It's not clear to me that the virtue theory of metabolism is a good example for this post, since it's likely to be highly contentious.

Hi Eliezer,

Thanks for writing this! A few comments about this article (mostly minor, with one exception).

The famous statistician Fischer, who was also a smoker, testified before Congress that the correlation between smoking and lung cancer couldn't prove that the former caused the latter.

Fisher was specifically worried about hidden common causes. Fisher was also the one who brought the concept of a randomized experiment into statistics. Fisher was "curmudgeony," but it is not quite fair to use him as an exemplar of the "keep causality out of our statistics" camp.

Causal models (with specific probabilities attached) are sometimes known as "Bayesian networks" or "Bayes nets", since they were invented by Bayesians and make use of Bayes's Theorem.

Graphical causal models and Bayesian networks are not the same thing (this is a common misconception). A distribution that factorizes according to a DAG is a Bayesian network (this is just a property of a distribution -- nothing about causality). You can further say that a graphical model is causal if an additional set of properties holds. For example, you can (loosely) say that in a causal model all parents are "direct causes." If you want to say that formally, you would talk about the truncated factorization and do(.). Without interventions there is no interventionist causal model.

All this discipline was invented and systematized by Judea Pearl, Peter Spirtes, Thomas Verma, and a number of other people in the 1980s and you should be quite impressed by their accomplishment, because before then, inferring causality from correlation was thought to be a fundamentally unsolvable problem.

I often find myself in a weird position of having to point folks to people other than Pearl. I think it's commendable that you looked into other people in the field. The big early names in causality that you did not mention are Haavelmo (1950s) and Sewall Wright (this guy is amazing -- he figured so many things out correctly in the 1920s). Special cases of non-causal graphical models (Ising models, Hidden Markov models, etc.), along with factorizations and propagation algorithms were known long before Pearl in other communities.

P.S. Since I am in Cali.: if folks at SI are interested in new developments on the "learning causal structure from data" front, I could be bribed by the Cheeseboard to come by and give a talk.

That's a clear outline of the theory. I just want to note that the theory itself makes some assumptions about possible patterns of causation, even before you begin to select which causal graphs are plausible candidates for testing. Pearl himself stresses that without putting causal information in, you can't get causal information out from purely observational data.

For example, if overweight causes lack of exercise and lack of exercise causes overweight, you don't have an acyclic graph. Acyclicity of causation is one of the background assumptions here. Acyclicity of causation is reasonable when talking about point events in a universe without time-like loops. However, "weight" and "exercise level" are temporally extended processes, which makes acyclicity a strong assumption.

The statisticians who discovered causality were trying to find a way to distinguish, within survey data, the direction of cause and effect - whether, as common sense would have it, more obese people exercise less because they find physical activity less rewarding; or whether, as in the virtue theory of metabolism, lack of exercise actually causes weight gain due to divine punishment for the sin of sloth.

I recommend that Eliezer edit this post to remove this kind of provocation. The nature of the actual rationality message in this post is such that people are likely to link to it in the future (indeed, I found it via an external link myself). It even seems like something that may be intended to be part of a sequence. As it stands I expect many future references to be derailed and also expect to see this crop up prominently in lists of reasons to not take Eliezer's blog posts seriously. And, frankly, this reason would be a heck of a lot better than most others that are usually provided by detractors.

Being able to read off patterns of conditional dependence and independence is an art known as "D-separation", and if you're good at it you can glance at a diagram like this...

In order to get better at this, I recommend downloading and playing around with UnBBayes. Here's a brief video tutorial of the basic usage. The program is pretty buggy - for example, sometimes it randomly refuses to compile a file and then starts working after I close the file and reopen it - but that's more of a trivial inconvenience than a major problem.

What's great about UnBBayes is that it allows you to construct a network and then show how the probability flows around it; you can also force some variable to be true or false and see how this affects the surrounding probabilities. For example, here I've constructed a copy of the "Season" network from the post, filled it with conditional probabilities I made up, and asked the program to calculate the overall probabilities. (This was no tough feat - it took me maybe five minutes, most of which I spent on making up the probabilities.)

http://kajsotala.fi/Random/UnBBayesExample1.png

Let's now run through Eliezer's explanation:

...and see that, once we already know the Season

So, we know the season: let's say that we know it's fall. I tell the program to assume that it's fall, and ask it to propagate the effects of this throughout the network. We can see how this changes the probabilites of the different variables:

http://kajsotala.fi/Random/UnBBayesExample2.png

whether the Sprinkler is on and whether it is Raining are conditionally independent of each other - if we're told that it's Raining we conclude nothing about whether or not the Sprinkler is on.

The wording here is a little ambiguous, but Eliezer's saying that with our knowledge of the season, the variables of "Sprinkler" and "Rain" have become independent. Finding out that it rains shouldn't change the probability of the sprinkler being on. Let's test this by setting it to rain and again propagating the effects:

http://kajsotala.fi/Random/UnBBayesExample3.png

And indeed, the probability of it being wet increased, but the probability of the sprinkler being on didn't change.

But if we then further observe that the sidewalk is Slippery, then Sprinkler and Rain become conditionally dependent once more, because if the Sidewalk is Slippery then it is probably Wet and this can be explained by either the Sprinkler or the Rain but probably not both, i.e. if we're told that it's Raining we conclude that it's less likely that the Sprinkler was on.

So let's put the sidewalk to "slippery", and unset "rain" again. I've defined the network so that the sidewalk is never slippery unless it's wet, so setting the sidewalk to "slippery" forces the probability of "wet" to 100%.

http://kajsotala.fi/Random/UnBBayesExample4.png

Now let's see the effect on probabilities if we set it to rain - as Eliezer predicted, the probability of the sprinkler then goes down:

http://kajsotala.fi/Random/UnBBayesExample5.png

And vice versa, if we force the sprinkler to be on, the probability of it raining goes down:

http://kajsotala.fi/Random/UnBBayesExample6.png

(It occurs to me that even if the sidewalk wasn't wet, it could be slippery if it was covered by leaves, or with ice. So there should actually be an arrow from "season" to "slippery". That would be trivial to add.)

Another great thing about UnBBayes is that it not only helps you understand the direction of the probability flows, but also the magnitude of different kinds of changes. Depending on how you've set up the conditional probabilities, a piece of information can have a huge impact on another variable (the sidewalk being slippery always forces the probability of "wet" to 100%, regardless of anything else), or a rather minor one (when we already knew that it was fall and slippery, finding out that it rained only budged the probability of the sprinkler by about five percentage points). Eventually, the logic starts to become intuitive.

Build your own networks and play around with them!

After reading this post I was stunned. Now I think the central conclusion is wrong, though I still think it is a great post, and I will go back to being stunned if you convince me the conclusion is correct.

You've shown how to identify the correct graph structure from the data. But you've erred in assuming that the directed edges of the graph imply causality.

Imagine you did the same analysis, except instead of using O="overweight" you use W="wears size 44 or higher pants". The data would look almost the same. So you would reach an analogous conclusion: that wearing large pants causes one not to exercise. This seems obviously false unless your notion of causality is very different from mine.

In general, I think the following principle holds: inferring causality requires an intervention; it cannot be discovered from observational data alone. A researcher who hypothesized that W causes not-E could round up a bunch of people, have half of them wear big pants, observe the effect of this intervention on exercise rates, and then conclude that there is no causal effect.

(summary)

Correlation does not imply causation,

but

causation implies correlation,

and therefore

no correlation implies no causation

...which permits the falsification of some causal theories based on the absence of certain correlations.

Pretty good overall, but downvoted due to the inflammatory straw-manning of the physics diet. That kind of sloppy thinking just makes me think you have big blind-spots in your rationality. Maybe it's wrong, but it really has nothing to do with virtue or should-universes. To suggest otherwise is dishonest and rude. I usually don't care about rude, but this pissed me off.

I strongly agree with Vaniver's take

I recall reading some lovely quote on this (from somebody of the old camp, who believed that talk of 'causality' was naive), but I couldn't track it down in Pearl or on Google - if anyone knows of a good quote from the old school, may it be provided.

Maybe it's this one?

The law of causality, I believe, like much that passes muster among philosophers, is a relic of a bygone age, surviving, like the monarchy, only because it is erroneously supposed to do no harm. (Russell, 1913, p. 1).

It should be noted that Russell later reversed his skepticism about causality.

This is absolutely incredible. As a broadly numerate person who's never studied stats, I literally did not know that this could be done and while I'd heard about ways to work out what caused what from statistical information I thought they had to have far more assumptions in them. I'm slightly distressed that I studied A level Maths and don't know this, given that the concept at least can be taught in twenty minutes judging by the post and is massively open to testing (I thought this post didn't make sense at first and that there were other interpretations of the data, but when I expressed those in the terminology set out it because clear I was wrong.

Seriously, thanks.

I saw no discernible effect on my weight or my musculature

How did you ascertain possible effects on your musculature? If you used the inter-ocular trauma test, keep in mind that gradual changes in something you look at every day (e.g. yourself, a roommate, or a classmate/coworker) are much harder to discern that way. Did you try asking someone who hadn't seen you in a while, or comparing two photos of you taken a couple months apart?

Leaving only this causal graph:

Well, one hasn't ruled out the "virtuous" graph

until one computes that exercise and internet have a correlation.

By the pigeonhole principle (you can't fit 3 pigeons into 2 pigeonholes) there must be some joint probability distributions which cannot be represented in the first causal structure

Although this is a valid interpretation of the Pigeonhole Principle (PP) for some particular one-to-one cases, I think it misses the main point of it as relates to this particular example. You absolutely can fit 3 Pigeons into 2 Pigeonholes, and the standard (to my knowledge) intended takeaway from the PP is that you are gonna have to, if you want your pigeons holed. There might just not be a cute way to do it.

The idea being that in finite sets $A$ and $B$ with $card\left(A\right)>card\left(B\right)$ there is no injective $f$ from $A$ to $B$ (you could see this as losing information); but you absolutely can send things from $A$ to $B$, you just have to be aware that at least two originally (in $A$) different elements are going to be mapped onto the same element in $B$. This is an issue of losing uniqueness in the representation, not impossibility of the representation itself. It is even useful sometimes. 

A priori, it looks possible for a function to exist by which two different joint distributions would be mapped into the same causal structure in some nice, natural or meaningful way, in the sense that only related-in-some-cute-way joined distributions would share the same representation. If there are no such natural functions, there are definitely ugly ones. You can always cram all your pigeons in the first pigeonhole. They are all represented!

On the other hand, the full joint probability distribution would have 3 degrees of freedom - a free choice of (earthquake&recession), a choice of p(earthquake&¬recession), a choice of p(¬earthquake&recession), and then a constrained p(¬earthquake&¬recession) which must be equal to 1 minus the sum of the other three, so that all four probabilities sum to 1.0.

If you get to infinite stuff, it gets worse. You actually can inject $R²$ into $R$ (or in this case $R³$ —three degrees of freedom— into $R²$ — two degrees —), meaning that not only can you represent every 3D vector in 2D (wich we do all the time), but there are particular representations that wont be lossy, with every 3D object being uniquely represented. You won't lose that information! (the operative term here being that. You are most definitely losing something). 

If you substitute $R³$ and $R²$ for $[0,1]³$ and $[0,1]²$, to account for the fact that you are working in probabilities, this is still true. 

So, assuming there are infinitely many things to account for in the Universe, you would be able to represent the joint distribution as causal diagrams uniquely. If there aren't, you may be able to group them following "nice" relationships. If you can't do that, you can always cram them willy neely. There is no need for a joint distribution to not be represented. I'm not sure how this affects the following

This means the first causal structure is falsifiable; there's survey data we can get which would lead us to reject it as a hypothesis

Seemed weird to me that this hasn't been pointed out (after a rather superficial look at the comments), so I'm pretty sure that either I'm missing something  and there are already 2-3 similarly wrong, refuted comments on this , or this has actually been talked about already and I just haven't seen it. 

Edit: Just realized, in 

If you substitute $R³$ and $R²$ for $[0,1]³$ and $[0,1]²$, to account for the fact that you are working in probabilities, this is still true. 

should be the 1-norm unit balls from $R³$ and $R²$, $B^3$ and $B²$, so all the components sum up to 1, instead of the $[0,1]³,$$[0,1]²$. Pretty sure the injection still holds.

" [1] Somewhat to my own shame, I must admit to ignoring my own observations in this department - even after I saw no discernible effect on my weight or my musculature from aerobic exercise and strength training 2 hours a day 3 times a week, I didn't really start believing that the virtue theory of metabolism was wrong [2] until after other people had started the skeptical dogpile."

I am extremely skeptical of this portion, it would imply that Eliezer's body functions differently then literally every other person (myself included) I have ever known to make a serious attempt at working out.. 2 Hours 3 times a week? How long did you try this?

But the third model, in which recessions directly cause alarms, which only then cause burglars, does not have this property.

This is not the third model in your picture.

I recall reading some lovely quote on this (from somebody of the old camp, who believed that talk of 'causality' was naive), but I couldn't track it down in Pearl or on Google - if anyone knows of a good quote from the old school, may it be provided.

You thinking of this maybe, quoted in the Epilogue in Pearl's Causality?

Beyond such discarded fundamentals as 'matter' and 'force' lies still another fetish among the inscrutable arcana of modern science, namely, the category of cause and effect.

That's Karl Pearson, as in "Pearson's r", the correlation coefficient.

Causal networks have another name in computer science, in the context of compilers. They're the dataflow diagrams of loop-free array-free non-recursive functions represented in SSA form. (Functions that contain loops, arrays and recursion can still be reasoned about with causal networks, but only locally - you have to unroll everything, and that blows up quickly if you try to do it globally.)

A dataflow diagram is when you give each variable in a function a node, and draw an edge from the variable to each of the other variables that was used in its definition. SSA form is when you write a function in such a way that every variable is assigned to exactly once - ie, you can't give a var a value, use it, then give it a new value. Compilers automatically translate functions to SSA form by giving unique names to all the assignments, and defining a new variable (conventionally marked with a combination function phi) at each point where one of several of those names would be chosen based on branch conditions. Compilers use this form because it makes it easier to check which optimizations and transforms are valid before applying them.

This slammed into my "math is hard" block. I will return and read it, but it's going to be work.

But on pondering that, I think I realized why math is hard, compared to prose text that just presents itself as a fait accompli to my attention. (And why it is not hard, for some people who are savants.)

We are not executing mathematical computations. We are emulating a crude mathematical computer which takes the kind of explicit algorithms that are fed to students. No attempt is made to cultivate and tune a "feel" for the result (which is what executing computations would be like, since it's what the other hard computations we do - like reading - feel like).

Just putting that out there.

Reddit discussion (>=21 comments)

[1] Somewhat to my own shame, I must admit to ignoring my own observations in this department - even after I saw no discernible effect on my weight or my musculature from aerobic exercise and strength training 2 hours a day 3 times a week, I didn't really start believing that the virtue theory of metabolism was wrong [2] until after other people had started the skeptical dogpile.

Oh, speaking of which, I was amused the other day by http://well.blogs.nytimes.com/2012/10/10/are-you-likely-to-respond-to-exercise/ Apparently now there's even SNPs linked to non-response...

Well, that suggests that the probability of using Reddit, given that your weight is normal, is the same as the probability that you use Reddit, given that you're overweight. 47,222 out of 334,366 normal-weight people use Reddit, and 12,240 out of 88,376 overweight people use Reddit. That's about 14% either way

Nitpick, but I had busted out Excel at this point, and this is actually 16% either way.

It is obviously impossible to do the converse experiment where half the subjects are randomly assigned lower weights, since there's no known intervention which can cause weight loss.

There is exactly one such intervention that has been shown to cause persistent weight loss after the intervention period is over. (Starvation also causes weight loss, eventually, but only during the intervention period.)

I think it would be valuable if someone pointed out that a third party watching, without controlling, a scientist's controlled study is in pretty much the same situation as the three-column exercise/weight/internet use situation - they have instead exercise/weight/control group.

This "observe the results of a scientist's controlled study" thought experiment motivates and provides hope that one can sometimes derive causation from observation, where the current story arc makes a sortof magical leap.

This is great!

Tpyos:

• Shortly after the sentence, "We could consider three hypothetical causal diagrams over only these two variables", one of the "Earthquake --> Recession" tables gives p(¬e) as 0.70 when it should be 0.71 (so it and 0.29 sum to one).

• After the sentence, "So since all three variables are correlated, can we distinguish between, say, these three causal models?", this diagram I think is meant to have "Recession" on top and "Burglar" on the bottom. (Vaniver also noticed this one.)

Edit:

• The paragraph that starts with "Sure! First, we marginalize over the 'exercise' variable to get the table for just weight and Internet use" needs to have s/normal-weight/overweight/ run on it. (And maybe have a sentence added saying that you're getting the rest of the following table by doing the same math on the other three sets of people grouped by weight and Internet usage.)

If we know that there's a burglar, then we think that either an alarm or a recession caused it; and if we're told that there's an alarm, we'd conclude it was less likely that there was a recession, since the recession had been explained away.

Is this to say that a given node/observation/fact can only have one cause?

More concretely, lets say we have nodes x, y, and z, with causation arrows from x to z and from y to z.

.X...........Y

...\......./

.......Z

If z is just an "and" logic gate, that outputs a "True" value only when x is True and y is True, then it seems like it must be caused by both x and y.

Am I mixing up my abstractions here? Is there some reason why logic gate-like rules are disallowed by causal models?

I don't post here much (yet), and normally I feel fairly confident in my understanding of basic probability...

But I'm slightly lost here. "if the Sidewalk is Slippery then it is probably Wet and this can be explained by either the Sprinkler or the Rain but probably not both, i.e. if we're told that it's Raining we conclude that it's less likely that the Sprinkler was on." This sentence seems... Wrong. If we're told that it's Raining, we conclude that the chances of Sprinkler is... Exactly the same as it was before we learned that the sidewalk was wet.

This seems especially clear when there was an alarm, and we learn there was a burglar - p(B|A) = .9, so shouldn't our current p(E) go up to 0.1 * p(E|A) + p(E|~A)? Burglars burgling doesn't reduce the chance of earthquakes... Adding an alarm shouldn't change that.

What am I missing?

I think you've missed an important piece of this picture, or perhaps have not emphasized it as much as I would. The real real reason we can elucidate causation from correlation is that we have a prior that prefers simple explanations over complex ones, and so when some observed frequencies can be explained by a compact (simple) bayes net we take the arrows in that bayes net to be causation.

A fully connected bayes net (or equivalently, a causal graph with one hidden node pointing to all observed nodes) can represent any probability distribution whatsoever. Such a Bayes net can never be flat-out falsified. Rather it is our preference for simple explanations that sometimes gives us reason to infer structure in the world.

This contradicts nothing you've said, but I guess I read this article as suggesting there is some fundamental rule that gives us a crisp method for extracting causation from observations, whereas I would look at it as a special case of inference-with-prior-and-likelihood, just like in other forms of Bayesian reasoning.

Somewhat to my own shame, I must admit to ignoring my own observations in this department - even after I saw no discernible effect on my weight or my musculature from aerobic exercise and strength training 2 hours a day 3 times a week, I didn't really start believing that the virtue theory of metabolism was wrong until after other people had started the skeptical dogpile.

Wait, are you saying that aerobic exercise and strength training don't have any significant effect on weight?

All these conclusions seem to require simultaneity of causation. If earthquakes almost always caused recessions, but not until one year after the earthquake; and if recessions drastically increase the number of burglars, but not until one year after the recession; then drawing any of the conclusions you made from a survey taken at a single point in time would be entirely unwarranted. Doesn't that mean you’re essentially measuring entailment rather than causation via a series of physical events which take time to occur?

Also, the virtue theory of metabolism is so ridiculous that it seems only to be acting as a caricature here. Wouldn't the theory that “exercise normally metabolises fat and precursors of fat, reducing the amount of weight put on” result in a much more useful example? Or is there a subtext I'm missing here, like the excessive amount of fat-shaming done in many of the more developed nations?

A few minor clarity/readability points:

1. The second paragraph opening "The statisticians who discovered the nature of reality" reads rather oddly when taken out of the context of "The Fabric of Real Things".
2. When considering the three causal models of Burglars, Alarms and Recessions, tackling the models in a "First, third, second" order threw me on first reading. It would probably be easier to follow if the text and the diagram used the same order.
3. Perhaps giving each node a different pastel colour would make it easier to follow what is changing between different diagrams.

And this has probably been said, but using exercise and weight is probably distracting, since people already have opinions on the issue.

All in all though, a great article.

I followed most of the math but the part right before

multiplying together the conditional probability of each variable given the values of its immediate parents. (If a node has no parents, the probability table for it has just an unconditional probability, like "the chance of an earthquake is .01".)

has me puzzled. The variables used aren't explicitly mentioned elsewhere in the article, and while I think they have some conventional meaning I can't quiet remember what. The context let me make a decent guess, but I still feel a little fuzzy. Otherwise the post was pretty clear, much clearer than the other explanations I've seen.

accessibility error: Half the images on this page appear to not load.

Here is a spreadsheet with all the numbers for the Exercise example all crunched and the graph reasoning explained in a slightly different manner:

https://docs.google.com/spreadsheet/ccc?key=0ArkrB_7bUPTNdGhXbFd3SkxWUV9ONWdmVk9DcVRFMGc&usp=sharing

I think I spotted a mistake. I'm surprised no one noticed it earlier.

Eliezer says the data shows that Overweight and Internet both make exercise less likely.

That would imply that P(O|I & not E) should be less than P(O|not E); and that P(I|O & not E) should be less than P(I|not E). But actually, they're approximately equal!^[1]

(From the current data, P(O|E & I) is less than P(O|E): 5% vs 10%; P(I|E & O) is less than P(I|E): 6% vs 11%; which is what you'd expect from Overweight and Internet both making exercise less likely, but it's not all that O->E<-I implies.)

Edit: note that inferring the causal graph exactly the way it's done in the alarms/recessions/burglars example fails here due to the extra conditional independences. The data is generated in a way that doesn't show a correspondence to this graph if you follow a procedure identical to the one described in the post

1. ^{^}

   google sheet with calculations

↑ comment by DanielFilan · 2023-11-25T19:15:56.655Z · LW(p) · GW(p)

The data is generated in a way that doesn't show a correspondence to this graph if you follow a procedure identical to the one described in the post

 

When I calculate the conditional probability tables you get from the table of numbers in the post and multiply them out to get the joint distribution, my answers basically match with the table of numbers in the post.

 

Replies from: mikhail-samin

↑ comment by Mikhail Samin (mikhail-samin) · 2023-11-25T19:20:25.153Z · LW(p) · GW(p)

The point is, these probabilities don’t really correspond to that causal graph in a way described in the post. A script that simulates the causal graph: https://colab.research.google.com/drive/18pIMfKJpvlOZ213APeFrHNiqKiS5B5ve?usp=sharing

Replies from: DanielFilan

↑ comment by DanielFilan · 2023-11-26T22:56:08.468Z · LW(p) · GW(p)

I think they do correspond to the causal graph in the way described in the post. Your script simulates something more specific than the causal graph: you can fit the causal graph without being able to be fitted by the script.

Replies from: mikhail-samin

↑ comment by Mikhail Samin (mikhail-samin) · 2023-11-26T23:11:14.845Z · LW(p) · GW(p)

I think I don’t buy the story of a correct causal structure generating the data here in a way that supports the point of the post. If two variables, I and O, both make one value of E more likely than the other, that means the probability of I conditional on some value of E is different from the probability of I because I explains some of that value of E; but if you also know O, than this explains some of that value of E as well, and so P(I|E=x, O) should bd different.

The post describes this example:

This may seem a bit clearer by considering the scenario B->A<-E, where burglars and earthquakes both cause alarms. If we're told the value of the bottom node, that there was an alarm, the probability of there being a burglar is not independent of whether we're told there was an earthquake - the two top nodes are not conditionally independent once we condition on the bottom node

And if you apply this procedure to “not exercising”, we don’t see that absence of conditional independence, once we condition on the bottom node. Which means that “not exercising” is not at all explained away by internet (or being overweight)

Replies from: DanielFilan

↑ comment by DanielFilan · 2023-11-27T19:32:22.401Z · LW(p) · GW(p)

If two variables, I and O, both make one value of E more likely than the other, that means the probability of I conditional on some value of E is different from the probability of I because I explains some of that value of E

That's correct

but if you also know O, than this explains some of that value of E as well, and so P(I|E=x, O) should bd different.

This is generically true but doesn't have to be true for every value of x.

Here's one way to see why the graph in the post is right: look at all other casual graphs, and you will see they either fail to imply that I and O are independent (as our graph does), or imply independences or conditional independences that don't exist in the data.

Replies from: mikhail-samin

↑ comment by Mikhail Samin (mikhail-samin) · 2023-11-28T00:06:39.529Z · LW(p) · GW(p)

I don’t really get how this can be true for some values of x but not others if the variable is binary

Replies from: DanielFilan

↑ comment by DanielFilan · 2023-11-28T06:32:42.356Z · LW(p) · GW(p)

The post gives one example of how it can be true: the probabilities are compatible with the causal graph, I is independent of O given E = no, but I is not independent of O given E = yes.

Here's one way to see why the graph in the post is right: look at all other casual graphs, and you will see they either fail to imply that I and O are independent (as our graph does), or imply independences or conditional independences that don't exist in the data.

Have you tried this exercise?

(We have remnants of this type of reasoning in old-school "Correlation does not imply causation", without the now-standard appendix, "But it sure is a hint".)

Given the reasoning in this post and this post I think you can also infer that this old "Correlation does not imply causation" statement is not only flawed, but it's also outright wrong

And should instead just be "Correlation does imply causation, but doesn't tell which kind"

Let's consider a practical example. Since the question of exercise and weight has turned up, let's revisit it. First, let's collect some raw data (I can't use internet usage, since this poll is extremely biased on that axis).

For the purposes of this poll, "overweight" means a body mass index over 25. "Exercise" means at least 30 minutes a week, working hard at it, on a regular basis. "Diet" means that you actually think about the nutritional value of the food you eat, and consciously base your choice of food on that information in some significant way.

Select one of the following:

[pollid:183]

Once we have some data, we can then practice this skill on the results of the poll, and see whether (and if so, how) these variables are causally linked among poll respondants.

So, what if the causal diagram isn't simply dependent and/or contains loops? What if recessions cause burglars, and burglars disable alarms, and alarms cause recessions?

You also forgot about the graph that has confounding factors C and D; C affects exercise and weight, while D affects exercise and internet usage. Both of them make exercise more (or less) likely, and their other factor less (or more) likely; Weight and internet use remain uncorrelated, but remain negatively correlated with exercise.

Currently it looks like this page has lots of broken images, which are actually formulas. Can this be fixed? It's kind of hard to understand now.

If we know that there's a burglar, then we think that either an alarm or a recession caused it; and if we're told that there's an alarm, we'd conclude it was less likely that there was a recession, since the recession had been explained away.

Should that be "since the burglar had been explained away"? Or am I confused?

Edit: I was confused. The burglar was explained; the recession was explained away.

I've been enjoying this series so far, and I found this article to be particularly helpful. I did have a minor suggestion. The turnstile and the logical negation symbols were called out, and I thought it might be useful to explicitly breakdown the probability distribution equation. The current Less Wrong audience had little problem with it, certainly, but if you were showing it to someone new to this for the first time, they might not be acquainted with it. I was thinking something along the lines of this from stattrek:

"Generally, statisticians use a capital letter to represent a random variable and a lower-case letter, to represent one of its values. For example,

X represents the random variable X.
P(X) represents the probability of X.
P(X = x) refers to the probability that the random variable X is equal to a particular value, denoted by x. As an example, P(X = 1) refers to the probability that the random variable X is equal to 1."

(from http://stattrek.com/probability-distributions/probability-distribution.aspx)

Also, page 2 of A Student's Guide to Maxwell's Equations does a great job of diagramming Gauss' law for electrical fields, and I think it would be helpful if this were available to breakdown the right half of the equation, with the beginning reader seeing a breakdown of the equation.

http://books.google.com/books?id=I-x1MLny6y8C&printsec=frontcover#v=onepage&q&f=false

If this all was set aside in the footnote, the overall continuity of the article wouldn't be affected, and someone who might be intimidated at first by equations might see that these aren't so bad. With just a bit of exposition, more readers might be able to follow along with the entire argument, which I think could be introduced to someone with very little background.

To get that last graph, you have to show that internet and exercise are correlated...

This helped me understand what Instrumental Variables are, but Andrew Gelman's critique of instrumental variables has me confused again:

Suppose z is your instrument, T is your treatment, and y is your outcome. So the causal model is z -> T -> y. The trick is to think of (T,y) as a joint outcome and to think of the effect of z on each. For example, an increase of 1 in z is associated with an increase of 0.8 in T and an increase of 10 in y. The usual “instrumental variables” summary is to just say the estimated effect of T on y is 10/0.8=12.5, but I’d rather just keep it separate and report the effects on T and y separately.

In Piero’s example, this translates into two statements: (a) States with higher penalties for murder had higher penalties for defamation, and (b) States with higher penalties for murder had less reporting of corruption.

Fine. But I don’t see how this adds anything at all to my understanding of the defamation/corruption relationship, beyond what I learned from his simpler finding: States with higher penalties for defamation had less reporting of corruption.

If your model is z -> T -> y, and you show that z interacts with each of T and y, isn't the next step just to look at the relation between z and y, controlling for T? In other words, if it turns out that z still matters in predicting y once you have T in your model, then you don't have an instrumental variable. But if T screens off the effect of z in predicting y, then z is an instrumental variable, and only affects y through T.

Interesting article, thanks.

I agree with the general concept. I would be a bit more careful in the conclusions, however:
No visible correlation does not mean no causation - it is just a strong hint. In the specific example, the hint comes from a single parameter - the lack of significant correlation between internet & overweight when both exercise categories are added; together with the significant correlation of internet usage with the other two parameters.

With the proposed diagram, I get:
p(Internet)=.141
p(not Internet)=.859
p(Overweight)=.209
p(not Overweight)=.791

p(Ex|Int & Ov)=.10
p(Ex|Int & no OV)=.62
p(Ex|no Int & Ov)=.27
p(Ex|no Int & no Ov)=.85

This model has 6 free parameters - the insignificant correlation between overweight and internet is the only constraint. It is true that other models have to be more complex to explain data, but we know that our world is not a small toy simulation - there are causal connections everywhere, the question is just "are they negligible or not?".

I haven't read enough of Causality, but I think I get how to find a causal model from the examples above.

Basically, a model selection problem? P(Model|Data) = P(Data|Model)P(Model)/P(Data) ~ P(Data|Model)P(Model)?

Is P(Model) done in some objective sense, or is that left to the prior of the modeler? Or some combination of contextually objective and standard causal modeling priors (direction of time, locality, etc.)?

Any good powerpoint summary of Pearl's methods out there?

An earthquake is 0.8 likely to set off your burglar alarm; a burglar is 0.9 likely to set off your burglar alarm. And - we can't compute this model fully without this info - the combination of a burglar and an earthquake is 0.95 likely to set off the alarm;

I don't think that true. The earthquake can cause the burgler to have less control over his own movement and therefore increase the chance that he triggers the alarm.

This was a really good article overall; I just finished going through all the numbers in Excel and it makes a lot of sense.

The thing that is most counterintuitive to me is that it appears that the causal link between exercise and weight can ONLY be computed if you bring in a 3rd, seemingly irrelevant variable like internet usage. It looks like that variable has to be somehow correlated with at least one of the causal nodes - maybe it has to be correlated with one specific node... I am a little hazy on that.

I encourage readers to open an Excel file or something and, using Eliezer's made-up 'data' about exercise/weight/internet, exhaustively list all the possible causal graphs for those 3 variables, then falsify all of them until only the one remains. It really shows how nicely the technique works.

Now I am keen to find some controversial real-world causal hypothesis and test it using this method.

Early investigators in Artificial Intelligence, who were trying to represent all high-level events using primitive tokens in a first-order logic (for reasons of historical stupidity we won't go into) were stymied by the following apparent paradox:

[Description of a system with the following three theorems: ⊢ ALARM → BURGLAR, ⊢ EARTHQUAKE → ALARM, and ⊢ (EARTHQUAKE & ALARM) → NOT BURGLAR]

Which represents a logical contradiction.

This isn't a logical contradiction: perhaps what you mean is that we can deduce from this system that EARTHQUAKE is false. This would give us a contradiction in a modal system, if we also had the theorem ⊢ possibly(EARTHQUAKE), but as it stands it isn't yet contradictory.

Very nice and intuitive, thanks! This explanation is great.

(Though I've already spent a little while playing around with Bayes nets, and I don't know how large of a role that had in making this feel more intuitive to me.)

The concern of the philosophers is the idea of 'true causation' as independent from merely apparent causation. In particular, they have in mind the idea that even if the laws of the universe were deterministic there would be a sense in which certain events could be said to be causes of others even though mathematically, the configuration of the universe at any time completely entails it at all others. Frankly, this question arises out of half-baked arguments about whether events cause latter events or if god has predetermined and causes all events individually and I don't take it seriously.

My take is that there is no such thing as causation. Correlation is all there is and the fact that many correlations are usefully and compactlly described by Bayesian causal models is actually support for the idea that the ascription of causation reflects nothing more than how the arrows happen to point in those causal models we find most compelling. In other words I don't think it makes sense to look under your model to ask about what is truly causation but we should be clear that is what the philosophers mean.

Despite my great respect for Bayesian causal models it doesn't let us deduce causality from correlation and I can prove it.

Given results about k events (assume for simplicity they are binary True/False events) E_1...E_k (so E_1 might be burglary, E_2 earthquake, E_3 recession and a trial is each year) and any ordering < on 1..k there is a causal model such that E_i is a causal antecedant of E_j iff i < j that perfectly agrees with the given probabilities. In other words at the expense of potentially having every E_i with i <* j affect the probability of E_i I can have any causal order I want on the events and get the same results.

To see this is true start with whatever event we want to occur first, say E_{i1}. Now we compute the probabilities that the next event E{i2} occurs conditional on E{i1} and it's negation. For E{i3} we compute the probabilities that this event occurs conditional on all 4 outcomes for the pair E{i1}, E{i_2} and so on. This gives the correct probability to each set of outcomes and thus matches all observations. Alternatively, we can always make the E_i all dependent on some invisible common causes that match the appropriate priors.

True, these diagrams might be less simple in some sense than other diagrams we might draw but that doesn't mean they are false. Indeed, we might have very good general reasons for preferring some more complicated theory, e.g., even if a simpler causal model could explain the data but requires causal dependence on effects later in time reject it in favor of some more complicated model. This is a useful generalization we have about the world and following it helps us reach better predictions when we have limited data. Thus the mere number of arrows can't simply be minimized.

In other words all you've got is the same old crap about preferring the simpler theory where that has no principled mathematical definition and more or less means 'prefer whatever your priors say the causal model really looks like.' In other words we haven't gotten any closer to infering causation.

Just the opposite. The use of Bayesian causal models explains extremely well why, even if events are truly all effects caused by the choices of some unseen mover the notion of causation would be likely to evolve.