Mistakes with Conservation of Expected Evidence

Epistemic Status: I’ve really spent some time wrestling with this one. I am highly confident in most of what I say. However, this differs from section to section. I’ll put more specific epistemic statuses at the end of each section.

Some of this post is generated from mistakes I’ve seen people make (or, heard people complain about) in applying conservation-of-expected-evidence or related ideas. Other parts of this post are based on mistakes I made myself. I think that I used a wrong version of conservation-of-expected-evidence for some time, and propagated some wrong conclusions fairly deeply; so, this post is partly an attempt to work out the right conclusions for myself, and partly a warning to those who might make the same mistakes.

All of the mistakes I’ll argue against have some good insight behind them. They may be something which is usually true, or something which points in the direction of a real phenomenon while making an error. I may come off as nitpicking.

1. “You can’t predict that you’ll update in a particular direction.”

Starting with an easy one.

It can be tempting to simplify conservation of expected evidence to say you can’t predict the direction which your beliefs will change. This is often approximately true, and it’s exactly true in symmetric cases where your starting belief is 50-50 and the evidence is equally likely to point in either direction.

To see why it is wrong in general, consider an extreme case: a universal law, which you mostly already believe to be true. At any time, you could see a counterexample, which would make you jump to complete disbelief. That’s a small probability of a very large update downwards. Conservation of expected evidence implies that you must move your belief upwards when you don’t see such a counterexample. But, you consider that case to be quite likely. So, considering only which direction your beliefs will change, you can be fairly confident that your belief in the universal law will increase—in fact, as confident as you are in the universal law itself.

The critical point here is direction vs magnitude. Conservation of expected evidence takes magnitude as well as direction into account. The small but very probable increase is balanced by the large but very improbable decrease.

The fact that we’re talking about universal laws and counterexamples may fool you into thinking about logical uncertainty. You can think about logical uncertainty if you want, but this phenomenon is present in the fully classical Bayesian setting; there’s no funny business with non-Bayesian updates here.

Epistemic status: confidence at the level of mathematical reasoning.

2. “Yes requires the possibility of no.”

Scott’s recent post, yes requires the possibility of no, is fine. I’m referring to a possible mistake which one could make in applying the principle illustrated there.

“Those who dream do not know they dream, but when you are awake, you know you are awake.”—Eliezer, Against Modest Epistemology

Sometimes, look around, and ask myself whether I’m in a dream. When this happens, I generally conclude very confidently that I’m awake.

I am not similarly capable of determining that I’m dreaming. My dreaming self doesn’t have the self-awareness to question whether he is dreaming in this way.

(Actually, very occasionally, I do. I either end up forcing myself awake, or I become lucid in the dream. Let’s ignore that possibility for the purpose of the thought experiment.)

I am not claiming that my dreaming self is never deluded into thinking he is awake. On the contrary, I have those repeatedly-waking-up-only-to-find-I’m-still-dreaming dreams occasionally. In those cases, I vividly believe myself to be awake. So, it’s definitely possible for me to vividly believe I’m awake and be mistaken.

What I’m saying is that, when I’m asleep, I am not able to perform the actually good test, where I look around and really consciously consider whether or not I might be dreaming. Nonetheless, when I can perform that check, it seems quite reliable. If I want to know if I’m awake, I can just check.

A “yes-requires-the-possibility-of-no” mindset might conclude that my “actually good test” is no good at all, because it can’t say “no”. I believe the exact opposite: my test seems really quite effective, because I only successfully complete it while awake.

Sometimes, your thought processes really are quite suspect; yet, there’s a sanity check you can run which tells you the truth. If you’re deluding yourself, the general category of “things which you think are simple sanity checks you can run” is not trustworthy. If you’re deluding yourself, you’re not even going to think about the real sanity checks. But, that does not in itself detract from the effectiveness of the sanity check.

The general moral in terms of conservation of expected evidence is: “‘Yes’ only requires the possibility of silence.”. In many cases, you can meaningfully say yes without being able to meaningfully say no. For example, the axioms of set theory could prove their own inconsistency. They could not prove themselves consistent (without also proving themselves inconsistent). This does not detract from the effectiveness of a proof of inconsistency! Again, although the example involves logic, there’s nothing funny going on with logical uncertainty; the phenomenon under discussion is understandable in fully Bayesian terms.

Symbolically: as is always the case, you don’t really want to update on the raw proposition, but rather, the fact that you observed the proposition, to account for selection bias. Conservation of expected evidence can be written , but if we re-write it to explicitly show the “observation of evidence”, it becomes . It does not become . In English: evidence is balanced between making the observation and not making the observation, not between the observation and the observation of the negation.

Epistemic status: confidence at the level of mathematical reasoning for the core claim of this section. However, some applications of the idea (such as to dreams, my central example) depend on trickier philosophical issues discussed in the next section. I’m only moderately confident I have the right view there.

3. “But then what do you say to the Republican?”

I suspect that many readers are less than fully on board with the claims I made in the previous section. Perhaps you think I’m grossly overconfident about being awake. Perhaps you think I’m neglecting the outside view, or ignoring something to do with timeless decision theory.

A lot of my thinking in this post was generated by grappling with some points made in Inadequate Equilibria. To quote the relevant paragraph of against modest epistemology:

Or as someone advocating what I took to be modesty recently said to me, after I explained why I thought it was sometimes okay to give yourself the discretion to disagree with mainstream expertise when the mainstream seems to be screwing up, in exactly the following words: “But then what do you say to the Republican?”

Let’s put that in (pseudo-)conservation-of-expected-evidence terms: we know that just applying one’s best reasoning will often leave one overconfident in one’s idiosyncratic beliefs. Doesn’t that mean “apply your best reasoning” is a bad test, which fails to conserve expected evidence? So, should we not adjust downward in general?

In the essay, Eliezer strongly advises allowing yourself to have an inside view even when there’s an outside view which says inside views broadly similar to yours tend to be mistaken. But doesn’t that go against what he said in Ethical Injunctions?

Ethical Injunctions argues that there are situations where you should not trust your reasoning, and fall back on a general rule. You do this because, in the vast majority of cases of that kind, your oh-so-clever reasoning is mistaken and the general rule saves you from the error.

In Against Modest Epistemology, Eliezer criticizes arguments which rely on putting arguments in very general categories and taking the outside view:

At its epistemological core, modesty says that we should abstract up to a particular very general self-observation, condition on it, and then not condition on anything else because that would be inside-viewing. An observation like, “I’m familiar with the cognitive science literature discussing which debiasing techniques work well in practice, I’ve spent time on calibration and visualization exercises to address biases like base rate neglect, and my experience suggests that they’ve helped,” is to be generalized up to, “I use an epistemology which I think is good.” I am then to ask myself what average performance I would expect from an agent, conditioning only on the fact that the agent is using an epistemology that they think is good, and not conditioning on that agent using Bayesian epistemology or debiasing techniques or experimental protocol or mathematical reasoning or anything in particular.
Only in this way can we force Republicans to agree with us… or something.

He instead advises that we should update on all the information we have, use our best arguments, reason about situations in full detail:

If you’re trying to estimate the accuracy of your epistemology, and you know what Bayes’s Rule is, then—on naive, straightforward, traditional Bayesian epistemology—you ought to condition on both of these facts, and estimate P(accuracy|know_Bayes) instead of P(accuracy). Doing anything other than that opens the door to a host of paradoxes.

In Ethical Injunctions, he seems to warn against that very thing:

But surely… if one is aware of these reasons… then one can simply redo the calculation, taking them into account. So we can rob banks if it seems like the right thing to do after taking into account the problem of corrupted hardware and black swan blowups. That’s the rational course, right?
There’s a number of replies I could give to that.
I’ll start by saying that this is a prime example of the sort of thinking I have in mind, when I warn aspiring rationalists to beware of cleverness.

Now, maybe Eliezer has simply changed views on this over the years. Even so, that leaves us with the problem of how to reconcile these arguments.

I’d say the following: modest epistemology points out a simple improvement over the default strategy: “In any group of people who disagree, they can do better by moving their beliefs toward each other.” “Lots of crazy people think they’ve discovered secrets of the universe, and the number of sane people who truly discover such secrets is quite small; so, we can improve the average by never believing we’ve discovered secrets of the universe.” If we take a timeless decision theory perspective (or similar), this is in fact an improvement; however, it is far from the optimal policy, and has a form which blocks further progress.

Ethical Injunctions talks about rules with greater specificity, and less progress-blocking nature. Essentially, a proper ethical injunction is actually the best policy you can come up with, whereas the modesty argument stops short of that.

Doesn’t the “actually best policy you can come up with” risk overly-clever policies which depend on broken parts of your cognition? Yes, but your meta-level arguments about which kinds of argument work should be independent sources of evidence from your object-level confusion. To give a toy example: let’s say you really, really want 8+8 to be 12 due to some motivated cognition. You can still decide to check by applying basic arithmetic. You might not do this, because you know it isn’t to the advantage of the motivated cognition. However, if you do check, it is actually quite difficult for the motivated cognition to warp basic arithmetic.

There’s also the fact that choosing a modesty policy doesn’t really help the republican. I think that’s the critical kink in the conservation-of-expected-evidence version of modest epistemology. If you, while awake, decide to doubt whether you’re awake (no matter how compelling the evidence that you’re awake seems to be), then you’re not really improving your overall correctness.

So, all told, it seems like conservation of expected evidence has to be applied to the details of your reasoning. If your put your reasoning in a more generic category, it may appear that a much more modest conclusion is required by conservation of expected evidence. We can justify this in classical probability theory, though in this section it is even more tempting to consider exotic decision-theoretic and non-omnescience considerations than it was previously.

Epistemic status: the conclusion is mathematically true in classical Bayesian epistemology. I am subjectively >80% confident that the conclusion should hold in >90% of realistic cases, but it is unclear how to make this into a real empirical claim. I’m unsure enough of how ethical injunctions should work that I could see my views shifting significantly. I’ll mention pre-rationality as one confusion I have which seems vaguely related.

4. “I can’t credibly claim anything if there are incentives on my words.”

Another rule which one might derive from Scott’s Yes Requires the Possibility of No is: you can’t really say anything if pressure is being put on you to say a particular thing.

Now, I agree that this is somewhat true, particularly in simple cases where pressure is being put on you to say one particular thing. However, I’ve suffered from learned helplessness around this. I sort of shut down when I can identify any incentives at all which could make my claims suspect, and hesitate to claim anything. This isn’t a very useful strategy. Either “just say the truth”, or “just say whatever you feel you’re expected to say” are both likely better strategies.

One idea is to “call out” the pressure you feel. “I’m having trouble saying anything because I’m worried what you will think of me.” This isn’t always a good idea, but it can often work fairly well. Someone who is caving to incentives isn’t very likely to say something like that, so it provides some evidence that you’re being genuine. It can also open the door to other ways you and the person you’re talking to can solve the incentive problem.

You can also “call out” something even if you’re unable or unwilling to explain. You just say something like “there’s some thing going on”… or “I’m somehow frustrated with this situation”… or whatever you can manage to say.

This “call out” idea also works (to some extent) on motivated cognition. Maybe you’re worried about the social pressure on your beliefs because it might influence the accuracy of those beliefs. Rather than stressing about this and going into a spiral of self-analysis, you can just state to yourself that that’s a thing which might be going on, and move forward. Making it explicit might open up helpful lines of thinking later.

Another thing I want to point out is that most people are willing to place at least a little faith in your honesty (and not irrationally so). Just because you have a story in mind where they should assume you’re lying doesn’t mean that’s the only possibility they are—or should be—considering. One problematic incentive doesn’t fully determine the situation. (This one also applies internally: identifying one relevant bias or whatever doesn’t mean you should block off that part of yourself.)

Epistemic status: low confidence. I imagine I would have said something very different if I were more an expert in this particular thing.

5. “Your true reason screens off any other evidence your argument might include.”

In The Bottom Line, Eliezer describes a clever arguer who first writes the conclusion which they want to argue for at the bottom of a sheet of paper, and then comes up with as many arguments as they can to put above that. In the thought experiment, the clever arguer’s conclusion is actually determined by who can pay the clever arguer more. Eliezer says:

So the handwriting of the curious inquirer is entangled with the signs and portents and the contents of the boxes, whereas the handwriting of the clever arguer is evidence only of which owner paid the higher bid. There is a great difference in the indications of ink, though one who foolishly read aloud the ink-shapes might think the English words sounded similar.

Now, Eliezer is trying to make a point about how you form your own beliefs—that the quality of the process which determines which claims you make is what matters, and the quality of any rationalizations you give doesn’t change that.

However, reading that, I came away with the mistaken idea that someone listening to a clever arguer should ignore all the clever arguments. Or, generalizing further, what you should do when listening to any argument is try to figure out what process wrote the bottom line, ignoring any other evidence provided.

This isn’t the worst possible algorithm. You really should heavily discount evidence provided by clever arguers, because it has been heavily cherry-picked. And almost everyone does a great deal of clever arguing. Even a hardboiled rationalist will tend to present evidence for the point they’re trying to make rather than against (perhaps because that’s a fairly good strategy for explaining things—sampling evidence at random isn’t a very efficient way of conversing!).

However, ignoring arguments and attending only to the original causes of belief has some absurd consequences. Chief among them is: it would imply that you should ignore mathematical proofs if the person who came up with the proof only searched for positive proofs and wouldn’t have spend time trying to prove the opposite. (This ties in with the very first section—failing to find a proof is like remaining silent.)

This is bonkers. Proof is proof. And again, this isn’t some special non-Bayesian phenomenon due to logical uncertainty. A Bayesian can and should recognize decisive evidence, whether or not it came from a clever arguer.

Yet, I really held this position for a while. I treated mathematical proofs as an exceptional case, rather than as a phenomenon continuous with weaker forms of evidence. If a clever arguer presented anything short of a mathematical proof, I would remind myself of how convincing cherry-picked evidence can seem. And I’d notice how almost everyone mostly cherry-picked when explaining their views.

This strategy was throwing out data when it has been contaminated by selection bias, rather than making a model of the selection bias so that I could update on the data appropriately. It might be a good practice in scientific publications, but if you take it as a universal, you could find reasons to throw out just about everything (especially if you start worrying about anthropic selection effects).

The right thing to do is closer to this: figure out how convincing you expect evidence to look given the extent of selection bias. Then, update on the difference between what you see and what’s expected. If a clever arguer makes a case which is much better than what you would have expected they could make, you can update up. If it is worse that you’d expect, even if the evidence would otherwise look favorable, you update down.

My view also made me uncomfortable presenting a case for my own beliefs, because I would think of myself as a clever-arguer any time I did something other than recount the actual historical causes of my belief (or honestly reconsider my belief on the spot). Grognor made a similar point in Unwilling or Unable to Explain:

Let me back up. Speaking in good faith entails giving the real reasons you believe something rather than a persuasive impromptu rationalization. Most people routinely do the latter without even noticing. I’m sure I still do it without noticing. But when I do notice I’m about to make something up, instead I clam up and say, “I can’t explain the reasons for this claim.” I’m not willing to disingenuously reference a scientific paper that I’d never even heard of when I formed the belief it’d be justifying, for example. In this case silence is the only feasible alternative to speaking in bad faith.

While I think there’s something to this mindset, I no longer think it makes sense to clam up when you can’t figure out how you originally came around to the view which you now hold. If you think there are other good reasons, you can give them without violating good faith.

Actually, I really wish I could draw a sharper line here. I’m essentially claiming that a little cherry-picking is OK if you’re just trying to convince someone of the view which you see as the truth, so long as you’re not intentionally hiding anything. This is an uncomfortable conclusion.

Epistemic status: confident that the views I claim are mistaken are mistaken. Less confident about best-practice claims.

6. “If you can’t provide me with a reason, I have to assume you’re wrong.”

If you take the conclusion of the previous section too far, you might reason as follows: if someone is trying to claim X, surely they’re trying to give you some evidence toward X. If they claim X and then you challenge them for evidence, they’ll try to tell you any evidence they have. So, if they come up with nothing, you have to update down, since you would have updated upwards otherwise. Right?

I think most people make this mistake due to simple conversation norms: when navigating a conversation, people have to figure out what everyone else is willing to assume, in order to make sensible statements with minimal friction. So, we look for obvious signs of whether a statement was accepted by everyone vs rejected. If someone was asked to provide a reason for a statement they made and failed to do so, that’s a fairly good signal that the statement hasn’t been accepted into the common background assumptions for the conversation. The fact that other people are likely to use this heuristic as well makes the signal even stronger. So, assertions which can’t be backed up with reasons are likely to be rejected.

This is almost the opposite mistake from the previous section; the previous one was justifications don’t matter, whereas this idea is only justifications matter.

I think something good happens when everyone in a conversation recognizes that people can believe things for good reason without being able to articulate those reasons. (This includes yourself!)

You can’t just give everyone a pass to make unjustified claims and assert that they have strong inarticulable reasons. Or rather, you can give everyone a pass to do that, but you don’t have to take them seriously when they do it. However, in environments of high intellectual trust, you can take it seriously. Indeed, applying the usual heuristic will likely cause you to update in the wrong direction.

Epistemic status: moderately confident.


I think all of this is fairly important—if you’re like me, you’ve likely made some mistakes along these lines. I also think there are many issues related to conservation of expected evidence which I still don’t fully understand, such as explanation vs rationalization, ethical injunctions and pre-rationality. Tsuyoku Naritai!