Evidential Decision Theory, Selection Bias, and Reference Classes

by Qiaochu_Yuan 6y8th Jul 20135 min read128 comments

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See also: Does Evidential Decision Theory really fail Solomon's Problem?, What's Wrong with Evidential Decision Theory?

It seems to me that the examples usually given of decision problems where EDT makes the wrong decisions are really examples of performing Bayesian updates incorrectly. The basic problem seems to be that naive EDT ignores a selection bias when it assumes that an agent that has just performed an action should be treated as a random sample from the population of all agents who have performed that action. Said another way, naive EDT agents make some unjustified assumptions about what reference classes they should put themselves into when considering counterfactuals. A more sophisticated Bayesian agent should make neither of these mistakes, and correcting them should not in principle require moving beyond EDT but just becoming less naive in applying it. 

Elaboration

Recall that an EDT agent attempts to maximize conditional expected utility. The main criticism of EDT is that naively computing conditional probabilities leads to the conclusion that you should perform actions which are good news upon learning that they happened, as opposed to actions which cause good outcomes (what CDT attempts to do instead). For a concrete example of the difference, let's take the smoking lesion problem:

Smoking is strongly correlated with lung cancer, but in the world of the Smoker's Lesion this correlation is understood to be the result of a common cause: a genetic lesion that tends to cause both smoking and cancer. Once we fix the presence or absence of the lesion, there is no additional correlation between smoking and cancer.

Suppose you prefer smoking without cancer to not smoking without cancer, and prefer smoking with cancer to not smoking with cancer. Should you smoke?

In the smoking lesion problem, smoking is bad news, but it doesn't cause a bad outcome: learning that someone smokes, in the absence of further information, increases your posterior probability that they have the lesion and therefore cancer, but choosing to smoke cannot in fact alter whether you have the lesion / cancer or not. Naive EDT recommends not smoking, but naive CDT recommends smoking, and in this case it seems that naive CDT's recommendation is correct and naive EDT's recommendation is not. 

The naive EDT agent's reasoning process involves considering the following counterfactual: "if I observe myself smoking, that increases my posterior probability that I have the lesion and therefore cancer, and that would be bad. Therefore I will not smoke." But it seems to me that in this counterfactual, the naive EDT agent -- who smokes and then glumly concludes that there is an increased probability that they have cancer -- is performing a Bayesian update incorrectly, and that the incorrectness of this Bayesian update, rather than any fundamental problem with making decisions based on conditional probabilities, is what causes the naive EDT agent to perform poorly. 

Here are some other examples of this kind of Bayesian update, all of which seem obviously incorrect to me. They lead to silly decisions because they are silly updates. 

  • "If I observe myself throwing away expensive things, that increases my posterior probability that I am rich and can afford to throw away expensive things, and that would be good. Therefore I will throw away expensive things." (This example requires that you have some uncertainty about your finances -- perhaps you never check your bank statement and never ask your boss what your salary is.)
  • "If I observe myself not showering, that increases my posterior probability that I am clean and do not need to shower, and that would be good. Therefore I will not shower." (This example requires that you have some uncertainty about how clean you are -- perhaps you don't have a sense of smell or a mirror.)
  • "If I observe myself playing video games, that increases my posterior probability that I don't have any work to do, and that would be good. Therefore I will play video games." (This example requires that you have some uncertainty about how much work you have to do -- perhaps you write this information down and then forget it.) 

Selection Bias

Earlier I said that in the absence of further information, learning that someone smokes increases your posterior probability that they have the lesion and therefore cancer in the smoking lesion problem. But when a naive EDT agent is deciding what to do, they have further information: in the counterfactual where they're smoking, they know that they're smoking because they're in a counterfactual about what would happen if they smoked (or something like that). This information should screen off inferences about other possible causes of smoking, which is perhaps clearer in the bulleted examples above. If you consider what would happen if you threw away expensive things, you know that you're doing so because you're considering what would happen if you threw away expensive things and not because you're rich. 

Failure to take this information into account is a kind of selection bias: a naive EDT agent considering the counterfactual where they perform some action treats itself as a random sample from the population of similar agents who have performed such actions, but it is not in fact such a random sample! The sampling procedure, which consists of actually performing an action, is undoubtedly biased. 

Reference Classes

Another way to think about the above situation is that a naive EDT agent chooses inappropriate reference classes: when an agent performs an action, the appropriate reference class is not all other agents who have performed that action. It's unclear to me exactly what it is, but at the very least it's something like "other sufficiently similar agents who have performed that action under sufficiently similar circumstances." 

This is actually very easy to see in the smoker's lesion problem because of the following observation (which I think I found in Eliezer's old TDT writeup): suppose the world of the smoker's legion is populated entirely with naive EDT agents who do not know whether or not they have the lesion. Then the above argument suggests that none of them will choose to smoke. But if that's the case, then where does the correlation between the lesion and smoking come from? Any agents who smoke are either not naive EDT agents or know whether they have the lesion. In either case, that makes them inappropriate members of the reference class any reasonable Bayesian agent should be using.

Furthermore, if the naive EDT agents collectively decide to become slightly less naive and restrict their reference class to each other, they now find that smoking no longer gives any information about whether they have the lesion or not! This is a kind of reflective inconsistency: the naive recommendation not to smoke in the smoker's lesion problem has the property that, if adopted by a population of naive EDT agents, it breaks the correlations upon which the recommendation is based. 

The Tickle Defense

As it happens, there is a standard counterargument in the decision theory literature to the claim that EDT recommends not smoking in the smoking lesion problem. It is known as the "tickle defense," and runs as follows: in the smoking lesion problem, what an EDT agent should be updating on is not the action of smoking but an internal desire, or "tickle," prompting it to smoke, and once the presence or absence of such a tickle has been updated on it screens off any information gained by updating on the act of smoking or not smoking. So EDT + Tickles smokes on the smoking lesion problem. (Note that this prescription also has the effect of breaking the correlation claimed in the setup of the smoking lesion problem among a population of EDT + Tickles agents who don't know whether hey have the lesion or not. So maybe there's just something wrong with the smoking lesion problem.) 

The tickle defense is good in that it encourages ignoring less information than naive EDT, but it strikes me as a patch covering up part of a more general problem, namely the problem of how to choose appropriate reference classes when performing Bayesian updates (or something like that). So I don't find it a satisfactory rescuing of EDT. It doesn't help that there's a more sophisticated version known as the "meta-tickle defense" that recommends two-boxing on Newcomb's problem.

Sophisticated EDT?

What does a more sophisticated version of EDT, taking the above observations into account, look like? I don't know. I suspect that it looks like some version of TDT / UDT, where TDT corresponds to something like trying to update on "being the kind of agent who outputs this action in this situation" and UDT corresponds to something more mysterious that I haven't been able to find a good explanation of yet, but I haven't thought about this much. If someone else has, let me know.

Here are some vague thoughts. First, I think this comment by Stuart_Armstrong is right on the money:

I've found that, in practice, most versions of EDT are underspecified, and people use their intuitions to fill the gaps in one direction or the other.

A "true" EDT agent needs to update on all the evidence they've ever observed, and it's very unclear to me how to do this in practice. So it seems that it's difficult to claim with much certainty that EDT will or will not do a particular thing in a particular situation.

CDT-via-causal-networks and TDT-via-causal-networks seem like reasonable candidates for more sophisticated versions of EDT in that they formalize the intuition above about screening off possible causes of a particular action. TDT seems like it better captures this intuition in that it better attempts to update on the cause of an action in a hypothetical about that action (the cause being that TDT outputs that action). My intuition here is that it should be possible to see causal networks as arising naturally out of Bayesian considerations, although I haven't thought about this much either. 

AIXI might be another candidate. Unfortunately, AIXI can't handle the smoking lesion problem because it models itself as separate from the environment, whereas a key point in the smoking lesion problem is that an agent in the world of the smoking lesion has some uncertainty about its innards, regarded as part of its environment. Fully specifying sophisticated EDT might involve finding a version of AIXI that models itself as part of its environment. 

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