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Tyrrell_McAllister comments on What Bayesianism taught me - Less Wrong
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Is it because you deny that P(H | E) > P(H) in this case? Or do you acknowledge that P(H | ~E) < P(H) is true in this case, but you don't interpret it as meaning "the fact that I haven't seen a tiger in my trashcan is evidence against the existence of tigers."
If you deny that P(H | E) > P(H), this might be because your implicit prior knowledge already screens off E from H. Perhaps we should, following Jaynes, always keep track of your prior knowledge X. Then we should rewrite P(H | E) > P(H) as P(H | E & X) > P(H | X). But if your prior knowledge already includes, say, seeing tigers at the zoo, then the additional experience of seeing a tiger in your trashcan may not make tigers any more likely to exist. That is, you could have that P(H | E & X) = P(H | X).
In that case, if you've already seen tigers at the zoo, then their absence from your trashcan does not count as evidence against their existence.
In this case I don't think P(H | ~E) < P(H) applies.
/me looks into the socks drawer, doesn't find any tigers
/me adjusts downwards the possibility of tigers existing
/me looks into the dishwasher, doesn't find any tigers
/me further adjusts downwards the possibility of tigers existing
/me looks into the fridge, doesn't find any tigers
...
You get the idea.
Sorry, I think that I was editing my comment after you replied. (I have no excuse. I think what happened was that I was going to make a quick typofix, but the edit grew longer, and by the end I'd forgotten that I had already submitted the comment.)
How do you react to my conjecture that your background knowledge screens off (or seems to) the experience of seeing a tiger in your trashcan from the hypothesis that tigers exist?
I don't think screening off helps with the underlying problem.
Let's recall where we started. I commented on the expression "absence of evidence is evidence of absence" by saying "Only provided you have looked, and looked in the right place."
The first part should be fairly uncontroversial. If you don't look you don't get any new evidence, so there's no reason to update your beliefs.
Now, the second part, "the right place". In this thread Wes_W gives a numerical example that involves searching for tigers in houses and says that you need to search about 5 billion houses to drop your confidence to 90% -- and if you search a trillion houses and still don't find a tiger, "then you'd be insane to still claim that tigers probably do exist."
Well, let's take this example as given but change one little thing. Let's say I'm not looking for tigers -- instead, I heard that there are two big rocks, Phobos and Deimos, and I'm looking for evidence of their existence.
I search a house and I don't find them. I search 5 billion houses and I don't find them. I search a trillion houses and still don't find them. At this point would I be insane to believe Phobos and Deimos exist?
That is the issue of "looking in the right place".
I agree that the "looking" part is important: Looking and not finding evidence is a different kind of "absence of evidence" than just not looking.
I think it would indeed be pretty silly to maintain that a) they exist and b) each house has an independent 10^-9 chance of containing them, after searching a trillion houses and finding neither. But if you didn't place much credence in anything like b) in the first place, your confidence in a) may not be meaningfully altered. If you already thought Phobos and Deimos were moons of Mars, then you would have extremely minimal evidence against their existence. But again, we can construct a Paradox of the Heap-type setup where you search the solar system, one household-volume at a time, and if all of them come up empty you should end up thinking Phobos and Deimos probably aren't real, so each individual household-volume must be some degree of evidence.
My thought here - and perhaps we agree on this, in which case I'm happy to concede the point - is that the need to look in the right place is technically already covered by the relevant math, specifically by the different strengths of evidence. But for us puny humans that are doing this without explicit numerical estimates, and who aren't well-calibrated to nine significant figures, it's a good rule of thumb.
(This comment has been edited multiple times. My apologies for any confusion.)
Meaningfully? I thought we were counting infintesimals :-D
If we are talking about "meaningfully altered" (or what I'd call "detectable") then not finding a tiger in my rubbish bin does not meaningfully alter my beliefs and the absence of evidence is NOT evidence of absence.
I am not sure of that. First, we're concerned with statistics, not math (and I think this is a serious difference). Second, I haven't thought this through, but I suspect a big issue here is what exactly your belief is. To give a quick example, when you don't find a tiger in your garbage, is the tiger live and real or plush and a toy? When you're unsure about the existence of something, your idea of what exactly that something is can be fuzzy and that affects what kind of evidence you'll accept and where will you look for it.
As in "for most practical purposes, and with human computational abilities, this is no update at all". I'm not sure we can usefully say this isn't really evidence after all, or we run into Paradox of the Heap problems.
Let me give an example where I think "absence of evidence is evidence of absence" is applicable, even though I'm not sure anyone has ever looked in the right place: Bigfoot.
Bigfoot moves around. It is possible that all of our searches happen to have missed it, like the one-volume-at-a-time search mentioned above.
We don't really know much about Bigfoot, so it's hard to be sure if we've been looking in the right place. Nor are we quite sure what we're looking for.
And any individual hike through the woods has a very, very small chance of encountering Bigfoot, even if it does exist, so any looking that has happened by accident won't be especially rigorous.
Nevertheless, if Bigfoot DID exist, we would expect there to be some good photographs by now. No individual instance of not finding evidence for Bigfoot is particularly significant, but all of the searches combined failing to produce any good evidence for Bigfoot makes me reasonably confident that Bigfoot doesn't exist, and every year of continued non-findings would drive that down a little more, if I cared enough to keep track.
Similar reasoning is useful for, say, UFOs and the power of prayer. In both cases, it is plausible that none of our evidence is really "looking in the right place" (because aliens might have arbitrarily good evasion capabilities [although beware of Giant Cheesecake Fallacy], because s/he who demands a miracle shall not receive one and running a study on prayer is like demanding a miracle, etc), but the dearth of positive evidence is pretty important evidence of absence, and justifies low confidence in those claims until/unless some strong positive evidence shows up.
Oh, of course there are situation where "absence of evidence is evidence of absence" is applicable.
For a very simple example, consider belief in my second head. The absence of evidence that I have a second head is for me excellent evidence that I do not, in fact, have a second head.
The discussion is really about whether AoE=EoA is universal.
The second half of the sentence was the reason I was bringing it up in this context. We've looked, kinda, and not very systematically, and maybe not in the right places, but haven't found any evidence. Is it fair to call this evidence against paranormal claims?
It's complicated, I don't think this problem can be resolved in one or two sentences.
For example, there is clear relationship to how specific the claim/belief is. Lack of evidence is more important for very specific and easily testable claims ("I can bend this very spoon in front of your eyes") and less important for general claims ("some people can occasionally perform telekinesis").
Oh, and there's lot of evidence for paranormal claims. It's just that this evidence is contested. Some of it has been conclusively debunked, but not all.
Well, you'd do better to search all of those volumes at once. Doing it one volume at a time has a significant chance of failing to find the moons even if they exist, since the moons move over time, and therefore failing to find them isn't significant evidence of their nonexistence.
But that's largely orthogonal to your point.
Kinda hard to do, but more to the point, the assumption that a single search is sufficient (= nothing changes with time) may not be true.
In fact, if you want to update your beliefs with absence of evidence, then every time your glance sweeps across a volume of space which physically could hold a tiger you need to update your beliefs about non-existence of tigers.
And then you get into more trouble because if your beliefs in (non)existence of tigers are time-specific, as they should be, the evidence from the previous second might not be relevant to the (non)existence of tigers in the next second. You need specific assumptions about persistence of entities like tigers on certain time scales (e.g. tigers don't persist on the time scale where the unit of time is a billion years).
(nods) Systems that don't assign very low priors to such "evasive" events can easily wind up incorrigibly believing falsehoods, even if they process evidence properly.
Suppose the chance of finding a tiger somewhere in a given household, on a given day, is one in a billion. Or so say the pro-tigerians. The tiger denialist faction, of course, claims that statistic is made-up, and tigers don't actually exist. But one household in a trillion might hallucinate a tiger, on any given day.
Today, you search your entire house - the dishwasher AND the fridge AND the trashcan etc.
P(You find a tiger|tigers exist) = .000000001
P(You don't find a tiger|tigers don't exist) = .000000000001
P(You don't find a tiger|tigers exist) = .999999999
P(You don't find a tiger|tigers don't exist) = .999999999999
And suppose you are 99.9% confident that tigers exist - you think you could make statements like that a thousand times in a row, and be wrong only once. (Perhaps rattling off all the animals you know.) Your prior odds ratio is 999 to 1. So you take your prior odds, (.999/.001) and multiply by the likelihood ratio, (.999999999/.999999999999), to get a posterior odds ratio of 998.999999002 to 1. This is, clearly, a VERY small adjustment.
What if you search more households: how many would you have to search, without finding a tiger, before you dropped just to 90% confidence in tigers, where you still think tigers exist but would not willingly bet your life on it? If I've done the math right, about five billion. There probably aren't that many households in the world, so searching every house would be insufficient to get you down to just 90% confidence, much less 10% or whatever threshold you'd like to use for "tigers probably don't exist".
(And my one-in-a-billion figure is probably far too high, and so searching every household in the world should get you even less adjustment...)
But if you could search a trillion houses at those odds, and still never found a tiger - then you'd be insane to still claim that tigers probably do exist.
And if a trillion searches can produce such a shift, then each individual search can't produce no evidence. Just very little.
I've posted a comment that answers you here