Posts
Comments
The fact that people tend to specialize in one or the other does not mean that "the two have little to do with each other." Likewise, there are physicists who spend a lot of time working in foundations and interpretation of QM, and others who spend their time applying it to solve problems in solid state physics, nuclear physics, etc. They're working on different kinds of problems, but it's absurd to say that the two have "little to do with each other."
This is simply wrong. Bayesian statistics is just Bayesian probability theory. As is Bayesian epistemology. Bayesian probabilities are epistemic probabilities.
But you'd have to be one really stupid correctional officer to get an order to disable the cameras around Epstein's cell the night he was murdered, and not know who killed him after he dies.
I assume you mean "who ordered him killed."
Here's what a news report says happened:
A letter filed by Assistant US Attorneys Jason Swergold and Maurene Comey said "the footage contained on the preserved video was for the correct date and time, but captured a different tier than the one where Cell-1 was located", New York City media report.Prince Andrew spoke to the BBC in November about his links to Epstein
"The requested video no longer exists on the backup system and has not since at least August 2019 as a result of technical errors."
Could be a lot more subtle than that. Just ask for the wrong video. A little mess up in procedures. Maybe some operative clandestinely gets into the system and causes some "technical errors."
I'm not an expert on assassinations, and I suspect you aren't either. It seems to me that you're using the argument from lack of imagination -- "I can't think of a way to do it, therefore it can't be done." If, say, the CIA were behind Epstein and didn't want him to talk, is it unreasonable to suspect that they would know of many techniques to assassinate someone while covering their tracks that neither you nor I would have a clue about?
Note that I'm not claiming that there's a strong case that Epstein was assassinated, just that it's not so easy to rule out.
Domain methodsofrationality.com
I own the domain methodsofrationality.com, but I'm not really doing anything with it. If you want it, send me a message telling me what you plan to do with it. I'll give it to whomever, in my opinion, has the best use for it.
But when you do assert that basically the entire U.S. government has collaborated on murdering Epstein
Isn't this a straw man? If someone powerful wanted Epstein dead, how many people does that require, and how many of them even have to know why they're doing what they're doing? It seems to me that only one person -- the murderer -- absolutely has to be in on it. Other people could get orders that sound innocuous, or maybe just a little odd, without knowing the reasons behind them. And, of course, there are always versions of "Will no one rid me of this troublesome priest?" to ensure deniability.
The context is *all* applications of probability theory. Look, when I tell you that A or not A is a rule of classical propositional logic, we don't argue about the context or what assumptions we are relying on. That's just a universal rule of classical logic. Ditto with conditioning on all the information you have. That's just one of the rules of epistemic probability theory that *always* applies. The only time you are allowed to NOT condition on some piece of known information is if you would get the same answer whether or not you conditioned on it. When we leave known information Y out and say it is "irrelevant", what that means is that Pr(A | Y and X) = Pr(A | X), where X is the rest of the information we're using. If I can show that these probabilities are NOT the same, then I have proven that Y is, in fact, relevant.
You are simply assuming that what I've calculated is irrelevant. But the only way to know absolutely for sure whether it is irrelevant is to actually do the calculation! That is, if you have information X and Y, and you think Y is irrelevant to proposition A, the only way you can justify leaving out Y is if Pr(A | X and Y) = Pr(A | X). We often make informal arguments as to why this is so, but an actual calculation showing that, in fact, Pr(A | X and Y) != Pr(A | X) always trumps an informal argument that they should be equal.
Your "probability of guessing the correct card" presupposes some decision rule for choosing a particular card to guess. Given a particular decision rule, we could compute this probability, but it is something entirely different from "the probability that the card is a king". If I assume that's just bad wording, and that you're actually talking about the frequency of heads when some condition occurs, well now you're doing frequentist probabilities, and we were talking about *epistemic* probabilities.
But randomly awakening Beauty on only one day is a different scenario than waking her both days. A priori you can't just replace one with the other.
Yes, in exactly the same sense that *any* mathematical / logical model needs some justification of why it corresponds to the system or phenomenon under consideration. As I've mentioned before, though, if you are able to express your background knowledge in propositional form, then your probabilities are uniquely determined by that collection of propositional formulas. So this reduces to the usual modeling question in any application of logic -- does this set of propositional formulas appropriately express the relevant information I actually have available?
This is the first thing I've read from Scott Garrabant, so "otherwise reputable" doesn't apply here. And I have frequently seen things written on LessWrong that display pretty significant misunderstandings of the philosophical basis of Bayesian probability, so that gives me a high prior to expect more of them.
I'm not trying to be mean here, but this post is completely wrong at all levels. No, Bayesian probability is not just for things that are space-like. None of the theorems from which it derived even refer to time.
So, you know the things in your past, so there is no need for probability there.
This simply is not true. There would be no need of detectives or historical researchers if it were true.
If you partially observe a fact, then I want to say you can decompose that fact into the part that you observed and the part that you didn't, and say that the part you observed is in your past, while the part you didn't observe is space-like separated from you.
You can say it, but it's not even approximately true. If someone flips a coin in front of me but covers it up just before it hits the table, I observe that a coin flip has occurred, but not whether it was heads or tails -- and that second even is definitely within my past light-cone.
You may have cached that you should use Bayesian probability to deal with things you are uncertain about.
No, I cached nothing. I first spent a considerable amount of time understanding Cox's Theorem in detail, which derives probability theory as the uniquely determined extension of classical propositional logic to a logic that handles uncertainty. There is some controversy about some of its assumptions, so I later proved and published my own theorem that arrives at the same conclusion (and more) using purely logical assumptions/requirements, all of the form, "our extended logic should retain this existing property of classical propositional logical."
The problem is that the standard justifications of Bayesian probability are in a framework where the facts that you are uncertain about are not in any way affected by whether or not you believe them!
1) It's not clear this is really true. It seems to me that any situation that is affected by an agent's beliefs can be handled within Bayesian probability theory by modeling the agent.
2) So what?
Therefore, our reasons for liking Bayesian probability do not apply to our uncertainty about the things that are in our future!
This is a complete non sequitur. Even if I grant your premise, most things in my future are unaffected by my beliefs. The date on which the Sun will expand and engulf the Earth is in no way affected by any of my beliefs. Whether you will get luck with that woman at the bar next Friday is in no way affected by any of my beliefs. And so on,
path analysis requires scientific thinking, as does every exercise in causal inference. Statistics, as frequently practiced, discourages it, and encouraged "canned" procedures instead.
Despite Pearl's early work on Bayesian networks, he doesn't seem to be very familiar with Bayesian statistics -- the above comment really only applies to frequentist statistics. Model construction and criticism ("scientific thinking") is an important part of Bayesian statistics. Causal thinking is common in Bayesian statistics, because causal intuition provides the most effective guide for Bayesian model building.
I've worked implementing Bayesian models of consumer behavior for marketing research, and these are grounded in microeconomic theory, models of consumer decision making processes, common patterns of deviation from strictly rational choice, etc.
I don't believe that the term "probability" is completely unambiguous once we start including weird scenarios that fall outside the scope which standard probability was intended to address.
The intended scope is anything that you can reason about using classical propositional logic. And if you can't reason about it using classical propositional logic, then there is still no ambiguity, because there are no probabilities.
You know, it has not actually been demonstrated that human consciousness can be mimicked by Turing-equivalent computer.
The evidence is extremely strong that human minds are processes that occur in human brains. All known physical laws are Turing computable, and we have no hint of any sort of physical law that is not Turing computable. Since brains are physical systems, the previous two observations imply that it is highly likely that they can be simulated on a Turing-equivalent computer (given enough time and memory).
But regardless of that, the Sleeping Beauty problem is a question of epistemology, and the answer necessarily revolves around the information available to Beauty. None of this requires an actual human mind to be meaningful, and the required computations can be carried out by a simple machine. The only real question here is, what information does Beauty have available? Once we agree on that, the answer is determined.
In these kinds of scenarios we need to define our reference class and then we calculate the probability for someone in this class.
No, that is not what probability theory tells us to do. Reference classes are a rough technique to try to come up with prior distributions. They are not part of probability theory per se, and they are problematic because often there is disagreement as to which is the correct reference class.
When Sleeping Beauty wakes up and observes a sequence, they are learning that this sequence occurs on a on a random day
Right here is your error. You are sneaking in an indexical here -- Beauty doesn't know whether "today" is Monday or Tuesday. As I discussed in detail in Part 2, indexicals are not part of classical logic. Either they are ambiguous, which means you don't have a proposition at all, or the ambiguity can be resolved, which means you can restate your proposition in a form that doesn't involve indexicals.
What you are proposing is equivalent to adding an extra binary variable d to the model, and replacing the observation R(y, Monday) or R(y, Tuesday) with R(y, d). That in turn is the same as randomly choosing ONE day on which to wake Beauty (in the Tails case) instead of waking her both times.
This kind of oversight is why I really insist on seeing an explicit model and an explicit statement (as a proposition expressible in the language of the original model) of what new information Beauty receives upon awakening.
All this maths is correct, but why do we care about these odds? It is indeed true that if you had pre-committed at the start to guess if and only if you experienced the sequence 111
We care about these odds because the laws of probability tell us to use them. I have no idea what you mean by "precommitted at the start to guess if and only if..." I can't make any sense of this or the following paragraph. What are you "guessing"? Regardless, this is a question of epistemology -- what are the probabilities, given the information you have -- and those probabilities have specific values regardless of whether you care about calculating them.
Neal wants us the condition on all information, including the apparently random experiences that Sleeping Beauty will undergo before they answer the interview question. This information seems irrelevant, but Neal argues that if it were irrelevant that it wouldn't affect the calculation. If, contrary to expectations, it actually does, then Neal would suggest that we were wrong about its irrelevance.
This isn't just Neal's position. Jaynes argues the same in Probability Theory: The Logic of Science. I have never once encountered an academic book or paper that argued otherwise. The technical term for conditioning on less than all the information is "cherry-picking the evidence" :-).
Unfortunately, Ksavnhorn's post jumps straight into the maths and doesn't provide any explanation of what is going on.
Ouch. I thought I was explaining what was going on.
But the development of probability theory and the way that it is applied in practice were guided by implicit assumptions about observers.
I don't think that's true, but even if it is an accurate description of the history, that's irrelevant -- we have justifications for probability theory that make no assumptions whatsoever about observers.
You seemed to argue in your first post that selection effects were not routinely handled within standard probability theory.
No, I argued that this isn't a case of selection effects.
Certainly agreed as to logic (which does not include probability theory).
Why are you ignoring what I wrote about proofs that probability theory is either a or the uniquely determined extension of classical propositional logic to handle degrees of certainty? That places probability theory squarely in the logical camp. It is a logic.
in which we made certain implicit assumptions about observers
No, we made no such implicit assumptions. There are no assumptions, implicit or otherwise, about observers at all. If you think otherwise, show me where they occur in Cox's Theorem or in my theorem.
I'm going to wait to address that until you clarify what you mean by applying standard probability theory, since you offered a fairly narrow view of what this means in your original post, and seemed to contradict it in your point 3 in the comment.
I have no idea what you're talking about here.
My position is that "the information available" should not be interpreted as simply the existence of at least one agent making the same observations you are, while declining to make any inferences at all about the number of such agents (beyond that it is at least 1).
Um, there's only one agent here, but if by "agent" you mean the pair (person, day), then the above is just wrong -- it's very clearly part of the model that if the coin comes up Heads, there is exactly one day on which the remembered observations could be made, and if the coin comes up Tails, there are exactly two days on which the remembered observations could be made. I even worked out the probabilities that the observations occurred on just Monday, just Tuesday, or both Monday and Tuesday.
Listen, if you want to argue against my analysis, you need to
1. Propose a different model of what Beauty knows on Sunday, and/or
2. Propose a different proposition that expresses the additional information Beauty has on Monday/Tuesday and that accounts for her altered probabilities. This proposition should be possible to sensibly state and talk about on Sunday, Monday, Tuesday, or Wednesday, by either Beauty or one of the experimenters, and mean the same thing in all these cases.
When Sleeping Beauty wakes up and observes a sequence, they are learning that this sequence occurs on a on a random day out of those days when they are awake.
That would be a valid description if she were awakened only on one day, with that day chosen through some unpredictable process. That is not the case here, though.
What you're doing here is sneaking in an indexical -- "today" is either Monday if Heads, and "today" is either Monday or Tuesday if Tails. See Part 2 for a discussion of this issue. To the extent that indexicals are ambiguous, they cannot be used in classical propositions. The only way to show that they are unambiguous is to show that there is an equivalent way of expressing that same thing that doesn't use any indexical, and only uses well-defined entities -- in which case you might as well use the equivalent expression that has no indexical.
Yes, that is shown in Part 2.
From the OP: "honor requires recognition from others." That's not a component of the notion of honor I grew up with. Nor is the requirement of avenging insults.
This is a very, very different concept of honor than the one I grew up with. I was taught that honor means doing what is right (ethical, moral), regardless of personal cost. It meant being unfailingly honest, always keeping your word, doing your duty, etc. How others perceived you was irrelevant. One example of this notion of honor is the case of Sir Thomas More, who was executed by Henry VIII because his conscience would not allow him to cooperate with Henry's establishment of the Church of England. Another is the Dreyfus Affair and Colonel Georges Picquart, who suffered grave personal consequences for insisting on giving an honest report and refusing to go along with the framing of Alfred Dreyfus for espionage. (There's a wonderful movie about this, called Prisoner of Honor.)
...the standard formalization of probability... was not designed with anthropic reasoning in mind. It is usually taken for granted that the number of copies of you that will be around in the future to observe the results of experiments is fixed at exactly 1, and that there is thus no need to explicitly include observation selection effects in the formalism.
1. Logic, including probability theory, is not observer-dependent. Just as the conclusions one can obtain with classical propositional logic depend only on the information (propositional axioms) available, and not on any characteristic or circumstance of the reasoner, epistemic probabilities also depend only on the information available. Logic -- including probability theory -- was designed to be fully general. If you want to argue that probability theory is not, in its standard formulation, suitable for anthropic reasoning, you need to point out the specific points in its rationale that are incompatible with anthropic effects. As I have shown (preprint), all you have to assume to get probability theory from classical propositional logic is that certain properties of propositional logic are retained in the extended logic.
2. No, neither classical logic nor probability theory as the extension of classical propositional logic assumes anything about observers, or their numbers, or experiments, or what may happen in the future.
3. Selection effects are routinely handled within the framework of standard probability theory. You don't need to go beyond standard probability theory for this.
No, P(H | X2, M) is , and not . Recall that is the proposed model. If you thought it meant "today is Monday," I question how closely you read the post you are criticizing.
I find it ironic that you write "Dismissing betting arguments is very reminiscent of dismissing one-boxing in Newcomb's" -- in an earlier version of this blog post I brought up Newcomb myself as an example of why I am skeptical of standard betting arguments (not sure why or how that got dropped.) The point was that standard betting arguments can get the wrong answer in some problems involving unusual circumstances where a more comprehensive decision theory is required (perhaps FDT).
Re constructing rational agents: this is one use of probability theory; it is not "the point". We can discuss logic from a purely analytical viewpoint without ever bringing decisions and agents into the discussion. Logic and epistemology are legitimate subjects of their own quite apart from decision theory. And probability theory is the unique extension of classical propositional logic to handle intermediate degrees of plausibility.
You say you have read PTLOS and others. Have you read Cox's actual paper, or any or detailed discussions of it such as Paris's discussion in The Uncertain Reasoner's Companion, or my own "Constructing a Logic of Plausible Inference: A Guide to Cox's Theorem"? If you think that Cox's Theorem has too many arguable technical requirements, then I invite you to read my paper, "From Propositional Logic to Plausible Reasoning: A Uniqueness Theorem" (preprint here). That proof assumes only that certain existing properties of classical propositional logic be retained when extending the logic to handle degrees of plausibility. It does not assume any particular functional decomposition of plausibilities, nor does it even assume that plausibilities must be real numbers. As with Cox, we end up with the result that the logic must be isomorphic to probability theory. In addition, the theorem gives the required numeric value for a probability when contains, in propositional form, all of the background information we are using to assess the probability of . How much more "clear cut" do you demand the relationship between logic and probability be?
Regardless, for my argument about indexicals all that is necessary is that probability theory deals with classical propositions.
I responded to David Chapman's essay (https://meaningness.com/probability-and-logic) a couple of years ago here.
The point is that the meaning of a classical proposition must not change throughout the scope of the problem being considered. When we write A1, ..., An |= P, i.e. "A1 through An together logically imply P", we do not apply different structures to each of A1, ..., An, and P.
The trouble with using "today" in the Sleeping Beauty problem is that the situation under consideration is not limited to a single day; it spans, at a minimum, both Monday and Tuesday, and arguably Sunday and/or Wednesday also. Any properly constructed proposition used in discussing this problem should make sense and be unambiguous regardless of whether Beauty or the experimenters are uttering the proposition, and whether they are uttering it on Sunday, Monday, Tuesday, or Wednesday.
...the details of the experiment do provide context for "today." But as a random variable, not an explicit value.
You seem to think that "random" variables are special in some way that avoids the problems of indexicals. They are not. When dealing with epistemic probabilities, a "random" variable is any variable whose precise value is not known with complete certainty.
Still, there are ways to avoid using an indexical in a solution. I suggested one in a comment to part 1: use four Beauties, where each is left asleep under a different combination of {Coin,Day}. Three are wakened each day of the experiment. One of those three will be awakened only once during the experiment. They are asked, essentially, "what is the probability that you will be awake only once?" It was agreed that this question is equivalent the original problem.
This is not what I understood you to be proposing. As described here, I would say that this is not the same as the original question, and does not avoid using an indexical. You have simply camouflaged the indexical by omission when you write that "One of those three [who are awake today] will be awakened only once during the experiment."
The situation with indexicals is similar to the situation with "irrelevant" information. If there is any dispute over whether some information is irrelevant, you condition on it and see if it changes the answer. If it does, the judgment that the information was irrelevant was wrong.
Same thing with indexicals. You may claim that use of an indexical in a proposition is unambiguous. The only way to prove this is to actually remove the ambiguity -- replace it with a more explicit statement that lacks indexicals -- and see that this doesn't change anything. So for your burned paper analogy, "today" and "here" are replaced by "the day on which Carl wrote this note" and "the city of origin for the call for which Carl took this note". For the dice-throwing example, "What is the probability that today is Wednesday?" can be replaced by "What is the probability that the day on Beauty experiences is Wednesday" because there can only be one such day, in which her last memories before her last sleep were from Sunday.
When we try this for the SB problem, however, a nonzero probability of ambiguity remains. Neal gives one way of removing the ambiguity in terms of information to which Beauty actually has access, that is, her memories and experiences. Doing that leads to an answer that is close to, but not quite the same as, treating "today" as unambiguous. If Beauty has exactly the same experiences on both Monday and Tuesday, she cannot disambiguate "today".
That leaves you with a choice: either you must agree that "today" is ambiguous in this problem, or you need to propose a different way of rephrasing the statement "today is Monday" into a form that removes the indexicals and then condition on information Beauty actually has.
You're right, my argument wasn't quite right. Thanks for looking into this and fixing it.
I think a variation of my approach to resolving the betting argument for SB can also help deal with the very large universe problem. I've taken a look at the following setup:
- There are Experimenters scattered throughout the universe, where is very, very large. Each Experimenter tries to determine which of two hypotheses and about the universe are correct by running some experiment and collection some data. Let be the data collected, and let be the remaining information (experiences, memories) that could distinguish this Experimenter from others.
- It is possible to choose so large that the prior probability approaches one that there will be some Experimenter with that particular and , regardless of whether or is true. This means that the Experimenter's posterior probability for versus will update only slightly from its prior probability.
- And yet if the Experimenter has to make a choice based on whether or is true, and we weight the payoffs according to how many Experimenters there are with the same and (as done in my analysis for SB), then the maximum-expected-utility answer does not depend on : from the standpoint of decision-making, we can ignore the possibility of all those other Experimenters and just assume .
How do I do things like tables using the WYSIWYG interface? There doesn't seem to be any way to insert markdown in that interface. And once you've already been using WYSIWYG on an article, you can't really switch to markdown -- I tried, and it was a complete mess.
At any point in the history that Beauty remembers in step 2 of step 3, the proposition has a simple, single truth value.
No, it doesn't. This boils down to a question of identity. Absent any means of uniquely identifying the day -- such as, "the day in which a black marble is on the dresser" -- there is a fundamental ambiguity. If Beauty's remembered experiences and mental state are identical at a point in time on Monday and another point in time on Tuesday, then "today" becomes ill-defined for her.
In some instances of the experiment
What instances are you talking about? We're talking about a single experiment. We're talking about epistemic probabilities, not frequencies. You need to relinquish your frequentist mindset for this problem, as it's not a problem about frequentist probabilities.
to an awake Beauty, the "experiment" she sees consists of Sunday and a single day after it.
No, it doesn't. She knows quite well that if the coin lands Tails, she will awaken on two separate days. It doesn't matter that she can only remember one of them.
The problem that I am analyzing is the problem that Beauty was asked to analyze. Not what an outside observer sees.
Epistemic probabilities are a function, not of the person, but of the available information. Any other person given the same information must produce the same epistemic probabilities. That's fundamental.
No, "time" is an indexical.
Go read the quotes again. Are you a greater authority on this subject than the authors of the Stanford Encyclopedia of Philosphy?
you didn't answer my questions, about the variable Sleeping Beauty Problem.
They're irrelevant. You added an extra layer of randomness on top of the problem. Each of the four card outcomes leads to a problem equivalent to the first. But randomly choosing one of four problems equivalent to the first problem doesn't tell you what the solution to the first problem is.
I do not understand why you are so insistent on using "propositions" that include indexicals, especially when there is no need to do so -- we can express the information Beauty has in a way that does not involve indexicals. When we do so, we get an answer that is not quite the same as the answer you get when you play fast and loose with indexicals. Since you've never been able to point out a flaw in the argument -- all you've done is presented a different argument you like better -- you should consider this evidence that indexicals are, in fact, a problem, just like Epstein and others have said.
Note the clause "in general."
Now you're really stretching.
And over the duration of when Beauty considers the meaning of "today," it does not change.
That duration potentially includes both Monday and Tuesday.
"Today" means the same thing every time Beauty uses it.
This is getting ridiculous. "Today" means a different thing on every different day. That's why the article lists it as an indexical. Going back to the quote, the "discussion" is not limited to a single day. There are at least two days involved.
I notice you carefully ignored the quote from Epstein's book, which was very clear that a classical proposition must not contain indexicals.
It is true on Monday when Beauty is awake, and false on Sunday Night, on Tuesday whether or not Beauty is awake, and on Wednesday.
That's not a simple, single truth value; that's a structure built out of truth values.
The proposition "coin lands heads" is sometimes true, and sometimes false, as well.
No, it is not. It has the same truth value throughout the entire scenario, Sunday through Wednesday. On Sunday and Monday it is impossible to know what that truth value is, but it is either true that the coin will land heads, or false that it will land heads -- and by definition, that is the same truth value you'll assign after seeing the coin toss. In contrast, the truth of "it is Monday" keeps on changing throughout the scenario. Likewise, the truth of "the sensor detects white" changes throughout the scenario you are considering in your button-and-sensor example.
Day is an independent parameter that defines the randomness of the situation
I don't know what it means to "define the randomness of the situation." In any event, the point you are missing is that Day changes throughout the problem you are analyzing -- not just that there are different possible values for it, and you don't know which is the correct one, but at different points in the same problem it has different values.
Things like "today" and "now" are known as indexicals, and there is an entire philosophical literature on them because they are problematic for classical logic. Various special logics have been devised specifically to handle them. It would not have been necessary to devise such alternative logics if they posed no problem for classical logic. You can read about them in the article Demonstratives and Indicatives in The Internet Encyclopedia of Philosophy. Some excerpts:
In the philosophy of language, an indexical is any expression whose content varies from one context of use to another. The standard list of indexicals includes... adverbs such as “now”, “then”, “today”, “yesterday”, “here”, and “actually”...
Contemporary philosophical and linguistic interest in indexicals and demonstratives arises from at least four sources. ...(iii) Indexicals and demonstratives raise interesting technical challenges for logicians seeking to provide formal models of correct reasoning in natural language...
The problem with indexicals is that they have meanings that may change over the course of the problem being discussed. This is simply not allowed in classical logic. In classical logic, a proposition must have a stable, unvarying truth value over the entire argument. I'm going to appeal to authority here, and give you some quotes.
Section 3.2, "Meanings of Sentences", in Propositions, Stanford Encyclopedia of Philosophy:
The problem is this: it seems propositions, being the objects of belief, cannot in general be spatially and temporally unqualified. Suppose that Smith, in London, looks out his window and forms the belief that it is raining. Suppose that Ramirez, in Madrid, relying on yesterday’s weather report, awakens and forms the belief that it is raining, before looking out the window to see sunshine. What Smith believes is true, while what Ramirez believes is false. So they must not believe the same proposition. But if propositions were generally spatially unqualified, they would believe the same proposition. An analogous argument can be given to show that what is believed must not in general be temporally unqualified.
(Emphasis added.) The above is telling us that a "proposition" involving an indexical is not a single proposition, but a set of propositions that you get by specifying a particular time/location.
Classical Mathematical Logic: The Semantic Foundations of Logic, by Richard L. Epstein, is clear that indexicals are not allowed in classical logic. On p. 4, "Exercises for Sections A and B," one of the exercises is this:
Explain why we cannnot take sentence types as propositions if we allow the use of indexicals in our reasoning.
The explanation is given on the previous page (p. 3):
When we reason together, we assume that words will continue to be used in the same way. That assumption is so embedded in our use of language that it's hard to think of a word except as a type, that is, as a representative of inscriptions that look the same and utterances that sound the same. I don't know how to make precise what we mean by 'look the same' or 'sound the same.' But we know well enough in writing and conversation what it means for two inscriptions or utterances to be equiform.
Words are types. We will assume that throughout any particular discussion equiform words will have the same properties of interest to logic. We therefore identify them and treat them as the same word. Briefly, a word is a type.
This assumption, while useful, rules out many sentences we can and do reason with quite well. Consider 'Rose rose and picked a rose.' If words are types, we have to distinguish the three equiform inscriptions in this sentence, perhaps as 'Rose_{name} rose_{verb} and picked a rose_{noun}'.
Further, if we accept this agreement, we must avoid words such as 'I', 'my', 'now', or 'this', whose meaning or reference depends on the circumstances of their use. Such words, called indexicals, play an important role in reasoning, yet our demand that words be types requires that they be replaced by words that we can treat as uniform in meaning or reference throughout a discussion.
...
Propositions are types. In any discussion in which we use logic we'll consider a sentence to be a proposition only if any other sentence or phrase that is composed of the same words in the same order can be assumed to have the same properties of concern to logic during that discussion. We therefore identify equiform sentences or phrases and treat them as the same sentence. Briefly, a proposition is a type.
Notice the following statements made above:
- "words will continue to be used in the same way" They do not change meaning within the discussion.
- "equiform words will have the same properties of interest to logic" In particular, the same word used at different points in the argument must have the same meaning.
- "we must avoid words such as... 'now', ... whose meaning or reference depends on the circumstances of their use."
- "our demand that words be types requires that they be replaced by words that we can treat as uniform in meaning or reference throughout a discussion."
On the first read I didn't understand what you were proposing, because of the confusion over "If the two coins show the same face" versus "If the two coins are not both heads." Now that it's clear it should be "if the two coins are not both heads" throughout, and after rereading, I now see your argument.
The problem with your argument is that you still have "today" smuggled in: one of your state components is which way the nickel is lying "today." That changes over the course of the time period we are analyzing, so it does not give a legitimate proposition. To get a legitimate proposition we'll have to split it up into two propositions: "The nickel lies Heads up on Monday" and "The nickel lies Heads up on Tuesday".
So in truth, the actual four possible outcomes are HHT, HTH, THT, and TTH. None of these is ruled out by the mere fact of waking up. Not until Beauty receives sufficient sensory input to provide a label for "today" that is nearly certain to be unique do we arrive at a situation in which your analysis is approximately correct.
BTW, is this argument your own? Although I don't think it's right, it is an interesting argument. Is there a citation I should use if I want to reference it in future writing?
Your whole analysis rests on the idea that "it is Monday" is a legitimate proposition. I've responded to this many other places in the comments, so I'll just say here that a legitimate proposition needs to maintain the same truth value throughout the entire analysis (Sunday, Monday, Tuesday, and Wednesday). Otherwise it's a predicate. The point of introducing R(y,d) is that it's as close as we can get to what you want "it is Monday" to mean.
Are you really claiming that the statement "today is Monday" is not a sentence that is either true or false?
Yes. It does not have a simple true/false truth value. Since it is sometimes true and sometimes false, its truth value is a function from time to {true, false}. That makes it a predicate, not a proposition.
Or are you simply ignoring the fact that the frame of reference, within which Beauty is asked to assess the proposition "The coin lands Heads," is a fixed moment in time?
It is not a fixed moment in time; if it were, the SB problem would be trivial and nobody would write papers about it. The questions about day of week and outcome of coin toss are potentially asked both on Monday and on Tuesday. This makes the rest of your analysis invalid. You keep on asserting that "today is Monday" is evaluated at a fixed moment in time, when in reality it is evaluated at at least two separate moments in time with different answers.
You are asked to assess the probability of the proposition W, that the sensor will detect "white" when you first press the button. This is a valid proposition, even though it varies with time.
The sentence "the sensor detects white" is not a valid proposition; it is a predicate, because it is a function of time. Let's write for this predicate. But yes, the sentence "the sensor detects white when you first press the button" is a legitimate proposition, precisely because specifies a particular time for which is true, and so the truth value of the statement itself does not vary with time.
This gets us to the whole point of defining : saying "Beauty has a stream of experiences on day " is as close as we can get to identifying a specific moment in time corresponding to the "this" in "this is day ". The more nearly that uniquely identifies the day, the more nearly that can be interpreted to mean "this is day ".
By bayes rule, Pr (H | M) * Pr(X2 |H, M) / Pr(X2 |M) = Pr(H∣X2, M), which is not the same quantity you claimed to compute Pr(H∣X2).
That's a typo. I meant to write , not .
Second, the dismissal of betting arguments is strange.
I'll have more to say soon about what I think is the correct betting argument. Until then, see my comment in reply to Radford Neal about disagreement on how to apply betting arguments to this problem.
“probability theory is logically prior to decision theory.” Yes, this is the common view because probability theory was developed first and is easier but it’s not actually obvious this *has* to be the case.
I said logically prior, not chronologically prior. You cannot have decision theory without probability theory -- the former is necessarily based on the latter. In contrast, probability theory requires no reference to decision theory for its justification and development. Have you read any of the literature on how probability theory is either an or the uniquely determined extension of propositional logic to handle degrees of certainty? If not, see my references. Neither Cox's Theorem nor my theorem rely on any form of decision theory.
Third, dismissal of “not H and it’s Tuesday” as not propositions doesn’t make sense. Classical logic encodes arbitrary statements within AND and OR -type constructions. There isn’t a whole lot of restrictions on them.
I'll repeat my response to Jeff Jo: The standard textbook definition of a proposition is a sentence that has a truth value of either true or false. The problem with a statement whose truth varies with time is that it does not have a simple true/false truth value; instead, its truth value is a function from time to the set {true,false}. In logical terms, such a statement is a predicate, not a proposition. For example, "Today is Monday" corresponds to the predicate . It doesn't become a proposition until you substitute in a specific value for , e.g. "Unix timestamp 1527556491 is a Monday."
The paradox still stands for the moment when you wake up
You have not considered the possibility that the usual decision analysis applied to this problem is wrong. There is, in fact, disagreement as to what the correct decision analysis is. I will be writing more on this in a future post.
You seem to simply declare [Beauty's probability at the moment of awakening] to be ½, by saying:
The prior for H is even odds: Pr(H∣M)=Pr(¬H∣M)=1/2.
This is generally indistinguishable from the ½ position you dismiss that argues for that prior on the basis of “no new information.”
In fact, I explicitly said that at the instant of awakening, Beauty's probability is the same as the prior, because at that point she does not yet have any new information. As she receives sensory input, her probability for Heads decreases asymptotically to 1/2. All of this is just standard probability theory, conditioning on the new information as it arrives. I dismissed the naive halfer position because it incorrectly assumes that Beauty's sensory input is irrelevant to the determination of her probability for Heads.
You still don’t know how to handle the situation of being told that it’s Monday and needing to update your probability accordingly,
Uh, yes I do---it's just standard probability theory again. Just do the math. If Beauty finds out that it is Monday, her new information since Sunday changes from to just , and since the problem definition assumes that
we get equal posterior probabilities for and , which is generally accepted to be the right answer.
Yes, there is. I'll be writing about that soon.
Hmmm. I would be responsive to "that's a slur," but the follow-on "the preferred term is X" raises my hackles. The former is merely a request to be polite; the latter feels like someone is trying to dictate vocabulary to me.
None of this is about "versions of me"; it's about identifying what information you actually have and using that to make inferences. If the FNIC approach is wrong, then tell me what how Beauty's actual state of information differs from what is used in the analysis; don't just say, "it seems really odd."
As I understand it, FDT says that you go with the algorithm that maximizes your expected utility. That algorithm is the one that bets on 1:2 odds, using the fact that you will bet twice, with the same outcome each time, if the coin comes up tails.
The standard textbook definition of a proposition is this:
A proposition is a sentence that is either true or false. If a proposition is true, we say that its truth value is "true," and if a proposition is false, we say that its truth value is "false."
(Adapted from https://www.cs.utexas.edu/~schrum2/cs301k/lec/topic01-propLogic.pdf.)
The problem with a statement whose truth varies with time is that it does not have a simple true/false truth value; instead, its truth value is a function from time to the set .
As for the rest of your argument, my request is this: show me the math. That is, define the joint probability distribution describing what Beauty knows on Sunday night, and tell me what additional information she has after awakening on Monday/Tuesday. As I argued in the OP, purely verbal arguments are suspect when it comes to probability problems; it's too easy to miss something subtle.
BTW, in one place you say "if the two coins are not both showing Heads," and in another you say "if the two coins show the same face"; which is the one you intended?
In regards to betting arguments:
1. Traditional CDT (causal decision theory) breaks down in unusual situations. The standard example is the Newcomb Problem, and various alternatives have been proposed, such as Functional Decision Theory. The Sleeping Beauty problem presents another highly unusual situation that should make one wary of betting arguments.
2. There is disagreement as to how to apply decision theory to the SB problem. The usual thirder betting argument assumes that SB fails to realize that she is going to both make the same decision and get the same outcome on Monday and Tuesday. It has been argued that accounting for these facts means that SB should instead compute her expected utility for accepting the bet as
3. Your own results show that the standard betting argument gets the wrong answer of 1/3, when the correct answer is . At best, the standard betting argument gets close to the right answer; but if Beauty is sensorily impoverished, or has just awakened, then can be sufficiently large that the answer deviates substantially from .
BTW, I was a solid halfer until I read your paper. It was the first and only explanation I've ever seen of how Beauty's state of information after awakening on Monday/Tuesday differs from her state of information on Sunday night in a way that affects the probability of Heads.
With regards to your "Sailor's Child" problem:
It was not immediately obvious to me that this is equivalent to the SB problem. I had to think about it for some time, and I think there are some differences. One is, again the different answers of versus . I've concluded that the SC problem is equivalent to a variant of the SB problem where (1) we've guaranteed that Beauty cannot experience the same thing on both Monday and Tuesday, and (2) there is a second coin toss that determines whether Beauty is awakened on Monday or on Tuesday in the case that the first coin toss comes up Heads.
In any event, it was the calculation based on Beauty's new information upon awakening that I found convincing. I tried to disprove it, and couldn't.
for decision-theoretic purposes you want the probability to be 1/3 as soon as the AI wakes up on Monday/Tuesday.
That is based on a flawed decision analysis that fails to account for the fact that Beauty will make the same choice, with the same outcome, on both Monday and Tuesday (it treats the outcomes on those two days as independent).
the mismatch with frequency
There are no frequencies in this problem; it is a one-time experiment.
Probability is logically prior to frequency estimation
That's not what I said; I said that probability theory is logically prior to decision theory.
If your "probability" has zero application because your decision theory uses "likeliness weights" calculated an entirely different way, I think something has gone very wrong.
Yes; what's gone wrong is that you're misapplying the decision theory, or your decision theory itself breaks down in certain odd circumstances. Exploring such cases is the whole point of things like Newcomb's problem and Functional Decision Theory. In this case, it's clear that Beauty is going to make the same betting decision, with the same betting outcome, on both Monday and Tuesday (if the coin lands Tails). The standard betting arguments use a decision rule that fails to account for this.
I think if you've gone wrong somewhere, it's in trying to outlaw statements of the form "it is Monday today."
See my response to Dacyn below ("Classical propositions are simply true or false..."). Classical propositions do not change their truth value over time.
Classical propositions are simply true or false, although you may not know which. They do not change from false to true or vice versa, and classical logic is grounded in this property. "Propositions" such as "today is Monday" are true at some times and false at other times, and hence are not propositions of classical logic.
If you want a "proposition" that depends on time or location, then what you need is a predicate---essentially, a template that yields different specific propositions depending on what values you substitute into the open slots. "Today is Monday" corresponds to the predicate , where
The closest we can come to an actual proposition meaning "today is Monday" would be
where y is some memory state and means your memory state at time .
It's a useless and misleading modeling choice to condition on irrelevant data
Strictly speaking, you should always condition on all data you have available. Calling some data irrelevant is just a shorthand for saying that conditioning on it changes nothing, i.e., . If you can show that conditioning on does change the probability of interest---as my calculation did in fact show---then this means that is in fact relevant information, regardless of what your intuition suggests.
even worse to condition on the assumption the unstated irrelevant data is actually relevant enough to change the outcome.
There was no such assumption. I simply did the calculation, and thereby demonstrated that certain data believed to be irrelevant was actually relevant.
There are several misconceptions here:
1. Non-indexical conditioning is not "a way to do probability theory"; it is just a policy of not throwing out any data, even data that appears irrelevant.
2. No, you do not usually do probability theory on centered propositions such as "today is Monday", as they are not legitimate propositions in classical logic. The propositions of classical logic are timeless -- they are true, or they are false, but they do not change from one to the other.
3. Nowhere in the analysis do I treat a data point as "there exists a version of me which has received this data..."; the concept of "a version of me" does not even appear in the discussion. If you are quibbling over the fact that is only the stream of perceptions Beauty remembers experiencing as of time , instead of being the entire stream of perceptions up to time , then you can suppose that Beauty has perfect memory. This simplifies things---we can now let simply be the entire sequence of perceptions Beauty experiences over the course of the day, and define to mean " is the first elements of , for some "---but it does not alter the analysis.