Absence of Evidence Is Evidence of Absence

BTW, what would you consider evidence for a genuine attempt to lull the government into a false sense of security (in an analagous situation)?

It is impossible for A and ~A to both be evidence for B. If a lack of sabotage is evidence for a fifth column, then an actual sabotage event must be evidence against a fifth column.

This is not correct.

One explanation (call it A) for why there fails to be sabotage is that the Fifth Column is trying to be sneaky and inflict maximum damage later on when no one expects it. The probability of that is greater than 0, so it is a legitimate potential explanation for the apparent absence of sabotage. But, on further thought, there is this other possible explanation (call it B): the absence of a Fifth Column will produce an absence of sabotage. The probability of this is also greater than 0.

So here we have the event (Fifth Column exists) constituting evidence for (absence of sabotage) (perhaps the probability is low, but not zero). Surely it is fair to take it for granted that ~(Fifth Column exists) also constitutes evidence for (absence of sabotage). So that's an example where an event and its negation can potentially both be evidence for something.

I think what you really mean to say is that since P(no sabotage) = P(no sabotage | Fifth Column) P(Fifth Column) + P(no sabotage | no Fifth Column) P(no Fifth Column), and since no sabotage has been observed, making P(no sabotage) = 1, this must imply that P(no sabotage | Fifth Column) P(Fifth Column) = 1 - P(no sabotage | no Fifth Column) P(no Fifth Column).

If we then make the (perhaps unwarranted) assumption that the prior probabilities are equal, i.e. P(Fifth Column) = P(no Fifth Column), then when deciding via a maximum a posterior decision rule which hypothesis to believe, we wind up with P(no sabotage | Fifth Column) = 1 - P(no sabotage | no Fifth Column), and thus we simply select the hypothesis corresponding to whichever conditional probability is larger... and from this, our intuitions about basic logic would say it doesn't make sense to assign probabilities in such a way that (no Fifth Column) is less likely to cause no sabotage than (Fifth Column), and this is what creates the effect you are noting that some event A and its negation ~A shouldn't both be evidence for the same thing.

Compactly, it's only fair to claim that A and ~A cannot both be evidence for B in some very special situations. In general, though, A and ~A definitely can both serve as supporting evidence for B, it's just that they will corroborate B to different degrees and this may or may not be further offset by the prior distributions of A and ~A.

But it is important not to assert the incorrect generalization that "It is impossible for A and ~A to both be evidence for B."

As I understand it, there were at least three hypotheses under consideration: a) No members (or a negligibly small fraction) of the ethnic group in question will make any attempt at sabotage. b) There will be attempts at sabotage by members of the ethnic group in question, but without any particular organization or coordination. c) There is a well-disciplined covert organization which is capable of making strategic decisions about when and where to commit acts of sabotage.

The prior for A was very low, and any attempt by the Japanese government to communicate with saboteurs in the States could be considered evidence against it. Lack of sabotage is evidence for C over B.

The reply that it "is impossible for A and ~A to both be evidence for B" is to ignore what Frank said in favor of insisting on the very overgeneralization I think he was trying to point out. It's not impossible at all when we are being imprecise enough about the prior expectations involved, such as when we lump all moments in a sustained effort together.

Here's an example to illustrate what I'm saying: Say you are a parent of a 10 year old boy who generally wants to stay up past his bedtime. His protests vary from occasional temper tantrums to the usual slumped-shoulders expression of disappointment that bedtime has finally arrived. Under normal circumstances, the expectation is that he would give at least some evidence of wanting to stay up later. We'll call this resistance "A," and A is evidence for "B": his desire and motivation to stay up later. What shall we say when ~A happens? That is, what shall we say when the boy one day suddenly goes enthusiastically to bed? That he has given up his desire? That it is impossible for this to be evidence of his continued desire and motivation? Of course not. It is exactly what we would expect a motivated and reasonably intelligent person to do: try different and probably more effective strategies. If we generalize the ongoing experience of the little boy's quest to say up later, A and ~A are both evidence of B. "It is impossible for A and ~A to both be evidence for B" is simply not narrow enough to be a true statement, and using it in that way can easily amount to a bad counterargument.

Rather, we need to be specific about each situation. What I think we should pay attention to here is the prior expectation of B. With a high enough prior, A and ~A could either (but not both) be evidence of B. But if we are not being specific to each precise situation, the generalization "it is impossible for A and ~A to both be evidence of B" can be a very subtle straw man, because the person being argued against may not be relying on the assumption that A and ~A are equal evidence for B at the same time and in the same situation.

Returning to the Japanese Fifth Column argument, unlike the little boy in my example the Japanese (and, in general, descendants from countries that go to war with their current country of citizenship) do not have a consistent track record of wartime sabotage. Also, there isn't any reason to think they would not generally be more loyal to their country of citizenship than the country of their parents, grandparents, or even of their own childhood. So there is no particularly strong expectation that they would commit sabotage... and thus no such expectation that some mysterious lack of sabotage is itself a sign of a new strategic attempt as part of a sustained effort. The argument should come down to the prior expectation of Japanese sabotage. That seems to be the crux of it to me.

It seems to me the weakness in Frank's argument also lies in the basic premise that we should expect the Japanese to commit sabotage. And I believe the governor would need to rely on that premise, or a similar one, in order to sustain his argument beyond what Eliezer presented.

But. The inside information premise seems nearly undefeatable to me. We can't comment on information we don't have. I think that is always a possibility with controversial official responses that most people would prefer to deny. If the person whose claims you are evaluating has secret but pertinent information you don't have access to, then it can be very difficult to offer a fair analysis. For one, you will have very different yet subjectively valid prior expectations.

I think you're right, but there's an adjustment (an update, isn't it called?) warranted in two directions.

The absence of sabotage decreases the likelihood of the fifth column existing at all.

But if there is a fifth column, it could be reasonably predicted that there would be evidence of sabotage unless there was an attempt to keep a low profile. If they were to favor this hypothesis for other reasons, as in the classified data mentioned by Frank, then the lack of apparent sabotage would also increase the probability that if the unlikely fifth column DID exist, it would be one which is keeping a low profile. I grant, of course, at the same time, the decreased probability of there being any kind of fifth column in the first place.

Can we be sure that he did not just assign a very strong prior distribution to the existence of Fifth Column? In that case, if we model Warren's decision as binary hypothesis testing with a MAP rule, say, then maybe it occurred to Warren that the raw conditional probabilities satisfied this inequality P(no sabotage | imminent Fifth Column threat) < P(no sabotage | no imminent Fifth Column threat).

But perhaps, for Warren, P(imminent Fifth Column threat) >> P( no imminent Fifth Column threat).

In this scenario, he reasoned that it was so likely that there was a Fifth Column threat that it outweighed the ease with which (absence of Fifth Column) can account for (absence of sabotage), and led him to choose the hypothesis that a Fifth Column was a better explanation for lack of sabotage.

In that case, the issue becomes the strength in the prior belief. Similar reasoning can be applied to McCarthy, or to those suggesting we're due for another terrorist attack.

I guess what I am saying is like this: maybe someone just believes we're due for another terrorist attack very strongly (perhaps for irrational reasons, but reasons that have nothing to do with a witnessed lack of terrorist attacks). Then you present them with the evidence that no terrorist activity has been witnessed, say. Instead of this updating their prior to a better posterior that assigns less belief to imminence of terrorist attacks, they actually feel capable of explaining the absence of terrorist activities in light of their strong prior.

I do agree that it would then be nonsensical to take that conclusion and treat it like a new observation. As if: Fifth Column -> they absolutely must exist and be planning something -> invent a reason why strength of belief in prior is justified -> Fifth Column's existence explains absence of sabotage -> further absence of sabotage now feeds back as ever-more-salient corroborating evidence of original prior.

Perhaps more focus should be placed on the role of the prior in all of this, rather than outright misinterpretations of evidence.

I suspect a part of the appeal of this saying comes from a mental unease with conflicting evidence.

I am quite sure you're onto something here. A similar effect occurs when people try to argue that a given intervention has no downsides at all; none at all? Really? It will be absolutely free and have beneficial effects on everyone in the world? Why aren't we doing it already then?

People aren't used to thinking in terms of cost-benefit analysis, where you say "Yes, it has downsides A, B, C; but it also has upsides W, X, Y, Z, and on balance it's a good idea." They think that merely by admitting that the downsides exist you have given up the game. (Politics is the mind-killer?)

That's a good point. And clearly court standards for evidence are not the same as Bayesian standards; in court lots of things don't count that should (like base rate probabilities), and some things count more than they should (like eyewitness testimony).

Really, the issue here is whether evidence has to increase probability (of existence or nonexistence) by a positive amount or a non-negative amount. The difference between those two sets is the very important "zero."

You are interested in the question: "Are there extra-solar planets?", with possibilities "Yes" and "No". You wonder how to answer the question, and decide to try the experiment "look with my naked eyes." You sensibly decide that if you can see any extra-solar planets, then it's not less likely that there are extra-solar planets, and if you can't see extra-solar planets, then it's not more likely that there are extra-solar planets. The strength of those effects is determined by the quality of the experiment; in this case, that strength is 0.

The specific fallacy in question is saying that all outcomes of an experiment make a claim more likely- that is inconsistent with how probability works. Similarly, one can argue that you should have a good estimate of the quality of an experiment before you get the results. That estimate doesn't have to be perfect- you can look at the results and say "I'm going to doublecheck to make sure I didn't screw up the experiment"- but changing your bet after you lose should not be allowed.

There is another way: Look really really hard with tools that would be expected to work. If you find something? Yay, your hypothesis is confirmed. If you don't? You'd better start doubting your hypothesis.

You already do this in many situations I'm sure. If someone said, "You have a million dollars!" and you looked in your pockets, your bank accounts, your stock accounts (if any), etc. and didn't find a million dollars in them (or collectively in all of them put together), you would be pretty well convinced that the million dollars you allegedly have doesn't exist. (In fact, depending on your current economic status you might have a very low prior in the first place; I know I would.)

The fact that you can't see them when you look outside is evidence against their presence, it's just extremely weak evidence. See also the Raven paradox.

OK, make a lot of fun of this. Let's take it in context. 1) It is amazing that anyone would couch their argument in a logical manner at this point in civilization at all, even if the logic is wrong, so kudos. 2) This was not a logical action (the internment). It is a complicated human action which I imagine has a lot to do with the lack of trust between the Japanese (Americans) and Americans at the time. Evidently there were no Japanese (American) or American individuals who could broker a mutual trust at this time, so sad. 3) The determinants of whatever limited trust which did exist, if known,might be available to logical analysis, but I would think they are a very complex set of statements. These statements probably reflect all the (unknown) possibilities mentioned on this thread. One logical result might have been internment. We, now, far in the future believe that internment was if not wrong, at least unnecessary. Such is hindsight

The crime rate has gone up. This means that everything is getting worse and the police are ineffective.

The crime rate has gone up. This means that the police are getting better at catching formerly clandestine criminal behavior.

Seems reasonable to me; if there's the expected amount of crime in an area, then it's not too worthy of special attention. If there's a higher than usual amount of crime, then it's clearly worthy of special attention.

However, if there's a lower than usual amount of crime, then it's also worthy of special attention, because that indicates that something odd is happening there (or, it indicates that something has genuinely reduced the amount of crime and not just the metric, which is worth investigating and hopefully replicating).

Can you create an example in which you would draw a different conclusion by your procedure than by the procedure outlined in the post?

In that case, the two cancel each other out, and both types of evidence are worthless.

You can run it through Bayes Theorom.

This is like a test that predicts an outcome 70% of the time, but has a 70% false positive rate. The end result is a coin toss that is weighted to be 70% on one side, it doesn't help you find the truth in any way.

Welcome to Less Wrong!

A belief is useless unless it makes predictions. Making Beliefs Pay Rent gives a couple examples of "beliefs" that are useless because they don't make any predictions.

A belief from which you can derive any prediction is just as useless. Your Strength as a Rationalist and the beginning of A Technical Explanation of Technical Explanation give examples of people trying too hard to make their beliefs explain their observations; they fail to discover that their beliefs are incorrect.

Warren stated in the quote that the lack of any subversive activity was the most convincing factor of all the evidence he has that the 5th Column would soon commit subversive activity.

The problem here should be pretty obvious.

As soon as any subversive activity occurs, the evidence that the 5th Column is going to commit subversive activity clearly just went down! And since the lack of evidence was the strongest evidence for this fact, the fact that the "lack of evidence" is now 0 (either evidence exists, or no evidence exists, there are no degrees for this type of evidence) makes it impossible for the 5th Column to have committed the subversive activity!

The absurdity of this reasoning should be obvious, and it should be thrown out immediately. The lack of subversive activity was clearly not evidence that the 5th Column was planning something. It could not be. You might think the 5th Column was planning something based on other evidence, and that is perfectly fine, but your reasoning for the risk of a subversive activity cannot be based on the lack of any subversive activity. It must be based on other evidence or it invalidates itself.

While I think your reading is consistent with a very generous application of the principle of charity, I'm not certain it's appropriate in this case to so apply. Do you have any evidence that Warren was reasoning in this way rather than the less-charitable version, and if so, why didn't he say so explicitly?

It really seems like the simpler explanation is fear plus poor thinking.

That is a good question, but it doesn't help Warren's reasoning.

His reasoning was not that there was a high probability that they had committed acts of subversion that were undectectible. His reasoning was that because there was no evidence of subversion, this was evidence of future subversion.

This line of reasoning invalidates itself as soon as the first evidence of subversion is discovered, since the reason subversion was imminent was because there was no evidence of subversion.

In its most simple form, Warren was saying: "Because there is no evidence that the ball is blue, the ball is blue."

Proof means "extremely strong evidence". Absence of proof and absence of evidence are both evidence of absence. Their strength is determined by the probability with which we'd expect to see them, conditional on the thing existing and not existing.

Hi DevilMaster, welcome to LessWrong!

Generally, the answer to your question is Bayes' Theorem. This theorem is essentially the mathematical formulation of how evidence ought to be weighed when testing ideas. If the wikipedia article doesn't help you much, Eliezer has written an in-depth explanation of what it is and why it works.

The specific answer to your question can be revealed by plugging into this equation, and defining "proof". We say that nothing is ever "proven" to 100% certainty, because if it were (again, according to Bayes' Theorem), no amount of new evidence against it could ever refute it. So "proof" should be interpreted as "really, really likely". You can pick a number like "99.9% certain" if you like. But your best bet is to scrap the notion of absolute "proof" and start thinking in likelihoods.

You'll notice that an integral part of Bayes' Theorem is the idea of how strongly we would expect to see a certain piece of evidence. If the Hypothesis A is true, how likely is it that we'll see Evidence B? And additionally, how likely would it be to see Evidence B regardless of Hypothesis A?

For a piece of evidence to be strong, it has to be something that we would expect to see with much greater probability if a hypothesis is true than if it is false. Otherwise there's a good chance it's a fluke. Furthermore, if that evidence is something that we wouldn't expect to see much either way, than it's not very informative when we don't see it.

So you see how this bears on your examples. I'm not especially familiar with astronomy, so I don't know whether it's true that we haven't seen other galaxies with planets, or how powerful our telescopes are. But let's assume that what you've said is all true.

If we know our telescopes aren't powerful enough to see other planets, then the fact that they don't see any is virtually zero evidence. The probability of us seeing other planets is basically the same whether they're out there or not (because we won't see them either way), so our inability to see them doesn't count as evidence at all. This test doesn't actually tell us anything because we already know that it will tell us the same thing either way. It's like counting how many fingers you have to determine if the stock market will go up or down. You're gonna get "ten" no matter what, and this tells you nothing about the market.

The same reasoning applies to the bacteria example. If we're not more likely to see them given that they're real than we are given that they're not real, then our inability to see them is not evidence in either direction. The test is a bad one because it fails to distinguish one possibility from the other.

But all this isn't to say that it would be valid to reject these notions based on the absence of these evidences alone. There may be other tests we can run that would be more likely to come out one way or the other based on whether the hypothesis is true. So no, it wouldn't make sense to reject the existence of planets or bacteria, because in both of your examples people are using tests that are known to be useless.

The simple answer is that absence of proof towards a possibility is not proof that that the possibility cannot exist, merely that there is no actual proof either way. However, in this specific case, the absence of evidence pointing towards the existence of a fifth column that is engaging in sabotage is evidence that indicates that the fifth column does not exist. I agree that the specific terminology is a bit confusing, but that is the simple explanation as to your question.

If absence of proof is not proof of absence, but absence of evidence is evidence of absence, what makes proof different from evidence?

Proof is absolute, evidence is probabilistic.

Example: we currently have no evidence supporting the existence of planets orbiting stars in other galaxies, because our telescopes are not powerful enough to observe them. Should we take this as evidence that no galaxy except ours has planets around its stars?

No, absence of evidence is not evidence of absence if evidence is impossible, but it is evidence of absence if evidence is possible but absent.

(try saying that quickly 3 times :)

Benelliot and others have explained this well, but note that we do have direct evidence for planets in other galaxies. We've had it for about two years.

we currently have no evidence supporting the existence of planets orbiting stars in other galaxies, because our telescopes are not powerful enough to observe them. Should we take this as evidence that no galaxy except ours has planets around its stars?

Yes we do. We have evidence about how physics (ie. gravity) works and about the formation phases of the universe. That earth and the other planets here exists is evidence. We just didn't happen to have one particular kind of evidence (seeing them). And no, until we developed (recently) the ability to see evidence of them ourselves you would not have been entitled to that piece of evidence either. Because we should not have expected to see them. Seeing planets with tech that should not see them would have been evidence that something else was wrong.

Planets capable of interstellar travel form an accord that treats less advanced civilizations as nature preserves and agree not to contact or help them.

Sounds similar to the Federation's Prime Directive in Star Trek.

Most rationalists will likely agree that we would trade slightly more homogenized art and culture for cures to disease and death

I'm uncomfortable with the resemblance of this to an argument by definition. It also ignores the more reasonable view that "slightly more homogenized art and culture" isn't usually the worst consequence of more a powerful (I won't say "advanced") society trying to "help" a less powerful one. (Not that that increases the prior probability of space aliens.)

In the situation you describe, the settlement is weak evidence for the product not working. Weak evidence is still evidence. The flaw in "Absence of evidence is evidence of absence," is that the saying omits the detailed description of how to correctly weight the evidence, but this omission does not make the simple statement untrue.

if I have two hypotheses that both explain an "absence of evidence" occurrence equally well, then that occurrence does not give me reason to favor either hypothesis and is not "evidence of absence."

This statement is technically true, but not in the way you're using it.

Suppose Vibrams had been around for a thousand years. For a thousand years, people had been challenging their claims to health benefits in court. For a thousand years, time and again, Vibrams had been unable to credibly defend their claims. Would that make you any more skeptical of the claims in question, at least a little bit? If the answer is "yes", you are agreeing that some very large number of such events constitutes evidence against Vibrams. I don't see any way around concluding, from there, that at least one individual instance provides some nonzero amount of evidence - perhaps very small, but not zero.

"Vibrams work, but the effect is small and/or the experiment was shoddy" and "Vibrams don't work" explain the outcome nearly equally well. They cannot explain it precisely equally well: the first hypothesis would assign a higher P(claims defended) than the second, because even small effects are sometimes correctly detected, and even shoddy experiments sometimes aren't fatally flawed. So the second necessarily has a higher P(~claims defended) than the first. This difference is precisely the thing that makes (~claims defended) evidence for the second hypothesis.

Evidence is not proof. Depending on the ratios involved, it may constitute very weak evidence, sometimes weak enough that it's not even worth tracking for mere humans: a .0001% shift is lost in the noise when people aren't even calibrated to the nearest 10%.

If you have two hypotheses that both explain an "absence of evidence" precisely equally well, then you're looking at something completely uncorrelated: trying to deduce the existence of a Fifth Column from the result of a coin flip. And if they explain it only nearly, but not exactly equally well, then you have evidence of absence - although maybe not very much, and maybe not enough to actually push you into the other camp.

The idea of Bayesianism is that you think in terms of probability instead of true and false.

From the OP:

The absence of an observation may be strong evidence of absence or very weak evidence of absence, depending on how likely the cause is to produce the observation.

So, yes, absence of evidence can convincingly disprove a theory in some cases (although, as ChristianKI points out, Bayesians typically do not assign probabilities of 0 or 1 to any theory).

I found what you're writing very self-contradicting. If "Observing nothing carries no information", then you should not be able to use it to update belief. Any belief must be updated based on new information, I would say observing nothing carries the information that the action (sabotage) which your belief predicts to happen did not happen during the observation interval.