs as this process given the same inputs. &quot;</p> </blockquote> <p>The definition of &quot;algorithmic modelling culture&quot; is somewhat circular, as it just moves the ambiguity surrounding &quot;prediction&quot; to the word &quot;input&quot;. If by &quot;input&quot; they mean that the algorithm observes the value of an independent variable and makes a prediction for the dependent variable, then you are talking about a true prediction model, which may be useful for certain purposes (diagnosis, prognosis, etc) but which is unusable if you are interested in optimizing the outcome. </p> <p>If you instead claim that the &quot;input&quot; can also include observations about interventions on a variable, then your predictions will certainly fail unless the algorithm was trained in a dataset where someone actually intervened on X (i.e. someone did a randomized controlled trial), or unless you have a correct causal model.</p> <p>Machine learning algorithms are not magic, they do not solve the problem of confounding unless they have a correct causal model. The fact that these algorithms are good at predicting stuff in observational datasets does not tell you anything useful for the purposes of deciding what the optimal value of the independent variable is. </p> <p>In general, this paper is a very good example to illustrate why I keep insisting that machine learning people need to urgently read up on Pearl, Robins or Van der Laan. The field is in danger of falling into the same failure mode as epidemiology, i.e. essentially ignoring the problem of confounding. In the case of machine learning, this may be more insidious because the research is dressed up in fancy math and therefore looks superficially more impressive. </p> anders_h ubgdNBZNpePLFqKqN 2017-01-18T01:23:22.918Z Comment by Anders_H on Triple or nothing paradox https://www.lesswrong.com/posts/Eb6DKbTdnQ3R7triD/triple-or-nothing-paradox#sAs5y9nQnRofRKwa3 <p>Thanks for catching that, I stand corrected.</p> anders_h sAs5y9nQnRofRKwa3 2017-01-05T23:59:28.417Z Comment by Anders_H on Triple or nothing paradox https://www.lesswrong.com/posts/Eb6DKbTdnQ3R7triD/triple-or-nothing-paradox#z8emMZzxXZJouQKJF <p>The rational choice depends on your utility function. Your utility function is unlikely to be linear with money. For example, if your utility function is log (X), then you will accept the first bet, be indifferent to the second bet, and reject the third bet. Any risk-averse utility function (i.e. any monotonically increasing function with negative second derivative) reaches a point where the agent stops playing the game. </p> <p>A VNM-rational agent with a linear utility function over money will indeed always take this bet. From this, we can infer that linear utility functions do not represent the utility of humans. </p> <p>(EDIT: The comments by Satt and AlexMennen are both correct, and I thank them for the corrections. I note that they do not affect the main point, which is that rational agents with standard utility functions over money will eventually stop playing this game)</p> anders_h z8emMZzxXZJouQKJF 2017-01-05T22:52:14.657Z Comment by Anders_H on A quick note on weirdness points and Solstices [And also random other Solstice discussion] https://www.lesswrong.com/posts/HWMoS3sBc2LKyRtZ3/a-quick-note-on-weirdness-points-and-solstices-and-also#j56tRKRPfiE5HXdgB <blockquote> <p>Because I didn't perceive a significant disruption to the event, I was mentally bucketing you with people I know who severely dislike children and would secretly (or not so secretly) prefer that they not attend events like this at all; or that they should do so only if able to remain silent (which in practice means not at all.) I suspect Anders_H had the same reaction I did.</p> </blockquote> <p>Just to be clear, I did not attend Solstice this year, and I was mentally reacting to a similar complaint that was made after last year's Solstice event. At last year's event, I did not perceive the child to be at all noteworthy as a disturbance. From reading this thread, it seems that the situation may well have been different this year, and that my reaction might have been different if I had been there. I probably should not have commented without being more familiar with what happened at this year's event.</p> <p> I also note that my thinking around this may very well be biased, as I used to live in a group house with this child. </p> anders_h j56tRKRPfiE5HXdgB 2016-12-23T17:36:32.640Z Comment by Anders_H on A quick note on weirdness points and Solstices [And also random other Solstice discussion] https://www.lesswrong.com/posts/HWMoS3sBc2LKyRtZ3/a-quick-note-on-weirdness-points-and-solstices-and-also#sBJ238rRnshfjr4mt <p>While I understand that some people may feel this way, I very much hope that this sentiment is rare. The presence of young children at the event only adds to the sense of belonging to a community, which is an important part of what we are trying to &quot;borrow&quot; from religions. </p> anders_h sBJ238rRnshfjr4mt 2016-12-22T06:00:51.168Z Comment by Anders_H on Feature Wish List for LessWrong https://www.lesswrong.com/posts/LiCvC6XEHjJJ8eQ7h/feature-wish-list-for-lesswrong#mWZTkAdr9SPE6hASB <p>I'd like each user to have their own sub domain (I.e such that my top level posts can be accessed either from Anders_h.lesswrong.com or from LW discussion). If possible it would be great if users could customize the design of their sub domain, such that posts look different when accessed from LW discussion.</p> anders_h mWZTkAdr9SPE6hASB 2016-12-19T07:57:05.056Z Comment by Anders_H on This one equation may be the root of intelligence https://www.lesswrong.com/posts/r6eMwD6Hjp7RzEutC/this-one-equation-may-be-the-root-of-intelligence#vhEd3wHwvodsN5KkF <p>Given that this was posted to LW, you'd think this link would be about a different equation..</p> anders_h vhEd3wHwvodsN5KkF 2016-12-12T02:56:28.887Z Is Caviar a Risk Factor For Being a Millionaire? https://www.lesswrong.com/posts/eSosndJmKT3KmDRt8/is-caviar-a-risk-factor-for-being-a-millionaire <p>Today, my paper "Is caviar a risk factor for being a millionaire?" was published in the Christmas Edition of the BMJ (formerly the British Medical Journal). &nbsp;The paper is available at http://www.bmj.com/content/355/bmj.i6536 but it is unfortunately behind a paywall. I am hoping to upload an open access version to a preprint server but this needs to be confirmed with the journal first.&nbsp;</p> <p>In this paper, I argue that the term "risk factor" is ambiguous, and that this ambiguity causes pervasive methodological confusion in the epidemiological literature. I argue that many epidemiological papers essentially use an audio recorder to determine whether a tree falling in the forest makes a sound, without being clear about which definition of "sound" they are considering.</p> <p>Even worse, I argue that epidemiologists often try to avoid claiming that their results say anything about causality, by hiding behind "prediction models". When they do this. they often still control extensively for "confounding", a term which only has a meaning in causal models. I argue that this is analogous to stating that you are interested in whether trees falling in the forest causes any human to perceive the qualia of hearing, and then spending your methods section discussing whether the audio recorder was working properly.</p> <p>Due to space constraints and other considerations, I am unable to state these analogies explicitly in the paper, but it does include a call for a taboo on the word risk factor, and a reference to Rationality: AI to Zombies. To my knowledge, this is the first reference to the book in the medical literature.</p> <p>I will give a short talk about this paper at the Less Wrong meetup at the MIRI/CFAR office in Berkeley at 6:30pm tonight.</p> <p>(I apologize for this short, rushed announcement, I was planning to post a full writeup but I was not expecting this paper to be published for another week)&nbsp;</p> anders_h eSosndJmKT3KmDRt8 2016-12-09T16:27:14.760Z Comment by Anders_H on Open thread, Nov. 21 - Nov. 27 - 2016 https://www.lesswrong.com/posts/RfcLoNxvFFLis65Kn/open-thread-nov-21-nov-27-2016#769BMPbzB2fnbs5QT <p>The one-year embargo on my doctoral thesis has been lifted, it is now available at <a href="https://dash.harvard.edu/bitstream/handle/1/23205172/HUITFELDT-DISSERTATION-2015.pdf?sequence=1">https://dash.harvard.edu/bitstream/handle/1/23205172/HUITFELDT-DISSERTATION-2015.pdf?sequence=1</a> . To the best of my knowledge, this is the first thesis to include a Litany of Tarski in the introduction. </p> anders_h 769BMPbzB2fnbs5QT 2016-11-22T20:58:25.516Z Comment by Anders_H on On Trying Not To Be Wrong https://www.lesswrong.com/posts/3NadXc5mP84HcwCKG/on-trying-not-to-be-wrong#w7rwXE6W92m3M5AGg <p>Upvoted. I'm not sure how to phrase this without sounding sycophantic, but here is an attempt: Sarah's blog posts and comments were always top quality, but the last couple of posts seem like the beginning of something important, almost comparable to when Scott moved from squid314 to Slatestarcodex. </p> anders_h w7rwXE6W92m3M5AGg 2016-11-11T22:08:48.111Z Comment by Anders_H on Open Thread, Sept 5. - Sept 11. 2016 https://www.lesswrong.com/posts/ChSbgwhbsux4S2cBW/open-thread-sept-5-sept-11-2016#EZdBqMrAzi57QGmCL <p>Today, I uploaded a sequence of three working papers to my website at <a href="https://andershuitfeldt.net/working-papers/">https://andershuitfeldt.net/working-papers/</a></p> <p>This is an ambitious project that aims to change fundamental things about how epidemiologists and statisticians think about choice of effect measure, effect modification and external validity. A link to an earlier version of this manuscript was posted to Less Wrong half a year ago, the manuscript has since been split into three parts and improved significantly. This work was also presented in poster form at EA Global last month. </p> <p>I want to give a heads up before you follow the link above: Compared to most methodology papers, the mathematics in these manuscripts is definitely unsophisticated, almost trivial. I do however believe that the arguments support the conclusions, and that those conclusions have important implications for applied statistics and epidemiology.</p> <p>I would very much appreciate any feedback. I invoke &quot;Crocker's Rules&quot; (see <a href="http://sl4.org/crocker.html">http://sl4.org/crocker.html</a>) for all communication regarding these papers. Briefly, this means that I ask you, as a favor, to please communicate any disagreement as bluntly and directly as possible, without regards to social conventions or to how such directness may affect my personal state of mind.</p> <p>I have made a standing offer to give a bottle of Johnnie Walker Blue Label to anyone who finds a flaw in the argument that invalidates the paper, and a bottle of 10-year old Single Scotch Malt to anyone who finds a significant but fixable error; or makes a suggestion that substantially improves the manuscript.</p> <p>If you prefer giving anonymous feedback, this can be done through the link <a href="http://www.admonymous.com/effectmeasurepaper">http://www.admonymous.com/effectmeasurepaper</a> .</p> anders_h EZdBqMrAzi57QGmCL 2016-09-07T23:07:05.720Z Link: The Economist on Paperclip Maximizers https://www.lesswrong.com/posts/mtF6rcAjPtv9L3MTy/link-the-economist-on-paperclip-maximizers <p>I certainly was not expecting the Economist to publish a special report on paperclip maximizers (!).</p> <p>See&nbsp;<a title="Frankenstein's Paperclips" href="http://www.economist.com/news/special-report/21700762-techies-do-not-believe-artificial-intelligence-will-run-out-control-there-are?fsrc=scn/fb/te/pe/ed/frankensteinspaperclips">http://www.economist.com/news/special-report/21700762-techies-do-not-believe-artificial-intelligence-will-run-out-control-there-are?fsrc=scn/fb/te/pe/ed/frankensteinspaperclips</a></p> <p>&nbsp;</p> <p>As the title suggests, they are downplaying the risks of unfriendly AI, but just the fact that the Economist published this is significant</p> anders_h mtF6rcAjPtv9L3MTy 2016-06-30T12:40:33.942Z Comment by Anders_H on Secret Rationality Base in Europe https://www.lesswrong.com/posts/nY63KZQJR4Non64FD/secret-rationality-base-in-europe#pnYQB2YDwMwkXYvGF <p>This is almost certainly a small minority view, but from my perspective as a European based in the Bay Area who may be moving back to Europe next summer, the most important aspect would be geographical proximity to a decent university where staff and faculty can get away with speaking only English. </p> anders_h pnYQB2YDwMwkXYvGF 2016-06-17T19:47:36.910Z Comment by Anders_H on Why you should consider buying Bitcoin right now (Jan 2015) if you have high risk tolerance https://www.lesswrong.com/posts/tsoAwbDZ54GXtfcjh/why-you-should-consider-buying-bitcoin-right-now-jan-2015-if#3BrYxrT5iNmbnCEem <p>What do you mean by &quot;no risk&quot;? This sentence seems to imply that your decisions are influenced by the sunk cost fallacy. </p> <p>Try to imagine an alien who has been teleported into your body, who is trying to optimize your wealth. The fact that the coins were worth a third of their current price 18 months ago would not factor into the alien's decision. </p> anders_h 3BrYxrT5iNmbnCEem 2016-06-14T17:57:25.679Z Comment by Anders_H on Open thread, Jun. 13 - Jun. 19, 2016 https://www.lesswrong.com/posts/pR2STsnPYGGE7PvDe/open-thread-jun-13-jun-19-2016#jJLaLhymmabvySyDB <p>There may be an ethically relevant distinction between a rule that tells you to avoid being the cause of bad things, and a rule that says you should cause good things to happen. However, I am not convinced that causality is relevant to this distinction. As far as I can tell, these two concepts are both about causality. We may be using words differently, do you think you could explain why you think this distinction is about causality?</p> anders_h jJLaLhymmabvySyDB 2016-06-14T01:05:37.629Z Comment by Anders_H on The Valentine’s Day Gift That Saves Lives https://www.lesswrong.com/posts/CQJaESSPLvy8XmzpM/the-valentine-s-day-gift-that-saves-lives#HWQv9AhRHcZj3Spbu <p>It would seem that the existence of such contractors follows logically from the fact that you are able to hire people despite the fact that you require contractors to volunteer 2/3 of their time. </p> anders_h HWQv9AhRHcZj3Spbu 2016-05-18T17:30:51.862Z Comment by Anders_H on Open Thread May 16 - May 22, 2016 https://www.lesswrong.com/posts/2aQ6CpngDQijtuRrs/open-thread-may-16-may-22-2016#9fjRWk3wBNDLjWXzK <p>The Economist published a fascinating blog entry where they use evidential decision theory to establish that tattoo removal results in savings to the prison system. See <a href="http://www.economist.com/blogs/freeexchange/2014/08/tattoos-jobs-and-recidivism">http://www.economist.com/blogs/freeexchange/2014/08/tattoos-jobs-and-recidivism</a> . Temporally, this blog entry corresponds roughly to the time I lost my respect for the Economist. You can draw your own causal conclusions from this.</p> anders_h 9fjRWk3wBNDLjWXzK 2016-05-17T21:07:59.908Z Comment by Anders_H on How do you learn Solomonoff Induction? https://www.lesswrong.com/posts/chWFNGAuiv5476Dku/how-do-you-learn-solomonoff-induction#9RqbxxLNH2QQfBDvq <p>Solomonoff Induction is uncomputable, and implementing it will not be possible even in principle. It should be understood as an ideal which you should try to approximate, rather than something you can ever implement.</p> <p>Solomonoff Induction is just bayesian epistemology with a prior determined by information theoretic complexity. As an imperfect agent trying to approximate it, you will get most of your value from simply grokking Bayesian epistemology. After you've done that, you may want to spend some time thinking about the philosophy of science of setting priors based on information theoretic complexity. </p> anders_h 9RqbxxLNH2QQfBDvq 2016-05-17T18:21:35.480Z Comment by Anders_H on Lesswrong 2016 Survey https://www.lesswrong.com/posts/hCWKHLKSgW5afmoaR/lesswrong-2016-survey#TpTnwkxNuhDqPE92h <p>I took the survey</p> anders_h TpTnwkxNuhDqPE92h 2016-03-26T22:10:21.611Z Comment by Anders_H on Open thread, Mar. 14 - Mar. 20, 2016 https://www.lesswrong.com/posts/HNzyci3KRnaLjuRiA/open-thread-mar-14-mar-20-2016#aXiFZaSG7sGkAAzqH <p>Thanks. Good points. Note that many of those words are already established in the literature with same meaning. For the particular example of &quot;doomed&quot;, this is the standard term for this concept, and was introduced by Greenland and Robins (1986). I guess I could instead use &quot;response type 1&quot; but the word doomed will be much more effective at pointing to the correct concept, particularly for people who are familiar with the previous literature.</p> <p>The only new term I introduce is &quot;flip&quot;. I also provide a new definition of effect equality, and it therefore seems correct to use quotation marks in the new definition. Perhaps I should remove the quotation marks for everything else since I am using terms that have previously been introduced.</p> anders_h aXiFZaSG7sGkAAzqH 2016-03-21T17:28:37.253Z Comment by Anders_H on Open thread, Mar. 14 - Mar. 20, 2016 https://www.lesswrong.com/posts/HNzyci3KRnaLjuRiA/open-thread-mar-14-mar-20-2016#yufJQentiwhdx3rdo <blockquote> <p>Do you mean probability instead of probably?</p> </blockquote> <p>Yes. Thanks for noticing. I changed that sentence after I got the rejection letter (in order to correct a minor error that the reviewers correctly pointed out), and the error was introduced at that time. So that is not what they were referring to. </p> <blockquote> <p>If the reviewers don't succeed in understanding what you are saying you might have explained yourself in casual language but still failed.</p> </blockquote> <p>I agree, but I am puzzled by why they would have misunderstood. I spent a lot of effort over several months trying to be as clear as possible. Moreover, the ideas are very simple: The definitions are the only real innovation: Once you have the definitions, the proofs are trivial and could have been written by a high school student. If the reviewers don't understand the basic idea, I will have to substantially update my beliefs about the quality of my writing. This is upsetting because being a bad writer will make it a lot harder to succeed in academia. The primary alternative hypotheses for why they misunderstood are either (1) that they are missing some key fundamental assumption that I take for granted or (2) that they just don't want to understand.</p> anders_h yufJQentiwhdx3rdo 2016-03-20T17:00:36.339Z Comment by Anders_H on Open thread, Mar. 14 - Mar. 20, 2016 https://www.lesswrong.com/posts/HNzyci3KRnaLjuRiA/open-thread-mar-14-mar-20-2016#aujQ7NXMnSg7euxDk <p>Three days ago, I went through a traditional rite of passage for junior academics: I received my first rejection letter on a paper submitted for peer review. After I received the rejection letter, I forwarded the paper to two top professors in my field, who both confirmed that the basic arguments seem to be correct and important. Several top faculty members have told me they believe the paper will eventually be published in a top journal, so I am actually feeling more confident about the paper than before it got rejected.</p> <p>I am also very frustrated with the peer review system. The reviewers found some minor errors, and some of their other comments were helpful in the sense that they reveal which parts of the paper are most likely to be misunderstood. However, on the whole, the comments do not change my belief in the soundness of the idea, and in my view they mostly show that the reviewers simply didn’t understand what I was saying.</p> <p>One comment does stand out, and I’ve spent a lot of energy today thinking about its implications: Reviewer 3 points out that my language is “too casual”. I would have had no problem accepting criticism that my language is ambiguous, imprecise, overly complicated, grammatically wrong or idiomatically weird. But too casual? What does that even mean? I have trouble interpreting the sentence to mean anything other than an allegation that I fail at a signaling game where the objective is to demonstrate impressiveness by using an artificially dense and obfuscating academic language.</p> <p>From my point of view, “understanding” something <i>means</i> that you are able to explain it in a casual language. When I write a paper, my only objective is to allow the reader to understand what my conclusions are and how I reached them. My choice of language is optimized only for those objectives, and I fail to understand how it is even possible for it to be “too casual”.</p> <p>Today, I feel very pessimistic about the state of academia and the institution of peer review. I feel stronger allegiance to the rationality movement than ever, as my ideological allies in what seems like a struggle about what it means to do science. I believe it was Tyler Cowen or Alex Tabarrok who pointed out that the true inheritors of intellectuals like Adam Smith are not people publishing in academic journals, but bloggers who write in a causal language. I can’t find the quote but today it rings more true than ever.</p> <p>I understand that I am interpreting the reviewers choice of words in a way that is strongly influenced both by my disappointment in being rejected, and by my pre-existing frustration with the state of academia and peer review. I would very much appreciate if anybody could steelman the sentence “the writing is too casual”, or otherwise help me reach a less biased understanding of what just happened.</p> <p>The paper is available at <a href="https://rebootingepidemiology.files.wordpress.com/2016/03/effect-measure-paper-0317162.pdf">https://rebootingepidemiology.files.wordpress.com/2016/03/effect-measure-paper-0317162.pdf</a> . I am willing to send a link to the reviewers’ comments by private message to anybody who is interested in seeing it.</p> anders_h aujQ7NXMnSg7euxDk 2016-03-20T05:32:34.388Z Comment by Anders_H on Link: Evidence-Based Medicine Has Been Hijacked https://www.lesswrong.com/posts/9R7gcCGb8uHj5hRrF/link-evidence-based-medicine-has-been-hijacked#oqoaG534R3LvZ3BPv <p>I think the evidence for the effectiveness of statins is very convincing. The absolute risk reduction from statins will depend primarily on your individual baseline risk of coronary disease. From the information you have provided, I don't think your baseline risk is extraordinarily high, but it is also not negligible.</p> <p>You will have to make a trade-off where the important considerations are (1) how bothered you are by the side-effects, (2) what absolute risk reduction you expect based on your individual baseline risk, (3) the marginal price (in terms of side effects) that you are willing to pay for slightly better chance at avoiding a heart attack. I am not going to tell you how to make that trade-off, but I would consider giving the medications a try simply because it is the only way to get information on whether you get any side effects, and if so, whether you find them tolerable.</p> <p>(I am not licensed to practice medicine in the United States or on the internet, and this comment does not constitute medical advise)</p> anders_h oqoaG534R3LvZ3BPv 2016-03-16T22:43:42.576Z Link: Evidence-Based Medicine Has Been Hijacked https://www.lesswrong.com/posts/9R7gcCGb8uHj5hRrF/link-evidence-based-medicine-has-been-hijacked <p>John Ioannidis has written a very insightful and entertaining article about the current state of the movement which calls itself "Evidence-Based Medicine". &nbsp;The paper is available ahead of print at&nbsp;<a href="http://www.jclinepi.com/article/S0895-4356(16)00147-5/pdf ">http://www.jclinepi.com/article/S0895-4356(16)00147-5/pdf</a>.</p> <p>As far as I can tell there is currently no paywall, that may change later, send me an e-mail if you are unable to access it.</p> <p>Retractionwatch interviews John about the paper here:&nbsp;http://retractionwatch.com/2016/03/16/evidence-based-medicine-has-been-hijacked-a-confession-from-john-ioannidis/</p> <p>(Full disclosure: John Ioannidis is a co-director of the Meta-Research Innovation Center at Stanford (METRICS), where I am an employee. I am posting this not in an effort to promote METRICS, but because I believe the links will be of interest to the community)</p> anders_h 9R7gcCGb8uHj5hRrF 2016-03-16T19:57:49.294Z Comment by Anders_H on If there was one element of statistical literacy that you could magically implant in every head, what would it be? https://www.lesswrong.com/posts/DtmFmKm86wPrxqCSg/if-there-was-one-element-of-statistical-literacy-that-you#ZxEJZXjgmDXQ75Q5T <p>Why do you want to be able to do that? Do you mean that you want to be able to look at a spreadsheet and move around numbers in your head until you know what the parameter estimates are? If you have access to a statistical software package, this would not give you the ability to do anything you couldn't have done otherwise. However, that is obvious, so I am going to assume you are more interested in groking some part of the underlying the epistemic process. But if that is indeed your goal, the ability to do the parameter estimation in your head seems like a very low priority, almost more of a party trick than actually useful. </p> anders_h ZxEJZXjgmDXQ75Q5T 2016-02-26T07:40:28.484Z Comment by Anders_H on Open Thread, Feb 8 - Feb 15, 2016 https://www.lesswrong.com/posts/9qXevmQAAg8prhxn2/open-thread-feb-8-feb-15-2016#JbcMD6iPqv2HgWoqj <p>I disagree with this. In my opinion QALYs are much superior to DALYs for reasons that are inherent to how the measures are defined. I wrote a Tumblr post in response to Slatestarscratchpad a few weeks ago, see <a href="http://dooperator.tumblr.com/post/137005888794/can-you-give-me-a-or-two-good-article-on-why">http://dooperator.tumblr.com/post/137005888794/can-you-give-me-a-or-two-good-article-on-why</a> . </p> anders_h JbcMD6iPqv2HgWoqj 2016-02-09T22:38:10.445Z Comment by Anders_H on The Fable of the Burning Branch https://www.lesswrong.com/posts/Ry5NJwQjfifFrCmE3/the-fable-of-the-burning-branch#vHQ3apbjTLtKZrSqm <p>Richard, I don't think Less Wrong can survive losing both Ilya and you in the same week. I hope both of you reconsider. Either way, we definitely need to see this as a wake-up call. This forum has been in decline for a while, but this week I definitely think it hit a breaking point.</p> anders_h vHQ3apbjTLtKZrSqm 2016-02-08T18:51:30.203Z Comment by Anders_H on Lesswrong Survey - invitation for suggestions https://www.lesswrong.com/posts/naQktZ5JScHTHChm2/lesswrong-survey-invitation-for-suggestions#TNqoKuayxM4QDKCHZ <p>How about asking &quot;What is the single most important change that would make you want to participate more frequently on Less Wrong?&quot;</p> <p>This question would probably not be useful for the census itself, but it seems like a great opportunity to brainstorm..</p> anders_h TNqoKuayxM4QDKCHZ 2016-02-08T18:49:04.965Z Comment by Anders_H on Disguised Queries https://www.lesswrong.com/posts/4FcxgdvdQP45D6Skg/disguised-queries#wE4xBtakNtMDaLHRA <p>I run the Less Wrong meetup group in Palo Alto. After we announced the events at Meetup.com, we often get a lot of guests who are interested in rationality but who have not read the LW sequences. I have an idea for a introductory session where we have the participants do a sorting exercise. Therefore, I am interested in getting 3D printed versions of rubes, bleggs and other items references in this post.</p> <p>Does anyone have any thoughts on how to do this cheaply? Is there sufficient interest in this to get a kickstarter running? I expect that these items may be of interest to other Less Wrong meetup groups, and possibly to CFAR workshops and/or schools? </p> anders_h wE4xBtakNtMDaLHRA 2016-02-07T19:45:59.705Z Comment by Anders_H on Clearing An Overgrown Garden https://www.lesswrong.com/posts/2gCjiASWJ94JRyznx/clearing-an-overgrown-garden#5ctpAW2a3KdELnidu <blockquote> <p>Less Wrong doesn't seem &quot;overgrown&quot; to me. It actually seems dried out and dying because the culture is so negative people don't want to post here. I believe Eliezer has talked about how whenever he posted something on LW, the comments would be full of people trying to find anything wrong with it.</p> </blockquote> <p>&quot;Overgrown&quot; was probably a bad analogy, I tried too hard to reference the idea of well-kept gardens. What I was trying to say is that there are too many hostile elements who are making this website an unwelcoming place, by unnecessary criticism, ad hominem attacks and downvotes; and that those elements should have been removed from the community earlier. I actually think we agree on this.</p> <blockquote> <p>I think trying to impose strict new censorship rules and social control over communication is more likely to deal the death blow to this website than to help it. LessWrong really needs an injection of positive energy and purpose. In the absence of this, I expect LW to continue to decline.</p> </blockquote> <p>OK, from reading this and other comments I accept that this was the weakest part of my post. Also, after re-reading the Wiki entry on Crocker's rule, I don't think I intended to suggest anything quite that extreme. Crocker's rules say that rudeness is acceptable simply in order to provide a precise and accurate signal of annoyance. This is certainly not what I had in mind. </p> <p>I apologize for my incorrect usage of the term &quot;Crocker's rules&quot;, and I recognize that this was probably not a good idea. I hope someone can come up with a policy that achieves the objectives I had in mind when I wrote that sentence. </p> anders_h 5ctpAW2a3KdELnidu 2016-01-31T04:50:39.820Z Comment by Anders_H on Clearing An Overgrown Garden https://www.lesswrong.com/posts/2gCjiASWJ94JRyznx/clearing-an-overgrown-garden#jDMwrqGEFbvdnjLEx <blockquote> <p>Would there be a way for people who already maintain blogs elsewhere to cross-post to their LW subdomain? (Would this even be desirable?)</p> </blockquote> <p>We would have to discuss this with people who run blogs elsewhere to find out what solutions would work for them. My preferred solution would be for people to import their old blog posts, and then redirect their domain to the LW subdomain. I do not know whether outside bloggers would find this acceptable. In some cases we may also have to consider the question of advertisement revenues. </p> <blockquote> <p>Do you envision LW2 continuing to include applied rationality type posts? Does that work with &quot;everything should work towards Aumann agreement&quot;?</p> </blockquote> <p>My apologies, I did not intend to declare that posts about applied rationality should be avoided. I guess my phrasing reveals my bias towards the &quot;epistemic&quot; part of this community rather than the &quot;instrumental&quot; side. My personal preference is to shift the community focus back towards epistemic rationality, but that is a separate discussion which I did not intend to raise here. The community should discuss this separately. </p> <blockquote> <blockquote> <p>users may not repeatedly bring up the same controversial discussion outside of their original context</p> </blockquote> <p>How could we track this, other than relying on mods to be like &quot;ugh, this poster again&quot;?</p> </blockquote> <p>There would have to be some moderator discretion on this issue. My personal view is that we should err on the side of allowing most content. This language was intended for extreme cases where the community consensus is clear that a line has been crossed, such as Eugine, AdvancedAtheist or Jim Donald. </p> <blockquote> <blockquote> <p>professionally edited rationality journal</p> </blockquote> <p>Woah. Is this really a thing that MIRI could (resources permitting) just like ... do?</p> </blockquote> <p>Yes, they can certainly do this if they have the resources. Initially, academics may not take the journal seriously and it definitely will not be indexed in academic databases. If the quality is sufficiently high, it is conceivable that this may change.</p> anders_h jDMwrqGEFbvdnjLEx 2016-01-29T23:05:56.726Z Clearing An Overgrown Garden https://www.lesswrong.com/posts/2gCjiASWJ94JRyznx/clearing-an-overgrown-garden <p>(tl;dr: In this post, I make some concrete suggestions for LessWrong 2.0.)</p> <p><strong>Less Wrong 2.0</strong></p> <p>A few months ago, Vaniver posted some ideas about how to reinvigorate Less Wrong. Based on comments in that thread and based on personal discussions I have had with other members of the community, I believe there are several different views on why Less Wrong is dying. The following are among the most popular hypotheses:</p> <p>(1) Pacifism has caused our previously well-kept garden to become overgrown</p> <p>(2) The aversion to politics has caused a lot of interesting political discussions to move away from the website</p> <p>(3) People prefer posting to their personal blogs.</p> <p>With this background, I suggest the following policies for Less Wrong 2.0. &nbsp;This should be seen only as a starting point for discussion about the ideal way to implement a rationality forum. Most likely, some of my ideas are counterproductive. If anyone has better suggestions, please post them to the comments.</p> <p><strong>Moderation Policy:</strong></p> <p>There are four levels of users: &nbsp;</p> <ol> <li>Users</li> <li>Trusted Users&nbsp;</li> <li>Moderators</li> <li>Administrator</li> </ol> <div>Users may post comments and top level posts, but their contributions must be approved by a moderator.</div> <div><br /></div> <div>Trusted users may post comments and top level posts which appear immediately. Trusted user status is awarded by 2/3 vote among the moderators</div> <div><br /></div> <div>Moderators may approve comments made by non-trusted users. There should be at least 10 moderators to ensure that comments are approved within an hour of being posted, preferably quicker. If there is disagreement between moderators, the matter can be discussed on a private forum. Decisions may be altered by a simple majority vote.</div> <div><br /></div> <div>The administrator (preferably Eliezer or Nate) chooses the moderators.</div> <div><br /></div> <p><strong>Personal Blogs:</strong></p> <div> <div><br /></div> <div>All users are assigned a personal subdomain, such as Anders_H.lesswrong.com. When publishing a top-level post, users may click a checkbox to indicate whether the post should appear only on their personal subdomain, or also in the Less Wrong discussion feed. The commenting system is shared between the two access pathways. Users may choose a design template for their subdomain. However, when the post is accessed from the discussion feed, the default template overrides the user-specific template. The personal subdomain may include a blogroll, an about page, and other information. Users may purchase a top-level domain as an alias for their subdomain</div> </div> <div><br /></div> <div><strong>Standards of Discourse and Policy on Mindkillers:</strong></div> <div><br /></div> <div>All discussion in Less Wrong 2.0 is seen explicitly as an attempt to exchange information for the purpose of reaching Aumann agreement. In order to facilitate this goal, communication must be precise. Therefore, all users agree to abide by Crocker's Rules for all communication that takes place on the website. &nbsp;</div> <div><br /></div> <div>However, this is not a license for arbitrary rudeness. &nbsp;Offensive language is permitted only if it is necessary in order to point to a real disagreement about the territory. Moreover, users may not repeatedly bring up the same controversial discussion outside of their original context.</div> <div><br /></div> <div>Discussion of politics is explicitly permitted as long as it adheres to the rules outlined above. All political opinions are permitted (including opinions which are seen as taboo by society as large), as long as the discussion is conducted with civility and in a manner that is suited for dispassionate exchange of information, and suited for accurate reasoning about the consequences of policy choice. By taking part in any given discussion, all users are expected to pre-commit to updating in response to new information.</div> <div><br /></div> <div><strong>Upvotes:</strong></div> <div><br /></div> <div>Only trusted users may vote. There are two separate voting systems. &nbsp;Users may vote on whether the post raises a relevant point that will result in interesting discussion (quality of contribution) and also on whether they agree with the comment (correctness of comment). The first is a property both of the comment and of the user, and is shown in their user profile. &nbsp;The second scale is a property only of the comment.&nbsp;</div> <div><br /></div> <div>All votes are shown publicly (for an example of a website where this is implemented, see for instance dailykos.com). &nbsp;Abuse of the voting system will result in loss of Trusted User Status.&nbsp;</div> <div><br /></div> <div><strong>How to Implement This</strong></div> <div><strong><br /></strong></div> <div>After the community comes to a consensus on the basic ideas behind LessWrong 2.0, my preference is for MIRI to implement it as a replacement for Less Wrong. However, if for some reason MIRI is unwilling to do this, and if there is sufficient interest in going in this direction, I offer to pay server costs. If necessary, I also offer to pay some limited amount for someone to develop the codebase (based on Open Source solutions).&nbsp;</div> <div><br /></div> <p><strong>Other Ideas:</strong></p> <div><br /></div> <div>MIRI should start a professionally edited rationality journal (For instance called "Rationality") published bi-monthly. Users may submit articles for publication in the journal. Each week, one article is chosen for publication and posted to a special area of Less Wrong. This replaces "main". Every two months, these articles are published in print in the journal. &nbsp;</div> <div><br /></div> <div>The idea behind this is as follows:</div> <div>(1) It will incentivize users to compete for the status of being published in the journal.</div> <div>(2) It will allow contributors to put the article on their CV.</div> <div>(3) It may bring in high-quality readers who are unlikely to read blogs. &nbsp;</div> <div>(4) Every week, the published article may be a natural choice for discussion topic at Less Wrong Meetup</div> anders_h 2gCjiASWJ94JRyznx 2016-01-29T22:16:56.620Z Comment by Anders_H on Study partner matching thread https://www.lesswrong.com/posts/2jY3m8rfLES9FWD8E/study-partner-matching-thread#HnFf8FakvFDqqmwkN <p>I'm a postdoctoral scholar at METRICS and I'd be happy to talk to you about this. Get in touch by e-mail or private message. Also, I'm giving a talk about a new research idea at the METRICS internal lab meeting this coming Monday at 12:00 at Stanford. You are welcome to attend if you want to meet the METRICS group (but the professors are probably going to be busy and may not have time to talk with you)</p> anders_h HnFf8FakvFDqqmwkN 2016-01-27T18:25:03.029Z Comment by Anders_H on Rationality Quotes Thread January 2016 https://www.lesswrong.com/posts/C8XRdyTvjp7Npqq63/rationality-quotes-thread-january-2016#CkucXeSR3Q8w4ns8e <p>Whoops, my apologies. Thanks for noticing. Corrected</p> anders_h CkucXeSR3Q8w4ns8e 2016-01-26T00:30:46.609Z Comment by Anders_H on Rationality Quotes Thread January 2016 https://www.lesswrong.com/posts/C8XRdyTvjp7Npqq63/rationality-quotes-thread-january-2016#4gPha9M8Ym4qjBWWA <p><strong>This note is for readers who are unfamiliar with The_Lion:</strong></p> <p>This user is a troll who has been banned multiple times from Less Wrong. He is unwanted as a participant in this community, but we are apparently unable to prevent him from repeatedly creating new accounts. Administrators have extensive evidence for sockpuppetry and for abuse of the voting system. The fact that The_Lion's comment above is heavily upvoted is almost certainly entirely due to sockpuppetry. It does not reflect community consensus</p> anders_h 4gPha9M8Ym4qjBWWA 2016-01-25T22:09:46.502Z Comment by Anders_H on Rationality Quotes Thread January 2016 https://www.lesswrong.com/posts/C8XRdyTvjp7Npqq63/rationality-quotes-thread-january-2016#MYDvsRabsuDaqMjgj <p>As expected, my karma fell by 38 points and my &quot;positive percentage&quot; fell from 97% to 92% shortly after leaving this comment </p> anders_h MYDvsRabsuDaqMjgj 2016-01-25T21:41:27.703Z Comment by Anders_H on Rationality Quotes Thread January 2016 https://www.lesswrong.com/posts/C8XRdyTvjp7Npqq63/rationality-quotes-thread-january-2016#bHJdzThAYkwymAT8K <p> Cloud Atlas is my favorite movie ever and I recommend it to anyone reading this. In fact, it is my opinion that it is one of the most important pieces of early 21st century art.</p> <p>The downvote is however <em>not</em> for your bad taste in movies, but for intentionally misgendering Lana. More generally, you can consider it payback for your efforts to make Less Wrong an unwelcoming place. I care about this community, and you are doing your best to break it. </p> <p>At this stage, I call for an IP ban.</p> anders_h bHJdzThAYkwymAT8K 2016-01-25T18:26:51.467Z Comment by Anders_H on Map:Territory::Uncertainty::Randomness – but that doesn’t matter, value of information does. https://www.lesswrong.com/posts/CnwSsKuKCg8fLC4vr/map-territory-uncertainty-randomness-but-that-doesn-t-matter#qAgzhWbKyGnmbj5ay <blockquote> <p>So thinking probabilities existing as &quot;things itself&quot; taken to the extreme could lead one to the conclusion that one cant say much for example about single-case probabilities. </p> </blockquote> <p>Thinking probabilities can exists in the territory leads to no such conclusion. Thinking probabilities exist <em>only</em> in the territory may lead to such a conclusion, but that is a strawman of the points that are being made. </p> <p>It would be insane to deny that frequencies exist, or that they can be represented by a formal system derived from the Kolmogorov (or Cox) axioms. </p> <p>It would also be insane to deny that beliefs exist, or that they can be represented by a formal system derived from the Kolmogorov (or Cox) axioms.</p> <p>I think this confusion would all go away if people stopped worrying about the semantic meaning of the word &quot;probability&quot; and just specified whether they are talking about frequency or belief. It puzzles me when people insist that the formal system can only be isomorphic to one thing, and it is truly bizarre when they take sides in a holy war over which of those things it &quot;really&quot; represents. A rational decision maker genuinely needs both the concept of frequency <em>and</em> the concept of belief. </p> <p>For instance, an agent may need to reason about the proportion (frequency) P of Everett branches in which he survives if he makes a decision, and also about how certain he is about his estimate of that probability. Let's say his beliefs about the probability P follow a beta distribution, or any other distribution bounded by 0 and 1. In order to make a decision, he may do something like calculate a new probability Q, which is the expected value of P under his prior. You can interpret Q as the agent's beliefs about the probability of dying, but it also has elements of frequency.</p> <p>You can make the untestable claim that all Everett branches have the same outcome, and therefore that Q is determined exclusively by your uncertainty about whether you will live or die in all Everett branches. This would be Bayesian fundamentalism. You can also go to the other extreme and argue that Q is determined exclusively by P, and that there is no reason to consider uncertainty. That would be Frequentist fundamentalism. However, there is a spectrum between the two and there is no reason we should only allow the two edge cases to be possible positions. The truth is almost certainly somewhere in between.</p> anders_h qAgzhWbKyGnmbj5ay 2016-01-24T19:34:21.276Z Meetup : Palo Alto Meetup: Lightning Talks https://www.lesswrong.com/posts/eRcFKcSyrfLu5gYAr/meetup-palo-alto-meetup-lightning-talks <h2>Discussion article for the meetup : <a href='/meetups/1ku'>Palo Alto Meetup: Lightning Talks</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">02 February 2016 06:30:00PM (-0800)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">3911 Grove Avenue, Palo Alto</span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>Be prepared to talk for 5 minutes about any subject!</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/1ku'>Palo Alto Meetup: Lightning Talks</a></h2> anders_h eRcFKcSyrfLu5gYAr 2016-01-20T20:04:34.593Z Comment by Anders_H on Stupid Questions, 2nd half of December https://www.lesswrong.com/posts/xLtHTqw6RcBGk5wfy/stupid-questions-2nd-half-of-december#rsRQ9AJ3Ldz2qd57e <p>Sure, this is true, thanks for noticing. Sorry about the inaccurate/incorrect wording. It does however not affect the main idea. </p> anders_h rsRQ9AJ3Ldz2qd57e 2016-01-12T22:34:27.652Z Comment by Anders_H on Open Thread, January 11-17, 2016 https://www.lesswrong.com/posts/4hWA5TRH4L3bmQR7y/open-thread-january-11-17-2016#z4saLb4vEAM82uvG6 <p>I finally gave in and opened a Tumblr account at <a href="http://dooperator.tumblr.com/">http://dooperator.tumblr.com/</a> . This open-thread comment is just to link my identity on Less Wrong with my username on websites where I do not want my participation to be revealed by a simple Google search for my name, such as SlateStarCodex and Tumblr. </p> anders_h z4saLb4vEAM82uvG6 2016-01-12T18:18:51.679Z Comment by Anders_H on Stupid Questions, 2nd half of December https://www.lesswrong.com/posts/xLtHTqw6RcBGk5wfy/stupid-questions-2nd-half-of-december#Yg5abe2P8X9d3sBL6 <p>There is a difference between &quot;How sure are you that if we looked at the coin now, it is heads?&quot; and &quot;How sure are you that if we looked at the coin only once, at the end of the experiment, it is heads?&quot;</p> <p>In the first variant, the thirder position is unambiguously true. </p> <p>I the second variant, I suspect that you really need more precision in the words to answer it. I think a halfer interpretation of this question is at least plausible under some definitions of &quot;how sure&quot;</p> <p>Unless &quot;how sure&quot; refers explicitly to well specified bet, many attempts to define it will end up being circular. </p> anders_h Yg5abe2P8X9d3sBL6 2016-01-12T01:51:56.644Z Comment by Anders_H on Variations on the Sleeping Beauty https://www.lesswrong.com/posts/KM5WcMse3c7ChTXie/variations-on-the-sleeping-beauty#69qd6XPFnamRXpXMs <p>This post caused me to type up some old, unrelated thoughts about Sleeping Beauty. I posted it as a comment to the stupid questions thread at <a href="http://lesswrong.com/lw/n3v/stupid_questions_2nd_half_of_december/d14z">http://lesswrong.com/lw/n3v/stupid_questions_2nd_half_of_december/d14z</a> . I'd very much appreciate any feedback on this idea. This comment is just to catch the attention of readers interested in Sleeping Beauty who may not see the comment in the stupid questions thread.</p> anders_h 69qd6XPFnamRXpXMs 2016-01-11T09:33:43.744Z Comment by Anders_H on Stupid Questions, 2nd half of December https://www.lesswrong.com/posts/xLtHTqw6RcBGk5wfy/stupid-questions-2nd-half-of-december#mNBHqfSYbctCJS89P <p>I have an intuition that I have dissolved the sleeping beauty paradox as semantic confusion about the word &quot;probability&quot;. I am aware that my reasoning is unlikely to be accepted by the community, but I am unsure what is wrong with it. I am posting this to the &quot;stupid questions&quot; thread to see if helps me gain any insight either on Sleeping Beauty or on the thought process that led to me feeling like I've dissolved the question. </p> <p>When the word &quot;probability&quot; is used to describe the beliefs of an agent, we are really talking about how that agent would bet, for instance in an ideal prediction market. However, if the rules of the prediction market are unclear, we may get semantic confusion. </p> <p>In general, when you are asked &quot;What is the probability that the coin came up heads&quot; we interpret this as &quot;how much are you willing to pay for a contract that will be worth 1 dollar if the coin came up heads, and nothing if it came up tails&quot;. This seems straight forward, but in the sleeping beauty problem, the agent may make the same bet multiple times, which introduces ambiguity. </p> <p>Person 1 may interpret then the question as follows: &quot;Every time you wake up, there is a new one dollar bill on the table. How much are you willing to pay for a contract that gives you the dollar if the coin came up heads?&quot;. In this interpretation, you get to keep all the dollars you won throughout the experiment.</p> <p>In contrast, person 2 may interpret the question as follows &quot;There is one dollar on the table. Every time you wake up, you are given a chance to revise the price you are willing to pay for the contract, but all earlier bets are cancelled such that only the last bet counts&quot;. In this interpretation, there is only one dollar to be won.</p> <p>Person 1 will conclude that the probability is 1/3, and person 2 will conclude that the probability is 1/2. However, once they agree on what bet they are asked to make, the disagreement is dissolved. </p> <p>The first definition is probably better matched to current usage of the word. This gives most rationalists a strong intuition that the thirder position is &quot;correct&quot;. However, if you want to know which definition is most useful or applicable, this really depends on the disguised query, and on which real world scenario the parable is meant to represent. If the payoff utility is only determined once (at the end of the experiment), then the halfer definition could be more useful?</p> <p>ETA: After reading the Wikipedia:Talk section for Sleeping Beauty, it appears that this idea is not original and that in fact a lot of people have reached the same conclusion. I should have read that before I commented...</p> anders_h mNBHqfSYbctCJS89P 2016-01-11T06:14:50.498Z Meetup : Palo Alto Meetup: Introduction to Causal Inference https://www.lesswrong.com/posts/FxPjMLXpSdZFS4htL/meetup-palo-alto-meetup-introduction-to-causal-inference <h2>Discussion article for the meetup : <a href='/meetups/1kb'>Palo Alto Meetup: Introduction to Causal Inference</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">19 January 2016 06:30:00PM (-0800)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">3911 Grove Avenue, Palo Alto</span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>Anders Huitfeldt will give an introductory talk about causal reasoning and the distinction between causal inference and statistical inference</p> <p>The meetup is at 6:30pm on Tuesday Jan 19th at the group house Tesseract in Palo Alto. Allergen notes: Cats and Dogs</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/1kb'>Palo Alto Meetup: Introduction to Causal Inference</a></h2> anders_h FxPjMLXpSdZFS4htL 2016-01-03T02:22:37.793Z Meetup : Palo Alto Meetup: The Economics of AI https://www.lesswrong.com/posts/aK4wfNo4FGWNtkMtQ/meetup-palo-alto-meetup-the-economics-of-ai <h2>Discussion article for the meetup : <a href='/meetups/1ka'>Palo Alto Meetup: The Economics of AI</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">05 January 2016 06:30:00PM (-0800)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">3911 Grove Avenue, Palo Alto</span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>Edward will give a talk about the Economics of AI and its implications for the Singularity Hypothesis</p> <p>The meetup is at 6:30pm on Tuesday Jan 5th at the group house Tesseract in Palo Alto. Allergen notes: Cats and Dogs</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/1ka'>Palo Alto Meetup: The Economics of AI</a></h2> anders_h aK4wfNo4FGWNtkMtQ 2016-01-03T02:20:41.540Z Comment by Anders_H on Rationality Quotes Thread December 2015 https://www.lesswrong.com/posts/RFdAPGxyrN2nRKFcL/rationality-quotes-thread-december-2015#gPj9AL6bySjuvzNyR <p>The original quote said to rate <em>each intervention</em> by how much it helped or hurt the situation, i.e. its <em>individual-level causal effect</em>. None of those study designs will help you with that: They may be appropriate if you want to estimate the average effect across multiple similar situations, but that is not what you need here.</p> <p>This is a serious question. How do you plan to rate the effectiveness of things like the decision to intervene in Libya, or the decision not to intervene in Syria, under profound uncertainty about what would have happened if the alternative decision had been made?</p> anders_h gPj9AL6bySjuvzNyR 2016-01-01T18:19:55.895Z Comment by Anders_H on Rationality Quotes Thread December 2015 https://www.lesswrong.com/posts/RFdAPGxyrN2nRKFcL/rationality-quotes-thread-december-2015#BfCc459CxBLzh6HJz <blockquote> <p>Rate how much each intervention (or decision not to intervene) helped or hurt the situation, in retrospect, on a &gt;scale from -10 to +10.</p> </blockquote> <p>How do you plan to do this without counterfactual knowledge?</p> anders_h BfCc459CxBLzh6HJz 2016-01-01T00:16:10.285Z Comment by Anders_H on In defense of philosophy https://www.lesswrong.com/posts/Zfegawh3akENy5m5z/in-defense-of-philosophy#5SuRus4sta8bhP3kq <blockquote> <p>Contrarily to LukeProg, knowledge of the Gettier Problem improves one's epistemology because it shows that knowledge equals justified true belief is not a viable stance.</p> </blockquote> <p>Consider two agents who are communicating with each other in an attempt to reach Aumann Agreement. These agents will certainly need precise words for the following concepts:</p> <p>Reality: &quot;Is statement A true?&quot;</p> <p>Belief: &quot;Does agent M believe that statement A is true?&quot; and &quot;With what probability does Agent M believe that statement A is true?&quot;</p> <p>Map/Territory correspondence: &quot;Does Agent M's belief that Statement A is true correspond to reality?&quot;</p> <p>Calibration: &quot;Are Agent M's beliefs well calibrated?&quot;</p> <p>Epistemic process: &quot;What method did agent M use to generate his posterior beliefs?&quot; &quot;Is that method reliable?&quot;</p> <p>Gettier problems show that you won't be able to project these five dimensions onto a single binary. Which is true but not very insightful. Moreover, I can't imagine that the ability to reach Aumann agreement will ever depend on the definition of &quot;knowledge&quot;. Therefore, this is mostly an empty semantics discussion. </p> anders_h 5SuRus4sta8bhP3kq 2015-12-23T06:24:36.201Z Post-doctoral Fellowships at METRICS https://www.lesswrong.com/posts/j9w2GAe6pKsSvWz7x/post-doctoral-fellowships-at-metrics <div>The Meta-Research Innovation Center at Stanford (METRICS) is hiring post-docs for 2016/2017. The full announcement is available at <a href="http://metrics.stanford.edu/education/postdoctoral-fellowships">http://metrics.stanford.edu/education/postdoctoral-fellowships</a>. Feel free to contact me with any questions; I am currently a post-doc in this position.</div> <div><br /></div> <div>METRICS is a research center within Stanford Medical School. It was set up to study the conditions under which the scientific process can be expected to generate accurate beliefs, for instance about the validity of evidence for the effect of interventions.</div> <div><br /></div> <div>METRICS was founded by Stanford Professors Steve Goodman and John Ioannidis in 2014, after Givewell connected them with the Laura and John Arnold Foundation, who provided the initial funding. See <a href="http://blog.givewell.org/2014/04/23/meta-research-innovation-centre-at-stanford-metrics/">http://blog.givewell.org/2014/04/23/meta-research-innovation-centre-at-stanford-metrics/</a> for more details.</div> <div><br /></div> anders_h j9w2GAe6pKsSvWz7x 2015-11-12T19:13:12.419Z On stopping rules https://www.lesswrong.com/posts/yfpCedvCaxvhzyemG/on-stopping-rules <p>(tl;dr: In this post I try to explain why I think the stopping rule of an experiment matters. It is likely that someone will find a flaw in my reasoning. That would be a great outcome as it would help me change my mind. &nbsp;Heads up: If you read this looking for new insight you may be disappointed to only find my confusion)</p> <p>&nbsp;</p> <p><em>(Edited to add: Comments by Manfred and Ike seem to point correctly to the critical flaws in my reasoning. I will try to update my intuition over the next few days)</em></p> <p>&nbsp;</p> <p>&nbsp;</p> <p>In the post <a title="Don't You Care If It Works Part 1" href="/lw/mjr/dont_you_care_if_it_works_part_1/&quot;">"Don't You Care If It Works Part 1"</a> on the Main section of this website, Jacobian writes:</p> <p>&nbsp;</p> <blockquote> <p><span style="line-height: 19.5px; text-align: justify; font-family: Calibri;">A few weeks ago I started reading&nbsp;</span><a style="color: #8a8a8b; font-family: Arial, Helvetica, sans-serif; line-height: 19.5px; text-align: justify;" href="/lw/mt/beautiful_probability/"><span style="color: #0563c1; font-family: Calibri;">beautiful probability</span></a><span style="line-height: 19.5px; text-align: justify; font-family: Calibri;">&nbsp;and immediately thought that Eliezer is wrong about the stopping rule mattering to inference. I dropped everything and spent the next three hours&nbsp;</span><a style="color: #8a8a8b; font-family: Arial, Helvetica, sans-serif; line-height: 19.5px; text-align: justify;" href="/lw/mt/beautiful_probability/cj0r"><span style="color: #0563c1; font-family: Calibri;">convincing myself</span></a><span style="line-height: 19.5px; text-align: justify; font-family: Calibri;">&nbsp;that the stopping rule doesn't matter and I agree with Jaynes and Eliezer. As luck would have it, soon after that the stopping rule question was the topic of discussion at our local LW meetup. A couple people agreed with me and a couple didn't and tried to prove it with math, but most of the room seemed to hold a third opinion: they disagreed but didn't care&nbsp;to find out. I found that position quite mind-boggling. Ostensibly, most people are in that room because we read the sequences and thought that this EWOR (Eliezer's Way Of Rationality) thing is pretty cool. EWOR is an epistemology based on the mathematical rules of probability, and the dude who came up with it apparently does mathematics for a living trying to save the world.&nbsp;<em>It doesn't seem like a stretch to think that if you disagree with Eliezer on a question of probability math</em>, a question that he considers so obvious it requires no explanation,&nbsp;<em>that's a big frickin' deal!</em></span></p> </blockquote> <p style="text-align: justify; ">First, I'd like to point out that the mainstream academic term for Eliezer's claim is <a title="the Strong Likelihood Principle" href="https://en.wikipedia.org/wiki/Likelihood_principle">The Strong Likelihood Principle</a>. &nbsp;In the comments section, a vigorous discussion of stopping rules ensued.&nbsp;</p> <p style="text-align: justify; ">My own intuition is that the strong likelihood principle is wrong. &nbsp;Moreover, there exist a small number of people whose opinion I give higher level of credence than Eliezer's, and some of those people also disagree with him. For instance, I've been present in the room when a distinguished Professor of Biostatistics at Harvard stated matter-of-factly that the principle is trivially wrong. I also observed that he was not challenged on this by another full Professor of Biostatistics who is considered an expert on Bayesian inference.</p> <p style="text-align: justify; ">So at best, the fact that Eliezer supports the strong likelihood principle is a single data point, ie pretty weak Bayesian evidence. &nbsp;I do however value Eliezer's opinion, and in this case I recognize that I am confused. Being a good rationalist, I'm going to take that as an indication that it is time for <a title="http://lesswrong.com/lw/us/the_ritual/#more" href="/lw/us/the_ritual/#more">The Ritual.</a> &nbsp;Writing this post is part of my "ritual": It is an attempt to clarify exactly why I think the stopping condition matters, and determine whether those reasons are valid. &nbsp; I expect a likely outcome is that someone will identify a flaw in my reasoning. This will be very useful and help improve my map-territory correspondence.</p> <p style="text-align: justify; ">--</p> <p style="text-align: justify; ">Suppose there are two coins in existence, both of which are biased: Coin A comes up heads with probability 2/3 and tails with probability 1/3, &nbsp;whereas Coin B comes up heads with probability 1/3. &nbsp; &nbsp; Someone gives me a coin without telling me which one, my goal is to figure out if it is Coin A or Coin B. &nbsp; My prior is that they are equally likely.</p> <p style="text-align: justify; ">There are two statisticians who both offer to do an experiment: &nbsp;Statistician 1 says that he will flip the coin 20 times and report the number of heads. &nbsp; &nbsp;Statistician 2 would really like me to believe that it is Coin B, and says he will terminate the experiment whenever there are more tails than heads. However, since Statistician 2 is kind of lazy and doesn't have infinite time, he also says that if he reaches 20 flips he is going to call it quits and give up.</p> <p style="text-align: justify; ">Both statisticians do the experiment, and both experiments end up with 12 heads and 8 tails. I trust both Statisticians to be honest about the experimental design and the stopping rules.&nbsp;</p> <p style="text-align: justify; ">In the experiment of Statistician 1, the probability of getting this outcome if you have Coin A was 0.1486, whereas the probability of getting this outcome if it was Coin B was 0.0092. &nbsp;The likelihood ratio is therefore&nbsp;16.1521 &nbsp; and the posterior probability of Coin A (after converting the prior to odds, applying the likelihood ratio and converting back to probability) is 0.94.</p> <p style="text-align: justify; ">In the experiment of Statistician 2, however, I can't just use the binomial distribution because there is an additional data point which is not Bernoulli, namely the number of coin flips. &nbsp;I therefore have to calculate, for both Coin A and Coin B, &nbsp;the probability that he would not terminate the experiment prior to the 20th flip, and that at that stage he would have 12 heads and 8 coins. &nbsp; &nbsp;Since the probability reaching 20 flips is much higher for Coin A than for Coin B, the likelihood ratio would be much higher than in the experiment of Statistician 1.&nbsp;</p> <p style="text-align: justify; ">&nbsp;</p> <p style="text-align: justify; ">This should not be unexpected: If Statistician B gives me data that supports the hypothesis which his stopping rule was designed to discredit, then that data is stronger evidence than similar data coming from the neutral Statistician A. &nbsp;</p> <p style="text-align: justify; ">In other words, the stopping rule matters. Yes, all the evidence in the trial is still in the likelihood ratio, but the likelihood ratio is different because there is an additional data point. &nbsp; Not considering this additional data point is statistical malpractice.&nbsp;</p> <p style="text-align: justify; ">&nbsp;</p> <p style="text-align: justify; ">&nbsp;</p> anders_h yfpCedvCaxvhzyemG 2015-08-02T21:38:08.617Z Meetup : Boston: Trigger action planning https://www.lesswrong.com/posts/oJ4rCNrPeTwrMMDcQ/meetup-boston-trigger-action-planning <h2>Discussion article for the meetup : <a href='/meetups/1dy'>Boston: Trigger action planning</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">21 June 2015 03:30:00PM (-0400)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">98 Elm Street, Somerville</span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>Mick Porter will be presenting on trigger action planning, which is a strategy taught by CFAR to systematize solutions to everyday problems.</p> <p>Cambridge/Boston-area Less Wrong meetups start at 3:30pm on the first and third Sunday of every month.</p> <p>The default location is at the Citadel Rationalist House in Porter Sq, at 98 Elm St, apt 1, Somerville (Occasionally, meetups take place at other locations, such as MIT or Harvard. This will be specified as needed)</p> <p>Our default schedule is as follows:</p> <p>—Phase 1: Arrival, greetings, unstructured conversation.</p> <p>—Phase 2: The headline event. This starts promptly at 4pm, and lasts 30-60 minutes.</p> <p>—Phase 3: Further discussion. We'll explore the ideas raised in phase 2, often in smaller groups.</p> <p>—Phase 4: Dinner.</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/1dy'>Boston: Trigger action planning</a></h2> anders_h oJ4rCNrPeTwrMMDcQ 2015-05-24T20:00:31.195Z Meetup : Boston: Making space in Interpersonal Interactions https://www.lesswrong.com/posts/Mvy282BBWnwZ69sJ3/meetup-boston-making-space-in-interpersonal-interactions <h2>Discussion article for the meetup : <a href='/meetups/1dx'>Boston: Making space in Interpersonal Interactions</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">07 June 2015 03:30:00PM (-0400)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">98 Elm Street, Somerville</span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>Kate (from the blog Gruntled and Hinged) is going to be talking about a concept called holding space. Specifically she's going to focus on talking with people who are having emotions when you disagree with their reasoning for having those emotions.</p> <p>Here's a cool intro: <a href="http://heatherplett.com/2015/03/hold-space/" rel="nofollow">http://heatherplett.com/2015/03/hold-space/</a></p> <p>As always, the talk will be followed by discussion, both about the feature presentation and general</p> <p>Cambridge/Boston-area Less Wrong meetups start at 3:30pm on the first and third Sunday of every month.</p> <p>The default location is at the Citadel Rationalist House in Porter Sq, at 98 Elm St, apt 1, Somerville (Occasionally, meetups take place at other locations, such as MIT or Harvard. This will be specified as needed)</p> <p>Our default schedule is as follows:</p> <p>—Phase 1: Arrival, greetings, unstructured conversation.</p> <p>—Phase 2: The headline event. This starts promptly at 4pm, and lasts 30-60 minutes.</p> <p>—Phase 3: Further discussion. We'll explore the ideas raised in phase 2, often in smaller groups.</p> <p>—Phase 4: Dinner.</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/1dx'>Boston: Making space in Interpersonal Interactions</a></h2> anders_h Mvy282BBWnwZ69sJ3 2015-05-24T19:58:56.123Z Meetup : Boston: How to Beat Perfectionism https://www.lesswrong.com/posts/gn6uQ23bG3sSQyxrK/meetup-boston-how-to-beat-perfectionism <h2>Discussion article for the meetup : <a href='/meetups/1dc'>Boston: How to Beat Perfectionism</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">17 May 2015 03:30:04PM (-0400)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">98 Elm Street, Somerville</span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>How to Beat Perfectionism (or at least reduce it)</p> <p>Are you a perfectionist? Do you want to be? Do you want to not be? Jesse Galef will talk on this matter, and his slides will be flawless.</p> <p>Join us at the Citadel for the talk and discussion!</p> <p>Cambridge/Boston-area Less Wrong meetups start at 3:30pm on the first and third Sunday of every month.</p> <p>The default location is at the Citadel Rationalist House in Porter Sq, at 98 Elm St, apt 1, Somerville (Occasionally, meetups take place at other locations, such as MIT or Harvard. This will be specified as needed)</p> <p>Our default schedule is as follows:</p> <p>—Phase 1: Arrival, greetings, unstructured conversation.</p> <p>—Phase 2: The headline event. This starts promptly at 4pm, and lasts 30-60 minutes.</p> <p>—Phase 3: Further discussion. We'll explore the ideas raised in phase 2, often in smaller groups.</p> <p>—Phase 4: Dinner.</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/1dc'>Boston: How to Beat Perfectionism</a></h2> anders_h gn6uQ23bG3sSQyxrK 2015-05-08T17:42:02.809Z Meetup : Boston: Unconference https://www.lesswrong.com/posts/BBGKLxYDQxj2GKW4s/meetup-boston-unconference-0 <h2>Discussion article for the meetup : <a href='/meetups/1be'>Boston: Unconference</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">05 April 2015 03:30:00PM (-0400)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">98 elm street somerville</span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>It's an Unconference! <a href="http://en.wikipedia.org/wiki/Unconference" rel="nofollow">http://en.wikipedia.org/wiki/Unconference</a> There will be short discussions on a variety of topics, as decided by attendees at the beginning of the meetup. Think about what topics you would like to discuss or talk about! Exact format TBD. Cambridge/Boston-area Less Wrong meetups start at 3:30pm on the 1st and 3rd Sunday at the Citadel in Porter Sq, at 98 Elm St, apt 1, Somerville. We also have last Wednesday meetups at the Citadel at 7pm. Our default schedule is as follows: —Phase 1: Arrival, greetings, unstructured conversation. This starts at 3:30; before then, Citadel residents will be busy. Looking forward to seeing you at 3:30! —Phase 2: The headline event. This starts promptly at 4pm, and lasts 30-60 minutes. —Phase 3: Further discussion. We'll explore the ideas raised in phase 2, often in smaller groups. —Phase 4: Dinner.</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/1be'>Boston: Unconference</a></h2> anders_h BBGKLxYDQxj2GKW4s 2015-03-19T16:58:45.803Z Prediction Markets are Confounded - Implications for the feasibility of Futarchy https://www.lesswrong.com/posts/xnC68ZfTkPyzXQS8p/prediction-markets-are-confounded-implications-for-the <p><em>(tl;dr: &nbsp;In this post, I show that prediction markets estimate non-causal probabilities, and can therefore not be used for decision making by rational agents following causal decision theory. &nbsp;I provide an example of a simple situation where such confounding leads to a society which has implemented futarchy making an incorrect decision)</em></p> <p>&nbsp;</p> <p>It is October 2016, and the US Presidential Elections are nearing. The most powerful nation on earth is about to make a momentous decision about whether being the brother of a former president is a more impressive qualification than being the wife of a former president. However, one additional criterion has recently become relevant in light of current affairs: &nbsp; Kim Jong-Un, Great Leader of the Glorious Nation of North Korea, is making noise about his deep hatred for Hillary Clinton. He also occasionally discusses the possibility of nuking a major US city. The US electorate, desperate to avoid being nuked, have come up with an ingenious plan: They set up a prediction market to determine whether electing Hillary will impact the probability of a nuclear attack.&nbsp;</p> <p>The following rules are stipulated: &nbsp;There are four possible outcomes, either "Hillary elected and US Nuked", "Hillary elected and US not nuked", "Jeb elected and US nuked", "Jeb elected and US not nuked". &nbsp; Participants in the market can buy and sell contracts for each of those outcomes, &nbsp;the contract which correponds to the actual outcome will expire at $100, all other contracts will expire at $0</p> <p>Simultaneously in a country far, far away, &nbsp;a rebellion is brewing against the Great Leader. &nbsp;The potential challenger not only appears not to have no problem with Hillary, he also seems like a reasonable guy who would be unlikely to use nuclear weapons. It is generally believed that the challenger will take power with probability 3/7; and will be exposed and tortured in a forced labor camp for the rest of his miserable life with probability 4/7. &nbsp; &nbsp; Let us stipulate that this information is known to all participants &nbsp;- I am adding this clause in order to demonstrate that this argument does not rely on unknown information or information asymmetry.&nbsp;</p> <p>A mysterious but trustworthy agent named "Laplace's Demon" has recently appeared, and informed everyone that, to a first approximation, &nbsp;the world is currently in one of seven possible quantum states. &nbsp;The Demon, being a perfect Bayesian reasoner with Solomonoff Priors, has determined that each of these states should be assigned probability 1/7. &nbsp; &nbsp; Knowledge of which state we are in will perfectly predict the future, with one important exception: &nbsp; It is possible for the US electorate to "Intervene" by changing whether Clinton or Bush is elected. This will then cause a ripple effect into all future events that depend on which candidate is elected President, but otherwise change nothing.&nbsp;</p> <p>The Demon swears up and down that the choice about whether Hillary or Jeb is elected has absolutely no impact in any of the seven possible quantum states. However, because the Prediction market has already been set up and there are powerful people with vested interests, it is decided to run the market anyways.&nbsp;</p> <p>&nbsp;Roughly, the demon tells you that the world is in one of the following seven states:</p> <p>&nbsp;</p> <table class="MsoTableGrid" style="border-collapse: collapse; border: none; mso-border-alt: solid windowtext .5pt; mso-yfti-tbllook: 1184; mso-padding-alt: 0in 5.4pt 0in 5.4pt;" border="1" cellspacing="0" cellpadding="0"> <tbody> <tr style="mso-yfti-irow: 0; mso-yfti-firstrow: yes; height: 49.5pt;"> <td style="width: 55.3pt; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 49.5pt;" width="74" valign="top"> <p class="MsoNormal">State</p> </td> <td style="width: 57.75pt; border: solid windowtext 1.0pt; border-left: none; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 49.5pt;" width="77" valign="top"> <p class="MsoNormal">Kim overthrown</p> </td> <td style="width: 63.4pt; border: solid windowtext 1.0pt; border-left: none; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 49.5pt;" width="85" valign="top"> <p class="MsoNormal">Election winner (if no intervention)</p> </td> <td style="width: 55.65pt; border: solid windowtext 1.0pt; border-left: none; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 49.5pt;" width="74" valign="top"> <p class="MsoNormal">US Nuked if Hillary elected</p> </td> <td style="width: 55.65pt; border: solid windowtext 1.0pt; border-left: none; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 49.5pt;" width="74" valign="top"> <p class="MsoNormal">US Nuked if Jeb elected</p> </td> <td style="width: 60.0pt; border: solid windowtext 1.0pt; border-left: none; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 49.5pt;" width="80" valign="top"> <p class="MsoNormal">US Nuked</p> </td> </tr> <tr style="mso-yfti-irow: 1; height: 11.85pt;"> <td style="width: 55.3pt; border: solid windowtext 1.0pt; border-top: none; mso-border-top-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">1</p> </td> <td style="width: 57.75pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="77" valign="top"> <p class="MsoNormal">No</p> </td> <td style="width: 63.4pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="85" valign="top"> <p class="MsoNormal">Hillary</p> </td> <td style="width: 55.65pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">Yes</p> </td> <td style="width: 55.65pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">Yes</p> </td> <td style="width: 60.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="80" valign="top"> <p class="MsoNormal">Yes</p> </td> </tr> <tr style="mso-yfti-irow: 2; height: 11.85pt;"> <td style="width: 55.3pt; border: solid windowtext 1.0pt; border-top: none; mso-border-top-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">2</p> </td> <td style="width: 57.75pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="77" valign="top"> <p class="MsoNormal">No</p> </td> <td style="width: 63.4pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="85" valign="top"> <p class="MsoNormal">Hillary</p> </td> <td style="width: 55.65pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">No</p> </td> <td style="width: 55.65pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">No</p> </td> <td style="width: 60.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="80" valign="top"> <p class="MsoNormal">No</p> </td> </tr> <tr style="mso-yfti-irow: 3; height: 11.85pt;"> <td style="width: 55.3pt; border: solid windowtext 1.0pt; border-top: none; mso-border-top-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">3</p> </td> <td style="width: 57.75pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="77" valign="top"> <p class="MsoNormal">No</p> </td> <td style="width: 63.4pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="85" valign="top"> <p class="MsoNormal">Jeb</p> </td> <td style="width: 55.65pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">Yes</p> </td> <td style="width: 55.65pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">Yes</p> </td> <td style="width: 60.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="80" valign="top"> <p class="MsoNormal">Yes</p> </td> </tr> <tr style="mso-yfti-irow: 4; height: 11.85pt;"> <td style="width: 55.3pt; border: solid windowtext 1.0pt; border-top: none; mso-border-top-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">4</p> </td> <td style="width: 57.75pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="77" valign="top"> <p class="MsoNormal">No</p> </td> <td style="width: 63.4pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="85" valign="top"> <p class="MsoNormal">Jeb</p> </td> <td style="width: 55.65pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">No</p> </td> <td style="width: 55.65pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">No</p> </td> <td style="width: 60.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="80" valign="top"> <p class="MsoNormal">No</p> </td> </tr> <tr style="mso-yfti-irow: 5; height: 11.85pt;"> <td style="width: 55.3pt; border: solid windowtext 1.0pt; border-top: none; mso-border-top-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">5</p> </td> <td style="width: 57.75pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="77" valign="top"> <p class="MsoNormal">Yes</p> </td> <td style="width: 63.4pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="85" valign="top"> <p class="MsoNormal">Hillary</p> </td> <td style="width: 55.65pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">No</p> </td> <td style="width: 55.65pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">No</p> </td> <td style="width: 60.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="80" valign="top"> <p class="MsoNormal">No</p> </td> </tr> <tr style="mso-yfti-irow: 6; height: 11.85pt;"> <td style="width: 55.3pt; border: solid windowtext 1.0pt; border-top: none; mso-border-top-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">6</p> </td> <td style="width: 57.75pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="77" valign="top"> <p class="MsoNormal">Yes</p> </td> <td style="width: 63.4pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="85" valign="top"> <p class="MsoNormal">Jeb</p> </td> <td style="width: 55.65pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">No</p> </td> <td style="width: 55.65pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="74" valign="top"> <p class="MsoNormal">No</p> </td> <td style="width: 60.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 11.85pt;" width="80" valign="top"> <p class="MsoNormal">No</p> </td> </tr> <tr style="mso-yfti-irow: 7; mso-yfti-lastrow: yes; height: 4.5pt;"> <td style="width: 55.3pt; border: solid windowtext 1.0pt; border-top: none; mso-border-top-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 4.5pt;" width="74" valign="top"> <p class="MsoNormal">7</p> </td> <td style="width: 57.75pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 4.5pt;" width="77" valign="top"> <p class="MsoNormal">Yes</p> </td> <td style="width: 63.4pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 4.5pt;" width="85" valign="top"> <p class="MsoNormal">Jeb</p> </td> <td style="width: 55.65pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 4.5pt;" width="74" valign="top"> <p class="MsoNormal">No</p> </td> <td style="width: 55.65pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 4.5pt;" width="74" valign="top"> <p class="MsoNormal">No</p> </td> <td style="width: 60.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 4.5pt;" width="80" valign="top"> <p class="MsoNormal">No</p> </td> </tr> </tbody> </table> <p><br />Let us use this table to define some probabilities: &nbsp; If one intervenes to make Hillary win the election, the probability of the US being nuked is 2/7 (this is seen from column 4). &nbsp;If one intervenes to make Jeb win the election, the probability of the US being nuked is 2/7 (this is seen from column 5). &nbsp; In the language of causal inference, these probabilities are Pr (Nuked| Do (Elect Clinton)] and Pr[Nuked | Do(Elect Bush)]. &nbsp;The fact that these two quantities &nbsp;are equal confirms the Demon&rsquo;s claim that the choice of President has no effect on the outcome. &nbsp;An agent operating under Causal Decision theory will use this information to correctly conclude that he has no preference about whether to elect Hillary or Jeb.&nbsp;</p> <p>However, if one were to condition on who actually was elected, we get different numbers: &nbsp;Conditional on being in a state where Hillary is elected, the probability of the US being nuked is 1/3; whereas conditional on being in a state where Jeb is elected, the probability of being nuked is &frac14;. &nbsp;Mathematically, these probabilities are Pr [Nuked | Clinton Elected] and Pr[Nuked | Bush Elected]. &nbsp;An agent operating under Evidentiary Decision theory will use this information to conclude that he will vote for Bush. &nbsp;Because evidentiary decision theory is wrong, he will fail to optimize for the outcome he is interested in.&nbsp;</p> <p>Now, let us ask ourselves which probabilities our prediction markets will converge to, ie which probabilities participants in the market have an incentive to provide their best estimate of. &nbsp;We defined our contract as "Hillary is elected and the US is nuked". &nbsp;The probability of this occurring in 1/7; &nbsp;if we normalize by dividing by the marginal probability that Hillary is elected, we get 1/3 which is equal to &nbsp;Pr [Nuked | Clinton Elected]. &nbsp; In other words, the prediction market estimates the wrong quantities.</p> <p>Essentially, what happens is structurally the same phenomenon as confounding in epidemiologic studies: &nbsp;There was a common cause of Hillary being elected and the US being nuked. &nbsp;This common cause - whether Kim Jong-Un was still Great Leader of North Korea - led to a correlation between the election of Hillary and the outcome, but that correlation is purely non-causal and not relevant to a rational decision maker.&nbsp;</p> <p>The obvious next question is whether there exists a way to save futarchy; ie any way to give traders an incentive to pay a price that reflects their beliefs about Pr (Nuked| Do (Elect Clinton)] &nbsp;instead of Pr [Nuked | Clinton Elected]). &nbsp; &nbsp;We discussed this question at the Less Wrong Meetup in Boston a couple of months ago. The only way we agreed will definitely solve the problem is the following procedure:&nbsp;</p> <p>&nbsp;</p> <ol> <li>The governing body makes an absolute pre-commitment that no matter what happens, the next President will be determined solely on the basis of the prediction market&nbsp;</li> <li>The following contracts are listed: &ldquo;The US is nuked if Hillary is elected&rdquo; and &ldquo;The US is nuked if Jeb is elected&rdquo;</li> <li>At the pre-specified date, the markets are closed and the President is chosen based on the estimated probabilities</li> <li>If Hillary is chosen, &nbsp;the contract on Jeb cannot be settled, and all bets are reversed. &nbsp;</li> <li>The Hillary contract is expired when it is known whether Kim Jong-Un presses the button.&nbsp;</li> </ol> <p>&nbsp;</p> <p>This procedure will get the correct results in theory, but it has the following practical problems: &nbsp;It allows maximizing on only one outcome metric (because one cannot precommit to choose the President based on criteria that could potentially be inconsistent with each other). &nbsp;Moreover, it requires the reversal of trades, which will be problematic if people who won money on the Jeb contract have withdrawn their winnings from the exchange.&nbsp;</p> <p>The only other option I can think of &nbsp;in order to obtain causal information from a prediction market is to &ldquo;control for confounding&rdquo;. &nbsp; If, for instance, the only confounder is whether Kim Jong-Un is overthrown, we can control for it by using Do-Calculus to show that Pr (Nuked| Do (Elect Clinton)] = Pr (Nuked| (Clinton elected, &nbsp;Kim Overthrown)* Pr (Kim Overthrown) + Pr (Nuked| (Clinton elected, &nbsp;Kim Not Overthrown)* Pr (Kim Not Overthrown). &nbsp; All of these quantities can be estimated from separate prediction markets. &nbsp;</p> <p>&nbsp;However, this is problematic for several reasons:</p> <p>&nbsp;</p> <ol> <li>There will be an exponential explosion in the number of required prediction markets, and each of them will ask participants to bet on complicated conditional probabilities that have no obvious causal interpretation.&nbsp;</li> <li>There may be disagreement on what the confounders are, which will lead to contested contract interpretations.</li> <li>The expert consensus on what the important confounders are may change during the lifetime of the contract, which will require the entire thing to be relisted. Etc. &nbsp; &nbsp;For practical reasons, therefore, &nbsp;this approach does not seem feasible.</li> </ol> <p>&nbsp;</p> <p>I&rsquo;d like a discussion on the following questions: &nbsp;Are there any other ways to list a contract that gives market participants an incentive to aggregate information on &nbsp;causal quantities? If not, is futarchy doomed?</p> <p>(Thanks to the Less Wrong meetup in Boston and particularly Jimrandomh for clarifying my thinking on this issue)</p> <div>(I reserve the right to make substantial updates to this text in response to any feedback in the comments)</div> anders_h xnC68ZfTkPyzXQS8p 2015-01-26T22:39:33.638Z Meetup : Boston: Antifragile https://www.lesswrong.com/posts/WxvwbspC2BupxCLgp/meetup-boston-antifragile <h2>Discussion article for the meetup : <a href='/meetups/18g'>Boston: Antifragile</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">04 January 2015 03:30:33PM (-0500)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">98 Elm Street, Somerville</span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>This Sunday, Jesse Galef will be reviewing the book Antifragile, by Nassim Nicholas Taleb, author of The Black Swan. Topics will include effective decision making, catastrophic risk, and pop culture references.</p> <p>Reviews of Antifragile:</p> <p>"The glossary alone offered more thought-provoking ideas than any other nonfiction book I read this year. That said, Antifragile is far from flawless."</p> <p>"As always, an imperfect, infuriating but intriguing book"</p> <p>"A big mixed bag of insights and misconceptions"</p> <p>Cambridge/Boston-area Less Wrong meetups start at 3:30pm, and have an alternating location:</p> <ul> <li><p>1st Sunday meetups are at Citadel in Porter Sq, at 98 Elm St, apt 1, Somerville.</p></li> <li><p>3rd Sunday meetups are in MIT's building 66 at 25 Ames St, room 156. Room number subject to change based on availability; signs will be posted with the actual room number.</p></li> </ul> <p>(We also have last Wednesday meetups at Citadel at 7pm.)</p> <p>Our default schedule is as follows:</p> <p>—Phase 1: Arrival, greetings, unstructured conversation.</p> <p>—Phase 2: The headline event. This starts promptly at 4pm, and lasts 30-60 minutes.</p> <p>—Phase 3: Further discussion. We'll explore the ideas raised in phase 2, often in smaller groups.</p> <p>—Phase 4: Dinner.</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/18g'>Boston: Antifragile</a></h2> anders_h WxvwbspC2BupxCLgp 2015-01-02T20:04:48.211Z Meetup : Boston: Self Therapy https://www.lesswrong.com/posts/iy3twixnP4toNGWNK/meetup-boston-self-therapy <h2>Discussion article for the meetup : <a href='/meetups/16v'>Boston: Self Therapy</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">16 November 2014 03:30:00PM (-0500)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">98 elm street somerville</span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>LOCATION: CITADEL</p> <p>Cognitive Behavioral Therapy? Dialectical Behavioral Therapy? Apps? Books? Mood-tracking? Kate runs through evidence-based self-therapy techniques and ways to implement them.</p> <p>Our own psychologist and social worker, Kate Donovan, will be speaking about self therapy. It should be both helpful and informative.</p> <p>Join us at the Citadel for the talk and discussion! Cambridge/Boston-area Less Wrong meetups start at 3:30pm, and have an alternating location:</p> <ul> <li><p>1st Sunday meetups are at Citadel in Porter Sq, at 98 Elm St, apt 1, Somerville.</p></li> <li><p>3rd Sunday meetups are in MIT's building 66 at 25 Ames St, room 156. Room number subject to change based on availability; signs will be posted with the actual room number.</p></li> </ul> <p>(We also have last Wednesday meetups at Citadel at 7pm.)</p> <p>Our default schedule is as follows:</p> <p>—Phase 1: Arrival, greetings, unstructured conversation. <br /> —Phase 2: The headline event. This starts promptly at 4pm, and lasts 30-60 minutes. <br /> —Phase 3: Further discussion. We'll explore the ideas raised in phase 2, often in smaller groups. <br /> —Phase 4: Dinner.</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/16v'>Boston: Self Therapy</a></h2> anders_h iy3twixnP4toNGWNK 2014-11-13T17:20:19.375Z Meetup : The Design Process https://www.lesswrong.com/posts/wMezbYn2kQKfG4amw/meetup-the-design-process <h2>Discussion article for the meetup : <a href='/meetups/162'>The Design Process</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">29 October 2014 07:00:00PM (-0400)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">98 Elm Street Somerville MA</span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>Ben Sancetta, a mechanical engineer, will discuss the process of designing a new product, such as when it's appropriate to do testing, get user feedback, or list product requirements. Following this, we'll have a discussion of our own processes for designing new things. Cambridge/Boston-area Less Wrong Wednesday meetups are once a month on the last Wednesday at 7pm at Citadel (98 Elm St Apt 1 Somerville, near Porter Square). All other meetups are on Sundays. Our default schedule is as follows: —Phase 1: Arrival, greetings, unstructured conversation. —Phase 2: The headline event. This starts promptly at 7:30pm, and lasts 30-60 minutes. —Phase 3: Further discussion. We'll explore the ideas raised in phase 2, often in smaller groups.</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/162'>The Design Process</a></h2> anders_h wMezbYn2kQKfG4amw 2014-10-24T03:37:24.473Z Meetup : Boston Meetup - New Location https://www.lesswrong.com/posts/ki6aH88QobnjYWvWF/meetup-boston-meetup-new-location <h2>Discussion article for the meetup : <a href='/meetups/15o'>Boston Meetup - New Location</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">19 October 2014 03:30:37PM (-0400)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">Harvard Science Center Room 109, 1 Oxford Street, Cambridge MA 02138</span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>IMPORTANT LOCATION CHANGE!</p> <p>Rather than meeting at MIT or at the Citadel, we'll be meeting at Harvard this week. Specifically, room 109 of the Science Center. Harvard Science Center 1 Oxford Street Cambridge MA 02138 Room 109 Ron will be speaking about active trading and financial options. This will discuss economics and markets more than our previous talk about personal finance.</p> <p>Cambridge/Boston-area Less Wrong meetups start at 3:30pm, and have an alternating location:</p> <ul> <li>1st Sunday meetups are at Citadel in Porter Sq, at 98 Elm St, apt 1, Somerville. </li> <li>3rd Sunday meetups are in MIT's building 66 at 25 Ames St, room 156. Room number subject to change based on availability; signs will be posted with the actual room number. (We also have last Wednesday meetups at Citadel at 7pm.) Our default schedule is as follows:</li> </ul> <p>—Phase 1: Arrival, greetings, unstructured conversation. —Phase 2: The headline event. This starts promptly at 4pm, and lasts 30-60 minutes. —Phase 3: Further discussion. We'll explore the ideas raised in phase 2, often in smaller groups. —Phase 4: Dinner.</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/15o'>Boston Meetup - New Location</a></h2> anders_h ki6aH88QobnjYWvWF 2014-10-15T04:39:13.533Z Meetup : Meta Meetup https://www.lesswrong.com/posts/KrwwuPijLY2M6ZcJQ/meetup-meta-meetup <h2>Discussion article for the meetup : <a href='/meetups/157'>Meta Meetup</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">05 October 2014 03:30:20AM (-0400)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">98 Elm Street Somerville</span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>We'll spend the first part of the meeting discussing the topics we'd like to see more of at future meetups. For the second part of the meetup, we'll socialize and get to know each other better. Cambridge/Boston-area Less Wrong meetups start at 3:30pm, and have an alternating location:</p> <ul> <li><p>1st Sunday meetups are at Citadel in Porter Sq, at 98 Elm St, apt 1, Somerville.</p></li> <li><p>3rd Sunday meetups are in MIT's building 66 at 25 Ames St, room 156. Room number subject to change based on availability; signs will be posted with the actual room number. (We also have last Wednesday meetups at Citadel at 7pm.)</p></li> </ul> <p>Our default schedule is as follows:</p> <p>—Phase 1: Arrival, greetings, unstructured conversation.</p> <p>—Phase 2: The headline event. This starts promptly at 4pm, and lasts 30-60 minutes.</p> <p>—Phase 3: Further discussion. We'll explore the ideas raised in phase 2, often in smaller groups.</p> <p>—Phase 4: Dinner.</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/157'>Meta Meetup</a></h2> anders_h KrwwuPijLY2M6ZcJQ 2014-10-02T16:49:44.048Z Meetup : Social Skills https://www.lesswrong.com/posts/p26zpACRzZPfdbKFF/meetup-social-skills <h2>Discussion article for the meetup : <a href='/meetups/14b'>Social Skills</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">24 September 2014 07:00:00PM (-0400)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">98 Elm Street, Somerville MA</span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>Come hear the amazing and charismatic Sam Rosen speak! Cambridge/Boston-area Less Wrong Wednesday meetups are once a month on the last Wednesday at 7pm at Citadel (98 Elm St Apt 1 Somerville, near Porter Square). All other meetups are on Sundays. Our default schedule is as follows: —Phase 1: Arrival, greetings, unstructured conversation. —Phase 2: The headline event. This starts promptly at 7:30pm, and lasts 30-60 minutes. —Phase 3: Further discussion. We'll explore the ideas raised in phase 2, often in smaller groups.</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/14b'>Social Skills</a></h2> anders_h p26zpACRzZPfdbKFF 2014-09-10T00:55:23.241Z Meetup : Passive Investing and Financial Independence https://www.lesswrong.com/posts/Q2TW6PDXLWw6ci9oH/meetup-passive-investing-and-financial-independence <h2>Discussion article for the meetup : <a href='/meetups/14a'>Passive Investing and Financial Independence</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">21 September 2014 03:30:00PM (-0400)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">25 Ames Street , Cambridge MA 02139</span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>Chase will be dispensing his financial wisdom.</p> <p>Cambridge/Boston-area Less Wrong meetups start at 3:30pm, and have an alternating location:</p> <ul> <li><p>1st Sunday meetups are at Citadel in Porter Sq, at 98 Elm St, apt 1, Somerville.</p></li> <li><p>3rd Sunday meetups are in MIT's building 66 at 25 Ames St, room 156. Room number subject to change based on availability; signs will be posted with the actual room number.</p></li> </ul> <p>(We also have last Wednesday meetups at Citadel at 7pm.)</p> <p>Our default schedule is as follows:</p> <p>—Phase 1: Arrival, greetings, unstructured conversation. <br /> —Phase 2: The headline event. This starts promptly at 4pm, and lasts 30-60 minutes. <br /> —Phase 3: Further discussion. We'll explore the ideas raised in phase 2, often in smaller groups. <br /> —Phase 4: Dinner.</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/14a'>Passive Investing and Financial Independence</a></h2> anders_h Q2TW6PDXLWw6ci9oH 2014-09-10T00:53:05.455Z Meetup : Prediction Markets and Futarchy https://www.lesswrong.com/posts/uryBPekq8SLLgeEf4/meetup-prediction-markets-and-futarchy <h2>Discussion article for the meetup : <a href='/meetups/13z'>Prediction Markets and Futarchy</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">07 September 2014 03:30:00PM (-0400)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">98 Elm St Apt 1, Somerville, MA </span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>I will give a talk about prediction markets and futarchy. The talk is intended as a basic introduction for people who are new to the concept. After my slides, I hope to have a discussion about whether futarchy is feasible.</p> <p>Cambridge/Boston-area Less Wrong meetups start at 3:30pm, and have an alternating location:</p> <p>1st Sunday meetups are at Citadel in Porter Sq, at 98 Elm St, apt 1, Somerville.</p> <p>3rd Sunday meetups are in MIT's building 66 at 25 Ames St, room 156. Room number subject to change based on availability; signs will be posted with the actual room number.</p> <p>(We also have last Wednesday meetups at Citadel at 7pm.)</p> <p>Our default schedule is as follows:</p> <p>—Phase 1: Arrival, greetings, unstructured conversation.</p> <p>—Phase 2: The headline event. This starts promptly at 4pm, and lasts 30-60 minutes.</p> <p>—Phase 3: Further discussion. We'll explore the ideas raised in phase 2, often in smaller groups.</p> <p>—Phase 4: Dinner.</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/13z'>Prediction Markets and Futarchy</a></h2> anders_h uryBPekq8SLLgeEf4 2014-09-02T14:13:33.018Z Meetup : Nick Bostrom Talk on Superintelligence https://www.lesswrong.com/posts/siMQxNzt35TkSqNNn/meetup-nick-bostrom-talk-on-superintelligence <h2>Discussion article for the meetup : <a href='/meetups/13y'>Nick Bostrom Talk on Superintelligence</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">04 September 2014 08:00:00PM (-0400)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">Emerson 105, Harvard University, Cambridge, MA</span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>What happens when machines surpass humans in general intelligence? Will artificial agents save or destroy us? In his new book - Superintelligence: Paths, Dangers, Strategies - Professor Bostrom explores these questions, laying the foundation for understanding the future of humanity and intelligent life. Q&amp;A will follow the talk.</p> <p><a href="http://harvardea.org/event/2014/09/04/bostrom/" rel="nofollow">http://harvardea.org/event/2014/09/04/bostrom/</a></p> <p>(This event is organized by Harvard Effective Altruism. It is not technically a Less Wrong Meetup, but the topic is highly relevant and most of the Boston area rationalist community will be there)</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/13y'>Nick Bostrom Talk on Superintelligence</a></h2> anders_h siMQxNzt35TkSqNNn 2014-09-02T14:09:43.925Z Meetup : The Psychology of Video Games https://www.lesswrong.com/posts/vrBugb5dFbwnxuSRF/meetup-the-psychology-of-video-games <h2>Discussion article for the meetup : <a href='/meetups/131'>The Psychology of Video Games</a></h2> <div class="meetup-meta"> <p> <strong>WHEN:</strong>&#32; <span class="date">17 August 2014 03:30:00PM (-0400)</span><br> </p> <p> <strong>WHERE:</strong>&#32; <span class="address">Cambridge, MA</span> </p> </div><!-- .meta --> <div id="" class="content"> <div class="md"><p>Kelly MacNeill will talk about the psychology of video games, starting at 4pm at MIT's Building 66</p> <p>Cambridge/Boston-area Less Wrong meetups start at 3:30pm, and have an alternating location:</p> <p>1st Sunday meetups are at Citadel in Porter Sq, at 98 Elm St, apt 1, Somerville.</p> <p>3rd Sunday meetups are in MIT's building 66 at 25 Ames St, room 156. Room number subject to change based on availability; signs will be posted with the actual room number.</p> <p>(We also have last Wednesday meetups at Citadel at 7pm.)</p> <p>Our default schedule is as follows:</p> <p>—Phase 1: Arrival, greetings, unstructured conversation.</p> <p>—Phase 2: The headline event. This starts promptly at 4pm, and lasts 30-60 minutes.</p> <p>—Phase 3: Further discussion. We'll explore the ideas raised in phase 2, often in smaller groups.</p> <p>—Phase 4: Dinner.</p></div> </div><!-- .content --> <h2>Discussion article for the meetup : <a href='/meetups/131'>The Psychology of Video Games</a></h2> anders_h vrBugb5dFbwnxuSRF 2014-08-11T04:58:57.663Z Ethical Choice under Uncertainty https://www.lesswrong.com/posts/wotzRhf2Y6dRPkiCG/ethical-choice-under-uncertainty <p><strong>Ethical Choice under Uncertainty:</strong></p> <p>Most discussions about utilitarian ethics are attempt to determine the goodness of an outcome. &nbsp;For instance, discussions may focus on whether it would be ethical to increase total utility by increasing the total number of individuals but reducing their average utility. &nbsp;Or, &nbsp;one could argue about whether we should give more weight to those who are worst off when we aggregate utility over individuals. &nbsp;</p> <p>These are all important questions. However, even if they were answered to everyone's satisfaction, the answers would not be sufficient to guide the choices of agents acting under uncertainty. To elaborate, &nbsp;I believe textbook versions of utilitarianism are unsatisfactory for the following reasons:&nbsp;</p> <ol> <li>Ethical theories that don't account for the agent's beliefs will have absurd consequences such as claiming that it is unethical to rescue a drowning child if the child goes on to become Hitler. &nbsp;Clearly, if we are interested in judging whether the agent is acting ethically, the only relevant consideration is his beliefs about the consequences at the time the choice is made. If we define "ethics" to require him to act on information from the future, it becomes impossible in principle to act ethically.</li> <li>In real life, there will be many situations where the agent makes a bad choice because he has incorrect beliefs about the consequences of his actions. &nbsp;For most people, if they were asked to judge the morality of a person who has pushed a fat man to his death, &nbsp;it is important to know whether the man believed he could save the lives of five children by doing so. &nbsp; Whether the belief is correct or not is not ethically relevant: &nbsp;There is a difference between stupidity and immorality. &nbsp;</li> <li>The real choices are never of the type &nbsp;"If you choose A, the fat man dies with probability 1, &nbsp;whereas if you choose B, the five children die with probability 1". &nbsp; Rather, they are of the type "If you choose A, the fat man dies with probability 0.5, the children die with probability 0.25 and they all die with probablity 0.25". &nbsp; Choosing between such options will require a formalization of the concept of risk aversion as an integral component of the ethical theory.&nbsp;</li> </ol> <p>I will attempt to fix this by providing the following definition of ethical choice, which is based on the same setup as Von Neumann Morgenstern Expected Utility Theory:</p> <p class="p1">An agent is making a decision, and can choose from a choice set A, with elements&nbsp; (a<sub>1</sub>, a<sub>2</sub>, a<sub>n</sub>). The possible outcome states of the world are contained in the set W, with elements (w<sub>1</sub>,w<sub>2</sub>,w<sub>m</sub>). &nbsp;The agent is uncertain about the consequences of his choice; he is not able to perfect predict whether choosing a<sub>1</sub> will lead to state w<sub>1</sub>, w<sub>2</sub> or w<sub>m</sub>. In other words, for every element of the choice set, he has a separate subjective probability distribution ("prior") on W.</p> <p class="p1">He also has a cardinal social welfare function f over possible states of the world. &nbsp; &nbsp;The social welfare function may have properties such as risk aversion or risk neutrality over attributes of W. &nbsp; Since the choice made by the agent is one aspect of the state of the world, the social welfare function may include terms for A.&nbsp;</p> <!--EndFragment--> <p class="p1">We define that the agent is acting "ethically" if he chooses the element of the choice set that maximizes the expected value of the social welfare function, under the agent's beliefs about the probability of each possible state of the world that could arise under that action:</p> <p>Max&nbsp;&Sigma;<sub>w</sub>&nbsp;Pr (W|a) * f(W, a)</p> <p>Note here that "risk aversion" is defined as the second derivative of the social welfare function. For details, I will unfortunately have to refer the reader to a textbook on Decision Theory, such as <a href="http://www.amazon.com/Theory-Choice-Underground-Classics-Economics/dp/0813375533">Notes on the Theory of Choice</a>.</p> <p class="p1">The advantage of this setup is that it allows us to define the ethical choice precisely, in terms of the intentions and beliefs of the agent. For example, if an individual makes a bad choice because he honestly has a bad prior about the consequences of his choice, we interpret him as acting stupidly, but not unethically. &nbsp;However, ignorance is not a complete "get out of jail for free" card: &nbsp;One element of the choice set is always "seek more information / update your prior". &nbsp;If your true prior says that you can maximize the expected social welfare function by updating your prior, the ethical choice is to seek more information &nbsp;(this is analogous to the decision theoretic concept <a href="http://en.wikipedia.org/wiki/Value_of_information">"value of information"</a>).&nbsp;</p> <p class="p2">At this stage, the &ldquo;social welfare function&rdquo; is completely unspecified. Therefore, this definition places only minor constraints on what we mean by the word &ldquo;ethics&rdquo;. &nbsp;Some ethical theories are special cases of this definition of ethical choice. &nbsp; For example, deontology is the special case where the social welfare function f(W,A) is independent of the state of the world, and can be simplified to f(A). &nbsp;(If the social welfare function is constant over W, the summation over the prior will cancel out)</p> <p class="p1">One thing that is ruled out by the definition, is outcome-based consequentialism, where an agent is defined to act ethically if his actions lead to good realized outcomes.&nbsp; Note that under this type of consequentialism, at the time a decision is made it is impossible for an agent to know what the correct choice is, because the ethical choice will depend on random events that have not yet taken place. &nbsp;This definition of ethics excludes strategies that cannot be followed by a rational agent acting solely on information from the past. This is a feature, not a bug.&nbsp;</p> <p class="p1">We now have a definition of acting ethically. However, it is not yet very useful: We have no way of knowing what the social welfare function looks like. The model simply rules out some pathological ethical theories that are not usable as decision theories, and gives us an appealing definition of ethical choice that allows us to distinguish "ignorance/stupidity" from "immorality".&nbsp;</p> <p class="p1">If nobody points out any errors that invalidate my reasoning, I will write another installment with some more speculative ideas about how we can attempt to determine what the social welfare function f(W,A) looks like</p> <p class="p1">&nbsp;</p> <p class="p1">--</p> <p>I have no expertise in ethics, and most my ideas will be obvious to anyone who has spent time thinking about decision theory. From my understanding of&nbsp;<a href="/lw/f3v/cake_or_death/">Cake or Death</a>&nbsp;, it looks like similar ideas have been explored here previously, but with additional complications that are not necessary for my argument. &nbsp; I am puzzled by the fact that this line of thinking is not a central component of most ethical discussions, because I don't believe that it is possible for a non-Omega agent to follow an ethical theory that does not explicitly account for uncertainty. My intuition &nbsp;is that unless there is a flaw in my reasoning,&nbsp;this is a neglected point that it would be important to draw people's attention to, in a simple form with as few complications as possible. &nbsp;Hence this post.&nbsp;</p> <p>This is a work in progress, I would very much appreciate feedback on where it needs more work.&nbsp;</p> <p class="p1">Some thoughts on where this idea needs more work:</p> <ul> <li>While agents who have bad priors about the consequences of their actions are defined to act stupidly and not unethically, I am currently unclear about how to interpret the actions of agents who have incorrect beliefs about the social welfare function. &nbsp;</li> <li>I am also unsure if this setup excludes some reasonable forms of ethics, such as a scenario where we model the agent is simultaneously trying to optimize the social welfare function and his own utility function. In such a setup, we may want to have a definition of ethics that involves the rate of substitution between the two things he is optimizing. &nbsp;However, it is possible that this can be handled within my model, by finding the right social welfare function. &nbsp;</li> </ul> <p>&nbsp;</p> anders_h wotzRhf2Y6dRPkiCG 2014-08-10T22:13:38.756Z Causal Inference Sequence Part II: Graphical Models https://www.lesswrong.com/posts/e8Refdq74jJ7Lf3ao/causal-inference-sequence-part-ii-graphical-models <p><span style="font-family: Arial, Helvetica, sans-serif; line-height: 19.5px; text-align: justify;">(Part 2 of a&nbsp;</span><a style="color: #8a8a8b; font-family: Arial, Helvetica, sans-serif; line-height: 19.5px; text-align: justify;" href="/lw/klh/sequence_announcement_applied_causal_inference/">Sequence on Applied Causal Inference</a><span style="font-family: Arial, Helvetica, sans-serif; line-height: 19.5px; text-align: justify;">. Follow-up to <a href="/lw/kli/causal_inference_sequence_part_1_basic/">Part 1</a>)</span></p> <p><strong>Saturated and Unsaturated Models</strong></p> <p>A model is a restriction on the possible states of the world: By specifying a model, you make a claim that you have knowledge about what the world does not look like. &nbsp;</p> <p>To illustrate this, if you have two binary predictors A and B, there are four groups defined by A and B, and four different values of E[Y|A,B]. &nbsp;Therefore, the regression<!--[if gte mso 9]><xml> <o:DocumentProperties> <o:Revision>0</o:Revision> <o:TotalTime>0</o:TotalTime> <o:Pages>1</o:Pages> <o:Words>6</o:Words> <o:Characters>37</o:Characters> <o:Company>Harvard University</o:Company> <o:Lines>1</o:Lines> <o:Paragraphs>1</o:Paragraphs> <o:CharactersWithSpaces>42</o:CharactersWithSpaces> <o:Version>14.0</o:Version> </o:DocumentProperties> <o:OfficeDocumentSettings> <o:AllowPNG /> </o:OfficeDocumentSettings> </xml><![endif]--> <!--[if gte mso 9]><xml> 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font-size:12.0pt; font-family:"Times New Roman"; color:black; mso-ansi-language:EN-US; mso-fareast-language:JA;} --> <!--[endif] --> <!--StartFragment-->E[Y|A,B] =&nbsp;&beta;<sub>0</sub>&nbsp;+&nbsp;&beta;<sub>1</sub>*A&nbsp; +&nbsp;&beta;<sub>2</sub>*B +&nbsp;&beta;<sub>3</sub>&nbsp;* A * B&nbsp;<!--EndFragment--> is not a real model : &nbsp;There are four parameters and four values of E[Y|A,B], so the regression is saturated. In other words, the regression does not make any assumptions about the joint distribution of A, B and Y. &nbsp; Running this regression in statistical software will simply give you exactly the same estimates as you would have obtained if you manually looked in each of the four groups defined by A and B, and estimated the mean of Y.</p> <p>If we instead fit the regression model<!--[if gte mso 9]><xml> <o:DocumentProperties> <o:Revision>0</o:Revision> <o:TotalTime>0</o:TotalTime> <o:Pages>1</o:Pages> <o:Words>6</o:Words> <o:Characters>37</o:Characters> <o:Company>Harvard University</o:Company> <o:Lines>1</o:Lines> <o:Paragraphs>1</o:Paragraphs> <o:CharactersWithSpaces>42</o:CharactersWithSpaces> <o:Version>14.0</o:Version> </o:DocumentProperties> <o:OfficeDocumentSettings> <o:AllowPNG /> </o:OfficeDocumentSettings> </xml><![endif]--> <!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:TrackMoves /> <w:TrackFormatting /> <w:PunctuationKerning /> 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mso-fareast-language:JA;} --> <!--[endif] --> <!--StartFragment-->E[Y|A,B] =&nbsp;&beta;<sub>0</sub>&nbsp;+&nbsp;&beta;<sub>1</sub>*A&nbsp; +&nbsp;&beta;<sub>2</sub>*B<span style="font-size: 12.0pt; font-family: &quot;mceinline&quot;,&quot;serif&quot;; mso-fareast-font-family: &quot;MS 明朝&quot;; mso-fareast-theme-font: minor-fareast; mso-bidi-font-family: Arial; color: windowtext; mso-ansi-language: #047F; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">&nbsp;</span>, we are making an assumption: We are assuming that there is no interaction between A and B on the average value of Y. &nbsp;In contrast to the previous regression, this is a true model: &nbsp; It makes the assumption that the value of&nbsp;&beta;<sub>3</sub>&nbsp;is 0. &nbsp;In other words, we are saying that the data did not come from a distribution where &beta;<sub>3&nbsp;</sub>is not equal to 0. If this assumption is not true, the model is wrong: &nbsp;We would have excluded the true state of the world</p> <p>In general, whenever you use models, think first about what the saturated model looks like, and then add assumptions by asking what parameters &nbsp;you can reasonably assume are equal to a specific value (such as zero). The same type of logic applies to graphical models such as directed acyclic graphs (DAGs). &nbsp;</p> <p>We will talk about two types of DAGs: &nbsp; Statistical DAGs are models for the joint distribution of the variables on the graph, whereas Causal DAGs are a special class of DAGs which can be used as models for the data generating mechanism. &nbsp;</p> <p><strong>Statistical DAGs</strong></p> <p>A Statistical DAG is a graph that allows you to encode modelling assumptions about the joint distribution of the individual variables. These graphs do not necessarily have any causal interpretation. &nbsp;</p> <p>On a Statistical DAG, we represent modelling assumptions by missing arrows. Those missing arrows define the DAG in the same way that the missing term for&nbsp;&beta;<sub>3</sub>&nbsp;defines the regression model above. &nbsp;If there is a directed arrow between any two variables on the graph, the DAG is saturated or complete. &nbsp;Complete DAGs make no modelling assumptions about the relationship between the variables, in the same way that a saturated regression model makes no modelling assumptions.&nbsp;</p> <p><strong>DAG Factorization</strong></p> <p>The arrows on DAGs are statements about how the joint distribution factorizes. To illustrate, consider the following complete DAG (where each individual patient in our study represents a realization of the joint distribution of the variables A, B, C and D. &nbsp;):</p> <p>&nbsp;</p> <p>&nbsp;</p> <p><img src="http://images.lesswrong.com/t3_klj_0.png" alt="" width="486" height="167" /></p> <p>&nbsp;</p> <p>&nbsp;Any joint distribution of A,B,C and D can be factorized algebraically according to the laws of probability as &nbsp;f(A,B,C,D) = f(D|C,B,A) * f(C|B,A) * f(B|A) * f(A). &nbsp; This factorization is always true, it does not require any assumptions about independence. &nbsp;By drawing a complete DAG, we are saying that we are not willing to make any further assumptions about how the distribution factorizes. &nbsp;</p> <p>Assumptions are represented by missing arrows: Every variable is assumed to be independent of the past, given its parents. &nbsp;Now, consider the following DAG with three missing arrows:</p> <p>&nbsp;</p> <p>&nbsp;</p> <p>&nbsp;</p> <p><img src="http://images.lesswrong.com/t3_klj_1.png" alt="" width="514" height="122" /></p> <p>&nbsp;</p> <p>&nbsp;This DAG is defined by the assumption that C is independent of the joint distribution of A and B, and that D is independent B, given A and C. &nbsp; If this assumption is true, the distribution can be factorized as f(A,B,C,D) = f(D|C, A) * f(C) * f(B|A) * f(A). &nbsp; &nbsp;Unlike the factorization of the complete DAG, the above is not a tautology. It is the algebraic representation of the independence assumption that is represented by the missing arrows. The factorization is the modelling assumption: &nbsp;When arrows are missing, you are really saying that you have a priori knowledge about how the distribution factorizes.&nbsp;</p> <p>&nbsp;</p> <p><strong>D-Separation</strong></p> <p>When we make assumptions such as the ones that define a DAG, other independences may automatically follow as logical implications. The reason DAGs are useful, is that you can use the graphs as a tool for reasoning about what independence statements are logical implications of the modelling assumptions. You could reason about this using algebra, but it is usually much harder. &nbsp;D-Separation is a simple graphical criterion that gives you an immediate answer to whether a particular statement about independence is a logical implication of the independences that define your model.</p> <p>Two variables are independent (in all distributions that are consistent with the DAG) if there is no open path between them. &nbsp;This is called &laquo;d-separation&raquo;. &nbsp;D-Separation is useful because it allows us to determine if a particular independence statement is true within our model. &nbsp; For example, if we want to know if A is independent of B given C, we check if A is d-separated from B on the graph where C is conditioned on</p> <p>A path between A and B is any set of edges that connect the two variables. For determining whether a path exists, the direction of the arrows does not matter: &nbsp;A--&gt;B--&gt;C and A--&gt;B&lt;--C are both examples of paths between A and C. &nbsp; &nbsp;Using the rules of D-separation, you can determine whether paths are open or closed. &nbsp;</p> <p>&nbsp;</p> <p><strong>The Rules of D-Separation</strong></p> <p><em>Colliders:</em></p> <p>If you are considering three variables, they can be connected in four different ways:</p> <p>&nbsp;A --&gt; B --&gt; C</p> <p>&nbsp;A &lt;-- B &lt;-- C</p> <p>&nbsp;A &lt;-- B --&gt; C</p> <p>&nbsp;A --&gt; B &lt;-- C</p> <p>&nbsp;</p> <ul> <li>In the first three cases, B is a non-collider.</li> <li>In the fourth case, &nbsp;B is a collider: The arrows from A and C "collide" in B.</li> <li>Non-Colliders are (normally) open, whereas colliders are (normally) closed</li> <li>Colliders are defined relative to a specific pathway. &nbsp;B could be a collider on one pathway, and a non-collider on another pathway&nbsp;</li> </ul> <p>&nbsp;</p> <p><em>Conditioning:</em></p> <p>If we compare individuals within levels of a covariate, that covariate is conditioned on. &nbsp;In an empirical study, this can happen either by design, or by accident. On a graph, we represent &ldquo;conditioning&rdquo; by drawing a box around that variable. &nbsp;This is equivalent to introducing the variable behind the conditioning sign in the algebraic notation</p> <p>&nbsp;</p> <ul> <li>If a non-collider is conditioned on, it becomes closed.</li> <li>If a collider is conditioned on, it is opened.</li> <li>If the descendent of a collider is conditioned on, the collider is opened</li> </ul> <p>&nbsp;</p> <p><em>Open and Closed Paths:</em></p> <p>&nbsp;</p> <ul> <li>A path is open if and only if all variables on the path are open.&nbsp;</li> <li>Two variables are D-separated if and only if there is no open path between them</li> <li>Two variables are D-separated conditional on a third variable if and only if there is no open path between them on a graph where the third variable has been conditioned on.</li> </ul> <p>&nbsp;</p> <p><em>Colliders:</em></p> <p>Many students who first encounter D-separation are confused about why conditioning on a collider opens it. &nbsp;Pearl uses the following thought experiment to illustrate what is going on:</p> <p>Imagine you live in a world where there is a sprinkler that sometimes randomly turns on, regardless of the weather. In this world, whether the sprinkler is on is independent of rain: &nbsp;If you notice that the sprinkler is on, this gives you no information about whether it rains.&nbsp;</p> <p>However, if the sprinkler is on, it will cause the grass to be wet. The same thing happens if it rains. Therefore, the grass being wet is a collider. &nbsp; Now imagine that you have noticed that the grass is wet. &nbsp;You also notice that the sprinkler is turned "off". &nbsp;In this situation, because you have conditioned on the grass being wet, &nbsp;the fact that the sprinkler is off allows you to conclude that it is probably raining. &nbsp;</p> <p><strong>Faithfulness</strong></p> <p>D-Separation says that if there is no open pathway between two variables, those variables are independent (in all distributions that factorize according to the DAG, ie, in all distributions where the defining independences hold). &nbsp;This immediately raises the question about whether the logic also runs in the opposite direction: &nbsp;If there is an open pathway between two variables, does that mean that they are correlated?</p> <p>The quick answer is that this does not hold, at least not without additional assumptions. &nbsp; DAGs are defined by assumptions that are represented by the missing arrows: &nbsp;Any joint distribution where those independences hold, can be represented by the DAG, even if there are additional independences that are not encoded. &nbsp; However, we usually think about two variables as correlated if they are connected: &nbsp;This assumption is called faithfulness</p> <p><strong>Causal DAGs</strong></p> <p>Causal DAGs are models for the data generating mechanism. The rules that apply to statistical DAGs - such as d-separation - are also valid for Causal DAGs. &nbsp;If a DAG is &laquo;causal&raquo;, we are simply making the following additional assumptions:&nbsp;</p> <ul> <li>The variables are in temporal (causal) order</li> <li>If two variables on the DAG share a common cause, the common cause is also shown on the graph</li> </ul> <p>If you are willing to make these assumptions, you can think of the Causal DAG as a map of the data generating mechanism. You can read the map as saying that all variables are generated by random processes with a deterministic component that depends only on the parents. &nbsp;</p> <p>For example, &nbsp;if variable Y has two parents A and U, the model says that Y<sup>a</sup> = &nbsp;f(A, U, *) where * is a random error term. &nbsp; The shape of the function f is left completely unspecified, hence the name "non-parametric structural equations model". &nbsp; The primary assumption in the model is that the error terms on different variables are independent. &nbsp;</p> <p>You can also think informally of the arrows as the causal effect of one variable on another: &nbsp;If we change the value of A, this change would propagate to downstream variables, but not to variables that are not downstream.</p> <p>Recall that DAGs are useful for reasoning about independences. &nbsp;Exchangeability assumptions are a special type of independence statements: They involve counterfactual variables. &nbsp;Counterfactual variables belong in the data generating mechanism, therefore, to reason about them, we will need Causal DAGs. &nbsp;</p> <p>A simplified heuristic for thinking about Causal DAGs is as follows: &nbsp; Correlation flows through any open pathway, but causation flows only in the forward direction. &nbsp;If you are interested in estimating the causal effect of A on Y, you have to quantify the sum of all forward-going pathways from A to Y. &nbsp; Any open pathway from A to Y which contains an arrow in the backwards direction will cause bias.&nbsp;</p> <p>In the next part in this sequence (which I hope to post next week), I will give a more detailed description of how we can use Causal DAGs to reason about bias in observational research, including confounding bias, selection bias and mismeasurement bias.&nbsp;</p> <p>(Feedback is greatly appreciated: &nbsp;I invoke Crocker's rules. &nbsp;The most important types of feedback will be if you notice anything that is wrong or misleading. &nbsp;I also greatly appreciate feedback on whether the structure of the text works, whether the sentences are hard to parse and whether there is any background information that needs to be included)</p> <div><br /></div> anders_h e8Refdq74jJ7Lf3ao 2014-08-04T23:10:02.285Z Causal Inference Sequence Part 1: Basic Terminology and the Assumptions of Causal Inference https://www.lesswrong.com/posts/JDWTro62tRAHzvhEH/causal-inference-sequence-part-1-basic-terminology-and-the <p class="Body"><strong></strong></p> <p class="Body">(Part 1 of the&nbsp;<a href="/lw/klh/sequence_announcement_applied_causal_inference/">Sequence on Applied Causal Inference</a>)&nbsp;</p> <p class="Body">&nbsp;</p> <p class="Body">In this sequence, I am going to present a theory on how we can learn about causal effects using observational data. &nbsp;As an example, we will imagine that you have collected information on a large number of Swedes - let us call them Sven, Olof, G&ouml;ran, &nbsp;Gustaf, Annica, &nbsp;Lill-Babs, Elsa and Astrid. For every Swede, you have recorded data on their gender, whether they smoked or not, and on whether they got cancer during the 10-years of follow-up. &nbsp; Your goal is to use this dataset to figure out whether smoking causes cancer. &nbsp;&nbsp;</p> <p class="Body">We are going to use the letter A as a random variable to represent whether they smoked. A can take the value 0 (did not smoke) or 1 (smoked). &nbsp;When we need to talk about the specific values that A can take, we sometimes use lower case a as a placeholder for 0 or 1. &nbsp; &nbsp;We use the letter Y as a random variable that represents whether they got cancer, and L to represent their gender.&nbsp;</p> <p class="Body"><strong>The data-generating mechanism and the joint distribution of variables</strong></p> <p class="Body">Imagine you are looking at this data set:</p> <table class="MsoNormalTable" style="width: 192pt; margin-left: 4.65pt; border-collapse: collapse;" border="0" cellspacing="0" cellpadding="0" width="256"> <tbody> <tr style="height: 15pt;"> <td style="border: 1pt solid windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:white">ID</span></p> </td> <td style="border-top-width: 1pt; border-right-width: 1pt; border-bottom-width: 1pt; border-style: solid solid solid none; border-top-color: windowtext; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:white">L</span></p> </td> <td style="border-top-width: 1pt; border-right-width: 1pt; border-bottom-width: 1pt; border-style: solid solid solid none; border-top-color: windowtext; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:white">A</span></p> </td> <td style="border-top-width: 1pt; border-right-width: 1pt; border-bottom-width: 1pt; border-style: solid solid solid none; border-top-color: windowtext; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:white">Y</span></p> </td> </tr> <tr style="height: 22.5pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 22.5pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:white">Name</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 22.5pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:white">Sex</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 22.5pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:white">Did they smoke?</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 22.5pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; color: white;">Did they get cancer?<strong></strong></span></p> </td> </tr> <tr style="height: 15pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Sven</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Male</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size:8.0pt;mso-ascii-font-family: Calibri;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Yes</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size:8.0pt;mso-ascii-font-family: Calibri;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Yes</span></p> </td> </tr> <tr style="height: 15pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Olof</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Male</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size:8.0pt;mso-ascii-font-family: Calibri;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">No</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size:8.0pt;mso-ascii-font-family: Calibri;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Yes</span></p> </td> </tr> <tr style="height: 15pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">G&ouml;ran</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Male</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size:8.0pt;mso-ascii-font-family: Calibri;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Yes</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size:8.0pt;mso-ascii-font-family: Calibri;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Yes</span></p> </td> </tr> <tr style="height: 15pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Gustaf</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Male</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size:8.0pt;mso-ascii-font-family: Calibri;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">No</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size:8.0pt;mso-ascii-font-family: Calibri;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">No</span></p> </td> </tr> <tr style="height: 15pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Annica</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Female</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size:8.0pt;mso-ascii-font-family: Calibri;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Yes</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size:8.0pt;mso-ascii-font-family: Calibri;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Yes</span></p> </td> </tr> <tr style="height: 15pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Lill-Babs</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Female</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size:8.0pt;mso-ascii-font-family: Calibri;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Yes</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size:8.0pt;mso-ascii-font-family: Calibri;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">No</span></p> </td> </tr> <tr style="height: 22.5pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 22.5pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Elsa</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 22.5pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Female</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 22.5pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size:8.0pt;mso-ascii-font-family: Calibri;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Yes</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 22.5pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size:8.0pt;mso-ascii-font-family: Calibri;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">No</span></p> </td> </tr> <tr style="height: 15pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Astrid</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size:8.0pt;mso-ascii-font-family:Calibri; mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">Female</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size:8.0pt;mso-ascii-font-family: Calibri;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">No</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size:8.0pt;mso-ascii-font-family: Calibri;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-hansi-font-family:Calibri; mso-bidi-font-family:&quot;Times New Roman&quot;;color:#2F2B20">No</span></p> </td> </tr> </tbody> </table> <p>&nbsp;</p> <p class="MsoNormal">&nbsp;</p> <p class="Body">This table records information about the&nbsp;<em>joint distribution</em>&nbsp;of the variables L, A and Y. &nbsp;By looking at it, you can tell that 1/4 of the Swedes were men who smoked and got cancer, 1/8 were men who did not smoke and got cancer, 1/8 were men who did not smoke and did not get cancer etc. &nbsp;</p> <p class="Body">You can make all sorts of statistics that summarize aspects of the joint distribution. &nbsp;One such statistic is the&nbsp;<em>correlation</em>&nbsp;between two variables. &nbsp;If "sex" is correlated with "smoking", it means that if you know somebody's sex, this gives you information that makes it easier to predict whether they smoke. &nbsp; If knowing about an individual's sex gives no information about whether they smoked, we say that sex and smoking are&nbsp;<em>independent</em>. &nbsp;We use the symbol ∐ to mean independence.&nbsp;</p> <p class="Body">When we are interested in causal effects, we are asking what would happen to the joint distribution if we intervened to&nbsp;<em>change&nbsp;</em>the value of a variable. &nbsp;For example, how many Swedes would get cancer in a hypothetical world where you intervened to make sure they all quit smoking? &nbsp;</p> <p class="Body">In order to answer this, we have to ask questions about the&nbsp;<em>data generating mechanism.&nbsp;</em>The data generating mechanism is the algorithm that assigns value to the variables, and therefore creates the joint distribution. We will think of the data as being generated by three different algorithms: One for L, one for A and one for Y. &nbsp; &nbsp;Each of these algorithms takes the previously assigned variables as input, and then outputs a value. &nbsp; &nbsp;</p> <p class="Body">Questions about the data generating mechanism include &ldquo;Which variable has its value assigned first?&rdquo;, &nbsp;&ldquo;Which variables from the past (observed or unobserved) are used as inputs&rdquo; and &ldquo;If I change whether someone smokes, how will that change propagate to other variables that have their value assigned later". &nbsp; &nbsp;The last of these questions can be rephrased as "What is the causal effect of smoking&rdquo;. &nbsp; &nbsp;</p> <p class="Body">The basic problem of causal inference is that the relationship between the set of possible data generating mechanisms, and the joint distribution of variables, is many-to-one: &nbsp; For any correlation you observe in the dataset, there are many possible sets of algorithms for L, A and Y that could all account for the observed patterns. For example, if you are looking at a correlation between cancer and smoking, you can tell a story about cancer causing people to take up smoking, or a story about smoking causing people to get cancer, or a story about smoking and cancer sharing a common cause. &nbsp;</p> <p class="Body">An important thing to note is that even if you have data on absolutely everyone, you still would not be able to distinguish between the possible data generating mechanisms. The problem is not that you have a limited sample. This is therefore&nbsp;<em>not a statistical problem.</em>&nbsp; What you need to answer the question, is not more people in your study, but&nbsp;<em>a priori causal information</em>. &nbsp;The purpose of this sequence is to show you how to reason about what prior causal information is necessary, and how to analyze the data if you have measured all the necessary variables.&nbsp;</p> <p class="Body"><strong>Counterfactual Variables and "God's Table":</strong></p> <p class="Body">The first step of causal inference is to translate the English language research question &laquo;What is the causal effect of smoking&raquo; into a precise, mathematical language.&nbsp; One possible such language is based on counterfactual variables.&nbsp; These counterfactual variables allow us to encode the concept of &ldquo;what would have happened if, possibly contrary to fact, the person smoked&rdquo;.</p> <p class="Body">We define one counterfactual variable called Y<sup>a=1</sup>&nbsp;which represents the outcome in the person if he smoked, and another counterfactual variable called Y<sup>a=0</sup>&nbsp;which represents the outcome if he did not smoke. Counterfactual variables such as Y<sup>a=0</sup>&nbsp;are mathematical objects that represent part of the data generating mechanism:&nbsp; The variable tells us what value the mechanism would assign to Y, if we intervened to make sure the person did not smoke. These variables are columns in an imagined dataset that we sometimes call &ldquo;God&rsquo;s Table&rdquo;:</p> <p class="Body">&nbsp;</p> <table class="MsoNormalTable" style="width: 240pt; margin-left: 4.65pt; border-collapse: collapse;" border="0" cellspacing="0" cellpadding="0" width="320"> <tbody> <tr style="height: 15pt;"> <td style="border: 1pt solid windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">ID</span></p> </td> <td style="border-top-width: 1pt; border-right-width: 1pt; border-bottom-width: 1pt; border-style: solid solid solid none; border-top-color: windowtext; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">A</span></p> </td> <td style="border-top-width: 1pt; border-right-width: 1pt; border-bottom-width: 1pt; border-style: solid solid solid none; border-top-color: windowtext; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Y</span></p> </td> <td style="border-top-width: 1pt; border-right-width: 1pt; border-bottom-width: 1pt; border-style: solid solid solid none; border-top-color: windowtext; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Y<sup>a=1</sup></span></p> </td> <td style="border-top-width: 1pt; border-right-width: 1pt; border-bottom-width: 1pt; border-style: solid solid solid none; border-top-color: windowtext; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Y<sup>a=0</sup></span></p> </td> </tr> <tr style="height: 84pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 84pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">&nbsp;</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 84pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;" align="center"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Smoking</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 84pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;" align="center"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Cancer</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 84pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;" align="center"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Whether they would have got cancer if they smoked</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 84pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Whether they would have got cancer if they didn't smoke</span></p> </td> </tr> <tr style="height: 15pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Sven</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> </tr> <tr style="height: 15pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Olof</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">0</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">0</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> </tr> <tr style="height: 15pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">G&ouml;ran</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">0</span></p> </td> </tr> <tr style="height: 15pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Gustaf</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">0</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">0</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">0</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">0</span></p> </td> </tr> </tbody> </table> <p class="Body">&nbsp;</p> <p>&nbsp;</p> <p>&nbsp;</p> <p>Let us start by making some points about this dataset.&nbsp; First, note that the counterfactual variables are variables just like any other column in the spreadsheet.&nbsp;&nbsp; Therefore, we can use the same type of logic that we use for any other variables.&nbsp; Second, note that in our framework, counterfactual variables are pre-treatment variables:&nbsp; They are determined long before treatment is assigned. The effect of treatment is simply to determine whether we see Y<sup>a=0</sup>&nbsp;or Y<sup>a=1</sup>&nbsp;in this individual.</p> <p class="Body">If you had access to God's Table, you would immediately be able to look up the average causal effect, by comparing the column Y<sup>a=1&nbsp;</sup>to the column Y<sup>a=0</sup>. &nbsp;However, the most important point about God&rsquo;s Table is that we cannot observe Y<sup>a=1&nbsp;</sup>and Y<sup>a=0</sup>. We only observe the joint distribution of observed variables, which we can call the &ldquo;Observed Table&rdquo;:</p> <p>&nbsp;</p> <table class="MsoNormalTable" style="width: 2in; margin-left: 4.65pt; border-collapse: collapse;" border="0" cellspacing="0" cellpadding="0" width="192"> <tbody> <tr style="height: 15pt;"> <td style="border: 1pt solid windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">ID</span></p> </td> <td style="border-top-width: 1pt; border-right-width: 1pt; border-bottom-width: 1pt; border-style: solid solid solid none; border-top-color: windowtext; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">A</span></p> </td> <td style="border-top-width: 1pt; border-right-width: 1pt; border-bottom-width: 1pt; border-style: solid solid solid none; border-top-color: windowtext; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #a9a57c;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Y</span></p> </td> </tr> <tr style="height: 15pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Sven</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> </tr> <tr style="height: 15pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Olof</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">0</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> </tr> <tr style="height: 15pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">G&ouml;ran</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #e2e1d7;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> </tr> <tr style="height: 15pt;"> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-left-width: 1pt; border-style: none solid solid; border-right-color: windowtext; border-bottom-color: windowtext; border-left-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Gustaf</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">0</span></p> </td> <td style="border-right-width: 1pt; border-bottom-width: 1pt; border-style: none solid solid none; border-right-color: windowtext; border-bottom-color: windowtext; width: 48pt; padding: 0in 5.4pt; height: 15pt; background: #f1f0ec;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">0</span></p> </td> </tr> </tbody> </table> <p class="Body">&nbsp;</p> <p class="Body">&nbsp;</p> <p class="Body">The goal of causal inference is to learn about God&rsquo;s Table using information from the observed table (in combination with a priori causal knowledge). &nbsp;In particular, we are going to be interested in learning about the distributions of Y<sup>a=1</sup>&nbsp;and Y<sup>a=0</sup>, and in how they relate to each other. &nbsp;</p> <p class="Body"><strong>&nbsp;</strong></p> <p class="Body"><strong>Randomized Trials</strong></p> <p class="Body">The &ldquo;Gold Standard&rdquo; for estimating the causal effect, is to run a randomized controlled trial where we randomly assign the value of A.&nbsp;&nbsp; This study design works because you select one random subset of the study population where you observe Y<sup>a=0</sup>, and another random subset where you observe Y<sup>a=1</sup>.&nbsp;&nbsp; You therefore have unbiased information about the distribution of both Y<sup>a=0</sup>and of Y<sup>a=1</sup>.&nbsp;</p> <p class="Body">An important thing to point out at this stage is that it is not necessary to use an unbiased coin to assign treatment, as long as your use the same coin for everyone. &nbsp; For instance, the probability of being randomized to A=1 can be 2/3.&nbsp; You will still see randomly selected subsets of the distribution of both Y<sup>a=0</sup>&nbsp;and Y<sup>a=1</sup>, you will just have a larger number of people where you see Y<sup>a=1</sup>.<sup>&nbsp;</sup>&nbsp; &nbsp; Usually, randomized trials use unbiased coins, but this is simply done because it increases the statistical power.&nbsp;</p> <p class="Body">Also note that it is possible to run two different randomized controlled trials:&nbsp; One in men, and another in women.&nbsp; The first trial will give you an unbiased estimate of the effect in men, and the second trial will give you an unbiased estimate of the effect in women.&nbsp; If both trials used the same coin, you could think of them as really being one trial. However, if the two trials used different coins, and you pooled them into the same database, your analysis would have to account for the fact that in reality, there were two trials. If you don&rsquo;t account for this, the results will be biased.&nbsp; This is called &ldquo;confounding&rdquo;. As long as you account for the fact that there really were two trials, you can still recover an estimate of the population average causal effect. This is called &ldquo;Controlling for Confounding&rdquo;.</p> <p class="Body">In general, causal inference works by specifying a model that says the data came from a complex trial, ie, one where nature assigned a biased coin depending on the observed past. &nbsp;For such a trial, there will exist a valid way to recover the overall causal results, but it will require us to think carefully about what the correct analysis is.&nbsp;</p> <p class="Body"><strong>Assumptions of Causal Inference</strong></p> <p class="Body">We will now go through in some more detail about why it is that randomized trials work, ie , the important aspects of this study design that allow us to infer causal relationships, or facts about God&rsquo;s Table, using information about the joint distribution of observed variables. &nbsp;</p> <p class="Body">We will start with an &ldquo;observed table&rdquo; and build towards &ldquo;reconstructing&rdquo; parts of God&rsquo;s Table. &nbsp;To do this, we will need three assumptions: These are positivity, consistency and (conditional) exchangeability:</p> <table class="MsoNormalTable" style="width: 2.0in; margin-left: 4.65pt; border-collapse: collapse; mso-yfti-tbllook: 1184; mso-padding-alt: 0in 5.4pt 0in 5.4pt;" border="0" cellspacing="0" cellpadding="0" width="192"> <tbody> <tr style="mso-yfti-irow: 0; mso-yfti-firstrow: yes; height: 15.0pt;"> <td style="width: 48.0pt; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; background: #A9A57C; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">ID</span></p> </td> <td style="width: 48.0pt; border: solid windowtext 1.0pt; border-left: none; mso-border-top-alt: solid windowtext .5pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #A9A57C; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">A</span></p> </td> <td style="width: 48.0pt; border: solid windowtext 1.0pt; border-left: none; mso-border-top-alt: solid windowtext .5pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #A9A57C; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Y</span></p> </td> </tr> <tr style="mso-yfti-irow: 1; height: 15.0pt;"> <td style="width: 48.0pt; border: solid windowtext 1.0pt; border-top: none; mso-border-left-alt: solid windowtext .5pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #E2E1D7; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Sven</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #E2E1D7; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #E2E1D7; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> </tr> <tr style="mso-yfti-irow: 2; height: 15.0pt;"> <td style="width: 48.0pt; border: solid windowtext 1.0pt; border-top: none; mso-border-left-alt: solid windowtext .5pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #F1F0EC; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Olof</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #F1F0EC; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">0</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #F1F0EC; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> </tr> <tr style="mso-yfti-irow: 3; height: 15.0pt;"> <td style="width: 48.0pt; border: solid windowtext 1.0pt; border-top: none; mso-border-left-alt: solid windowtext .5pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #E2E1D7; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">G&ouml;ran</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #E2E1D7; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #E2E1D7; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">1</span></p> </td> </tr> <tr style="mso-yfti-irow: 4; mso-yfti-lastrow: yes; height: 22.5pt;"> <td style="width: 48.0pt; border: solid windowtext 1.0pt; border-top: none; mso-border-left-alt: solid windowtext .5pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #F1F0EC; padding: 0in 5.4pt 0in 5.4pt; height: 22.5pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">Gustaf</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #F1F0EC; padding: 0in 5.4pt 0in 5.4pt; height: 22.5pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">0</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #F1F0EC; padding: 0in 5.4pt 0in 5.4pt; height: 22.5pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: right;" align="right"><span style="font-size: 8pt; font-family: Verdana, sans-serif;">0</span></p> </td> </tr> </tbody> </table> <p class="Body">&nbsp;</p> <p class="Body">&nbsp;</p> <p class="Body">&nbsp;</p> <p class="Body"><em>Positivity</em></p> <p class="Body">Positivity is the assumption that any individual has a positive probability of receiving all values of the treatment variable:&nbsp;&nbsp; Pr(A=a) &gt; 0 for all values of a. &nbsp;In other words, you need to have both people who smoke, and people who don't smoke. &nbsp;If positivity does not hold, you will not have any information about the distribution of Y<sup>a</sup>&nbsp;for that value of a, and will therefore not be able to make inferences about it.</p> <p class="Body">We can check whether this assumption holds in the sample, by checking whether there are people who are treated and people who are untreated. If you observe that in any stratum, there are individuals who are treated and individuals who are untreated, you know that positivity holds. &nbsp;</p> <p class="Body">If we observe a stratum where no individuals are treated (or no individuals are untreated), this can be either for statistical reasons (your randomly did not sample them) or for structural reasons (individuals with these covariates are deterministically never treated). &nbsp;As we will see later, our models can handle random violations, but not structural violations.</p> <p class="Body">In a randomized controlled trial, positivity holds because you will use a coin that has a positive probability of assigning people to either arm of the trial.</p> <p class="Body"><em>Consistency</em></p> <p class="Body">The next assumption we are going to make is that if an individual happens to have treatment (A=1), we will observe the counterfactual variable Y<sup>a=1</sup>&nbsp;in this individual. This is the observed table after we make the consistency assumption:</p> <table class="MsoNormalTable" style="width: 240.0pt; margin-left: -1.35pt; border-collapse: collapse; mso-yfti-tbllook: 1184; mso-padding-alt: 0in 5.4pt 0in 5.4pt;" border="0" cellspacing="0" cellpadding="0" width="320"> <tbody> <tr style="mso-yfti-irow: 0; mso-yfti-firstrow: yes; height: .25in;"> <td style="width: 65.25pt; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; background: #A9A57C; padding: 0in 5.4pt 0in 5.4pt; height: .25in;" width="87"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">ID</span></p> </td> <td style="width: 30.75pt; border: solid windowtext 1.0pt; border-left: none; mso-border-top-alt: solid windowtext .5pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #A9A57C; padding: 0in 5.4pt 0in 5.4pt; height: .25in;" width="41"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">A</span></p> </td> <td style="width: 48.0pt; border: solid windowtext 1.0pt; border-left: none; mso-border-top-alt: solid windowtext .5pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #A9A57C; padding: 0in 5.4pt 0in 5.4pt; height: .25in;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">Y</span></p> </td> <td style="width: 48.0pt; border: solid windowtext 1.0pt; border-left: none; mso-border-top-alt: solid windowtext .5pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #A9A57C; padding: 0in 5.4pt 0in 5.4pt; height: .25in;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">Y<sup>a=1</sup></span></p> </td> <td style="width: 48.0pt; border: solid windowtext 1.0pt; border-left: none; mso-border-top-alt: solid windowtext .5pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #A9A57C; padding: 0in 5.4pt 0in 5.4pt; height: .25in;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">Y<sup>a=0</sup></span></p> </td> </tr> <tr style="mso-yfti-irow: 1; height: 15.0pt;"> <td style="width: 65.25pt; border: solid windowtext 1.0pt; border-top: none; mso-border-left-alt: solid windowtext .5pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #E2E1D7; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="87"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">Sven</span></p> </td> <td style="width: 30.75pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #E2E1D7; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="41"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">1</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #E2E1D7; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">1</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #E2E1D7; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">1</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #E2E1D7; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">*</span></p> </td> </tr> <tr style="mso-yfti-irow: 2; height: 15.0pt;"> <td style="width: 65.25pt; border: solid windowtext 1.0pt; border-top: none; mso-border-left-alt: solid windowtext .5pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #F1F0EC; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="87"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">Olof</span></p> </td> <td style="width: 30.75pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #F1F0EC; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="41"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">0</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #F1F0EC; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">1</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #F1F0EC; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">*</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #F1F0EC; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">1</span></p> </td> </tr> <tr style="mso-yfti-irow: 3; height: 15.0pt;"> <td style="width: 65.25pt; border: solid windowtext 1.0pt; border-top: none; mso-border-left-alt: solid windowtext .5pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #E2E1D7; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="87"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">G&ouml;ran</span></p> </td> <td style="width: 30.75pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #E2E1D7; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="41"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">1</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #E2E1D7; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">1</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #E2E1D7; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">1</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #E2E1D7; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">*</span></p> </td> </tr> <tr style="mso-yfti-irow: 4; mso-yfti-lastrow: yes; height: 15.0pt;"> <td style="width: 65.25pt; border: solid windowtext 1.0pt; border-top: none; mso-border-left-alt: solid windowtext .5pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #F1F0EC; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="87"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">Gustaf</span></p> </td> <td style="width: 30.75pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #F1F0EC; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="41"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">0</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #F1F0EC; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">0</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #F1F0EC; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">*</span></p> </td> <td style="width: 48.0pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt; background: #F1F0EC; padding: 0in 5.4pt 0in 5.4pt; height: 15.0pt;" width="64"> <p class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 12pt;"><span style="font-size: 12pt; font-family: Arial, sans-serif;">0</span></p> </td> </tr> </tbody> </table> <p class="Body">&nbsp;</p> <p class="Body">&nbsp;</p> <p class="Body">&nbsp;</p> <p class="Body">&nbsp;</p> <p class="Body">&nbsp;Making the consistency assumption got us half the way to our goal.&nbsp; We now have a lot of information about Y<sup style="color: windowtext;">a=1</sup>&nbsp;and Y<sup style="color: windowtext;">a=0</sup>. However, half of the data is still missing.</p> <p class="Body">Although consistency seems obvious, it is an assumption, not something that is true by definition. &nbsp;We can expect the consistency assumption to hold if we have a well-defined intervention (ie, the intervention is a well-defined choice, not an attribute of the individual), and there is no causal interference (one individual&rsquo;s outcome is not affected by whether another individual was treated).</p> <p class="Body">Consistency may not hold if you have an intervention that is not well-defined:&nbsp; For example, there may be multiple types of cigarettes. When you measure Y<sup>a=1&nbsp;</sup>in people who smoked, it will actually be a composite of multiple counterfactual variables: &nbsp;One for people who smoked regular cigarettes (let us call that&nbsp;Y<sup>a=1*</sup>) and another for people who smoked e-cigarettes (let us call that&nbsp;Y<sup>a=1#</sup>)<sub>.&nbsp;</sub>&nbsp;&nbsp; Since you failed to specify whether you are interested in the effect of regular cigarettes or e-cigarettes, the construct<sub>&nbsp;</sub>Y<sup>a=1&nbsp;</sup>is a composite without any meaning, and people will be unable to use your results to predict the consequences of their actions.</p> <p class="Body"><em>Exchangeability</em></p> <p>To complete the&nbsp;table, we require an additional assumption on the nature of the&nbsp;data.&nbsp;We call this assumption &ldquo;Exchangeability&rdquo;.&nbsp; One possible exchangeability assumption is &ldquo;Y<sup>a=0</sup>&nbsp;∐ A and Y<sup>a=1</sup>&nbsp;∐ A&rdquo;.&nbsp;&nbsp; This is the assumption that says &ldquo;The data came from a randomized controlled trial&rdquo;. If this assumption is true, you will observe a random subset of the distribution of Y<sup>a=0</sup>&nbsp;in the group where A=0, and a random subset of the distribution of Y<sup>a=1</sup>&nbsp;in the group where A=1.</p> <p class="Body">Exchangeability is a statement about two variables being independent from each other. This means that having information about either one of the variables will not help you predict the value of the other. &nbsp;Sometimes, variables which are not independent are "conditionally independent". &nbsp;For example, it is possible that knowing somebody's race helps you predict whether they enjoy eating&nbsp;<a href="http://en.wikipedia.org/wiki/H%C3%A1karl">Hakarl</a>, an Icelandic form of rotting fish. &nbsp;However, it is also possible that this is just a marker for whether they were born in the ethnically homogenous Iceland. In such a situation, it is possible that once you already know whether somebody is from Iceland, also knowing their race gives you no additional clues as to whether they will enjoy Hakarl. &nbsp;In this case, the variables "race" and "enjoying hakarl" are conditionally independent, given nationality.&nbsp;</p> <p class="Body">The reason we care about conditional independence is that sometimes you may be unwilling to assume that marginal exchangeability Y<sup>a=1</sup>&nbsp;∐ A holds, but you are willing to assume conditional exchangeability Y<sup>a=1</sup>&nbsp;∐ A&nbsp; | L.&nbsp; In this example, let L be sex.&nbsp; The assumption then says that you can interpret the data as if it came from two different randomized controlled trials: One in men, and one in women. If that is the case, sex is a "confounder". (We will give a definition of confounding in Part 2 of this sequence. )</p> <p class="Body">If the data came from two different randomized controlled trials, one possible approach is to analyze these trials separately. This is called &ldquo;stratification&rdquo;.&nbsp; Stratification gives you effect measures that are conditional on the confounders:&nbsp; You get one measure of the effect in men, and another in women.&nbsp; Unfortunately, in more complicated settings, stratification-based methods (including regression) are always biased. In those situations, it is necessary to focus the inference on the marginal distribution of Y<sup>a</sup>.</p> <p class="Body"><strong>Identification</strong></p> <p class="Body">If marginal exchangeability holds (ie, if the data came from a marginally randomized trial), making inferences about the marginal distribution of Y<sup>a</sup>&nbsp;is easy: You can just estimate E[Y<sup>a</sup>] as E [Y|A=a].</p> <p class="Body">However, if the data came from a conditionally randomized trial, we will need to think a little bit harder about how to say anything meaningful about E[Y<sup>a</sup>]. This process is the central idea of causal inference. We call it &ldquo;identification&rdquo;:&nbsp; The idea is to write an expression for the distribution of a counterfactual variable, purely in terms of observed variables. &nbsp;If we are able to do this, we have sufficient information to estimate causal effects just by looking at the relevant parts of the joint distribution of observed variables.</p> <p class="Body">The simplest example of identification is standardization. &nbsp;As an example, we will show a simple proof:</p> <p class="Default">Begin by using the law of total probability to factor out the confounder, in this case L:</p> <p class="Default" style="margin-left: 38.25pt; text-indent: -.25in; mso-list: l0 level1 lfo1;">&middot;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;E(Y<sup>a</sup>) = &Sigma;&nbsp; E(Y<sup>a</sup>|L= l) * Pr(L=l)&nbsp;&nbsp;&nbsp; (The summation sign is over l)</p> <p class="Default" style="margin-left: 27.0pt; text-indent: -27.0pt;">We do this because we know we need to introduce L behind the conditioning sign, in order to be able to use our exchangeability assumption in the next step:&nbsp;&nbsp; Then, &nbsp;because Y<sup>a&nbsp;&nbsp;</sup>∐ A | L,&nbsp; we are allowed to introduce A=a behind the conditioning sign:</p> <p class="Default" style="margin-left: 38.25pt; text-indent: -.25in; mso-list: l0 level1 lfo1;">&middot;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;E(Y<sup>a</sup>) =&nbsp; &Sigma;&nbsp; E(Y<sup>a</sup>|A=a, L=l) * Pr(L=l)</p> <p class="Default" style="margin-left: 27.0pt; text-indent: -27.0pt;">Finally, use the consistency assumption:&nbsp;&nbsp; Because we are in the stratum where A=a in all individuals, we can replace Y<sup>a</sup>&nbsp;by Y</p> <p class="Default" style="margin-left: 38.25pt; text-indent: -.25in; mso-list: l0 level1 lfo1;">&middot;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;E(Y<sup>a</sup>) = &Sigma; E(Y|A=a, L=l) * Pr (L=l)</p> <p class="Default" style="margin-left: 38.25pt; text-indent: -.25in; mso-list: l0 level1 lfo1;">&nbsp;</p> <p class="Default"><span style="text-indent: -27pt;">We now have an expression for the counterfactual in terms of quantities that can be observed in the real world, ie, in terms of the joint distribution of A, Y and L. In other words, we have linked the data generating mechanism with the joint distribution &ndash; we have &ldquo;identified&rdquo;&nbsp; E(Y</span><sup style="text-indent: -27pt;">a</sup><span style="text-indent: -27pt;">). &nbsp;We can therefore estimate E(Y</span><sup style="text-indent: -27pt;">a</sup><span style="text-indent: -27pt;">)</span></p> <p class="Default">This identifying expression is valid if and only if L was the only confounder. If we had not observed sufficient variables to obtain conditional exchangeability, it would not be possible to identify the distribution of Y<sup>a</sup>&nbsp;: there would be intractable confounding.</p> <p class="Default">Identification is the core concept of causal inference: It is what allows us to link the data generating mechanism to the joint distribution, to something that can be observed in the real world.&nbsp;</p> <p class="Default">&nbsp;</p> <p class="Body"><strong>The difference between epidemiology and biostatistics</strong></p> <p class="Body">Many people see Epidemiology as &laquo;Applied Biostatistics&raquo;.&nbsp; This is a misconception. In reality, epidemiology and biostatistics are completely different parts of the problem.&nbsp; To illustrate what is going on, consider this figure:</p> <p class="Body"><img src="http://images.lesswrong.com/t3_kli_0.png" alt="" width="776" height="264" /></p> <p class="Body">&nbsp;</p> <p class="Body">&nbsp;</p> <p class="Body">The data generating mechanism first creates a joint distribution of observed variables.&nbsp; Then, we sample from the joint distribution to obtain data. Biostatistics asks:&nbsp; If we have a sample, what can we learn about the joint distribution?&nbsp; Epidemiology asks: &nbsp;If we have all the information about the joint distribution , what can we learn about the data generating mechanism?&nbsp;&nbsp; This is a much harder problem, but it can still be analyzed with some rigor.</p> <p class="Body">Epidemiology without Biostatistics is always impossible:&nbsp; It would not be possible to learn about the data generating mechanism without asking questions about the joint distribution. This usually involves sampling.&nbsp; Therefore, we will need good statistical estimators of the joint distribution.</p> <p class="Body">Biostatistics without Epidemiology is usually pointless:&nbsp; The joint distribution of observed variables is simply not interesting in itself. You can make the claim that randomized trials is an example of biostatistics without epidemiology.&nbsp; However, the epidemiology is still there. It is just not necessary to think about it, because the epidemiologic part of the analysis is trivial</p> <p class="Body">Note that the word &ldquo;bias&rdquo; means different things in Epidemiology and Biostatistics.&nbsp; In Biostatistics, &ldquo;bias&rdquo; is a property of a statistical estimator:&nbsp; We talk about whether&nbsp;<span style="color: #373737; font-family: 'Lucida Grande', 'Lucida Sans Unicode', 'Lucida Sans', 'Helvetica Neue', Helvetica, sans-serif; font-size: 12px; line-height: 18px; white-space: pre-wrap;">ŷ</span>&nbsp;is a biased estimator of E(Y<sup>&nbsp;</sup>|A).&nbsp;&nbsp; If an estimator is biased, it means that when you use data from a sample to make inferences about the joint distribution in the population the sample came from, there will be a systematic source of error.</p> <p class="Body">In Epidemiology, &ldquo;bias&rdquo; means that you are estimating the wrong thing:&nbsp; Epidemiological bias is a question about whether E(Y|A) is a valid identification of E(Y<sup>a</sup>).&nbsp;&nbsp; If there is epidemiologic bias, it means that you estimated something in the joint distribution, but that this something does not answer the question you were interested in. &nbsp;&nbsp;&nbsp;</p> <p class="Body">These are completely different concepts. Both are important and can lead to your estimates being wrong. It is possible for a statistically valid estimator to be biased in the epidemiologic sense, and vice versa.&nbsp;&nbsp; For your results to be valid, your estimator must be unbiased in both senses.</p> <p class="MsoNormal">&nbsp;</p> <p><br style="mso-special-character: line-break; page-break-before: always;" /></p> <p class="MsoNormal"><strong></strong></p> anders_h JDWTro62tRAHzvhEH 2014-07-30T20:56:31.866Z Sequence Announcement: Applied Causal Inference https://www.lesswrong.com/posts/tfwcCurc3x2qAmJnn/sequence-announcement-applied-causal-inference <p><strong>Applied Causal Inference for Observational Research</strong></p> <p>This sequence is an introduction to basic causal inference. &nbsp;It was originally written as auxiliary notes for a course in Epidemiology, but it is relevant to almost any kind of applied statistical research, including econometrics, sociology, psychology, political science etc. &nbsp;I would not be surprised if you guys find a lot of errors, and I would be very grateful if you point them out in the comments. This will help me improve my course notes and potentially help me improve my understanding of the material.&nbsp;</p> <p>For mathematically inclined readers, I recommend skipping this sequence and instead reading <a href="http://www.amazon.com/Causality-Reasoning-Inference-Judea-Pearl/dp/052189560X/ref=dp_ob_image_bk">Pearl's book on Causality</a>. &nbsp;There is also a lot of good material on causal graphs on<a href="http://wiki.lesswrong.com/wiki/Highly_Advanced_Epistemology_101_for_Beginners"> Less Wrong itself</a>. &nbsp; Also, note that my thesis advisor is writing a book that covers the same material in more detail, the first two parts are available for free&nbsp;<a href="http://www.hsph.harvard.edu/miguel-hernan/causal-inference-book/">at his website</a>.</p> <p>Pearl's book, Miguel's book and Eliezer's writings are all more rigorous and precise than my sequence. &nbsp;This is partly because I have a different goal: &nbsp;Pearl and Eliezer are writing for mathematicians and theorists who may be interested in contributing to the theory. &nbsp;Instead, &nbsp;I am writing for consumers of science who want to understand correlation studies from the perspective of a more rigorous epistemology. &nbsp;</p> <p>I will use Epidemiological/Counterfactual notation rather than Pearl's notation. I apologize if this is confusing. &nbsp;These two approaches refer to the same mathematical objects, it is just a different notation. Whereas Pearl would use the "Do-Operator"&nbsp;E[Y|do(a)], I use counterfactual variables&nbsp;&nbsp;E[Y<sup>a</sup>]. &nbsp;Instead of using Pearl's "Do-Calculus" for identification, I use Robins' G-Formula, which will give the same results.&nbsp;</p> <p>For all applications, I will use the letter "A" to represent "treatment" or "exposure" (the thing we want to estimate the effect of), &nbsp;Y to represent the outcome, L to represent any measured confounders, and U to represent any unmeasured confounders.&nbsp;</p> <p><strong>Outline of Sequence:</strong></p> <p>I hope to publish one post every week. &nbsp;I have rough drafts for the following eight sections, and will keep updating this outline with links as the sequence develops:</p> <p><strong><br /></strong></p> <p>Part 0: &nbsp;Sequence Announcement / Introduction (This post)</p> <p>Part 1: &nbsp;Basic Terminology and the Assumptions of Causal Inference</p> <p>Part 2: &nbsp;Graphical Models</p> <p>Part 3: &nbsp;Using Causal Graphs to Understand Bias</p> <p>Part 4: &nbsp;Time-Dependent Exposures</p> <p>Part 5: &nbsp;The G-Formula</p> <p>Part 6: &nbsp;Inverse Probability Weighting</p> <p>Part 7: &nbsp;G-Estimation of Structural Nested Models and Instrumental Variables</p> <p>Part 8: &nbsp;Single World Intervention Graphs, Cross-World Counterfactuals and Mediation Analysis</p> <p>&nbsp;</p> <p><strong>&nbsp;Introduction: Why Causal Inference?</strong></p> <p>The goal of applied statistical research is almost always to learn about causal effects.&nbsp; However, causal inference from observational is hard, to the extent that it is usually not even possible without strong, almost heroic assumptions.&nbsp;&nbsp; Because of the inherent difficulty of the task, many old-school investigators were trained to avoid making causal claims.&nbsp; Words like &ldquo;cause&rdquo; and &ldquo;effect&rdquo; were banished from polite company, and the slogan &ldquo;correlation does not imply causation&rdquo; became an article of faith which, when said loudly enough,&nbsp; seemingly absolved the investigators from the sin of making causal claims.</p> <p class="Body">However, readers were not fooled:&nbsp; They always understood that epidemiologic papers were making causal claims.&nbsp; Of course they were making causal claims; why else would anybody be interested in a paper about the correlation between two variables?&nbsp;&nbsp; For example, why would anybody want to know about the correlation between eating nuts and longevity, unless they were wondering if eating nuts would cause them to live longer?</p> <p class="Body">When readers interpreted these papers causally, were they simply ignoring the caveats, drawing conclusions that were not intended by the authors?&nbsp;&nbsp; Of course they weren&rsquo;t.&nbsp; The discussion sections of epidemiologic articles are full of &ldquo;policy implications&rdquo; and speculations about biological pathways that are completely contingent on interpreting the findings causally. Quite clearly, no matter how hard the investigators tried to deny it, they were making causal claims. However, they were using methodology that was not designed for causal questions, and did not have a clear language for reasoning about where the uncertainty about causal claims comes from.&nbsp;</p> <p class="MsoNormal">This was not sustainable, and inevitably led to a crisis of confidence, which culminated when some high-profile randomized trials showed completely different results from the preceding observational studies.&nbsp; In one particular case, when the Women&rsquo;s Health Initiative trial showed that post-menopausal hormone replacement therapy increases the risk of cardiovascular disease, the difference was so dramatic that many thought-leaders in clinical medicine completely abandoned the idea of inferring causal relationships from observational data.</p> <p class="MsoNormal">It is important to recognize that the problem was not that the results were wrong. The problem was that there was uncertainty that was not taken seriously by the investigators. A rational person who wants to learn about the world will be willing to accept that studies have errors of margin, but only as long as the investigators make a good-faith effort to examine what the sources of error are, and communicate clearly about this uncertainty to their readers.&nbsp; Old-school epidemiology failed at this. &nbsp;We are not going to make the same mistake. Instead, we are going to develop a clear, precise language for reasoning about uncertainty and bias.</p> <p class="MsoNormal">In this context, we are going to talk about two sources of uncertainty &ndash; &ldquo;statistical&rdquo; uncertainty and &ldquo;epidemiological&rdquo; uncertainty.&nbsp;</p> <p class="MsoNormal">We are going to use the word &ldquo;Statistics&rdquo; to refer to the theory of how we can learn about correlations from limited samples.&nbsp; For statisticians, the primary source of uncertainty is sampling variability. Statisticians are very good at accounting for this type of uncertainty: Concepts such as &ldquo;standard errors&rdquo;, &ldquo;p-values&rdquo; and &ldquo;confidence intervals&rdquo; are all attempts at quantifying and communicating the extent of uncertainty that results from sampling variability.</p> <p class="MsoNormal">The old school of epidemiology would tell you to stop after you had found the correlations and accounted for the sampling variability. They believed going further was impossible. However, correlations are simply not interesting. If you truly believed that correlations tell you nothing about causation, there would be no point in doing the study.</p> <p class="MsoNormal">Therefore, we are going to use the terms &ldquo;Epidemiology&rdquo; or &ldquo;Causal Inference&rdquo; to refer to the next stage in the process:&nbsp; Learning about causation from correlations.&nbsp; This is a much harder problem, with many additional sources of uncertainty, including confounding and selection bias. However, recognizing that the problem is hard does not mean that you shouldn't try, it just means that you have to be careful. As we will see, it is possible to reason rigorously about whether correlation really does imply causation <em>in your particular study</em>: You will just need a precise language. The goal of this sequence is simply to give you such a language.</p> <p class="Body">In order to teach you the logic of this language, we are going to make several controversial statements such as &laquo;The only way to estimate a causal effect is to run a randomized controlled trial&raquo; . You may not be willing to believe this at first, but in order to understand the logic of causal inference, it is necessary that you are at least willing to suspend your disbelief and accept it as true within the course.&nbsp;</p> <p class="Body">It is important to note that we are not just saying this to try to convince you to give up on observational studies in favor of randomized controlled trials. &nbsp; We are making this point because understanding it is necessary in order to appreciate what it means to control for confounding: It is not possible to give a coherent meaning to the word &ldquo;confounding&rdquo; unless one is trying to determine whether it is reasonable to model the data as if it came from a complex randomized trial run by nature.&nbsp;</p> <p class="Body">&nbsp;</p> <p class="Body">--</p> <p>When we say that causal inference is hard, what we mean by this is not that it is difficult to learn the basics concepts of the theory.&nbsp; What we mean is that even if you fully understand everything that has ever been written about causal inference, it is going to be very hard to infer a causal relationship from observational data, and that there will always be uncertainty about the results. This is why this sequence is not going to be a workshop that teaches you how to apply magic causal methodology. What we are interested in, is developing your ability to reason honestly about where uncertainty and bias comes from, so that you can communicate this to the readers of your studies. &nbsp;What we want to teach you about, is the epistemology that underlies epidemiological and statistical research with observational data.&nbsp;</p> <p>Insisting on only using randomized trials may seem attractive to a purist, it does not take much imagination to see that there are situations where it is important to predict the consequences of an action, but where it is not possible to run a trial. In such situations, there may be Bayesian evidence to be found in nature. This evidence comes in the form of correlations in observational data. When we are stuck with this type of evidence, it is important that we have a clear framework for assessing the strength of the evidence.&nbsp;</p> <p>&nbsp;</p> <p>--</p> <p>&nbsp;</p> <p>I am publishing Part 1 of the sequence at the same time as this introduction. I would be very interested in hearing feedback, particularly about whether people feel this has already been covered in sufficient detail on Less Wrong. &nbsp;If there is no demand, there won't really be any point in transforming the rest of my course notes to a Less Wrong format.&nbsp;</p> <p>Thanks to everyone who had a look at this before I published, including paper-machine and Vika, Janos, Eloise and Sam from the Boston Meetup group.&nbsp;</p> anders_h tfwcCurc3x2qAmJnn 2014-07-30T20:55:41.741Z