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September 29, 2007

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We really want to know: what are the typical filters applied in particular areas of life, and thus what evidence does testimony there give us? Doctors, lawyers, parents, lovers, teachers and so on - what filters do they collectively produce on the evidence they get?

We had a related discussion my blog a little while ago - your expert input would be most welcome.

Each statement that he makes is valid evidence - how could you not update your probabilities? ... But then the clever arguer can make you believe anything he chooses, if there is a sufficient variety of signs to selectively report. That doesn't sound right.

What's being overlooked is that your priors before hearing the clever arguer are not the same as your priors if there were no clever arguer.

Consider the case if the clever arguer presents his case and it is obviously inadequate. Perhaps he refers to none of the usual signs of containing a diamond and the signs he does present seem unusual and inconclusive. (Assume all the usual idealizations, ie no question that he knows the facts and presents them in the best light, his motives are known and absolute, he's not attempting reverse psychology, etc) Wouldn't it seem to you that here is evidence that box B does not contain the diamond as he says? But if no clever arguer were involved, it would be a 50/50 chance.

So the prior that you're updating for each point the clever arguer makes starts out low. It crosses 0.5 at the point where his argument is about as strong as you would expect given a 50/50 chance of A or B.

What lowers it when CA begins speaking? You are predictively compensating for the biased updating you expect to do when you hear a biased but correct argument. (Idealizations are assumed here too. If we let CA begin speaking and then immediately stop him, this shouldn't persuade anybody that the diamond is in box A on the grounds that they're left with the low prior they start with.)

The answer is less clear when CA is not assumed to be clever. When he presents a feeble argument, is it because he can have no good argument, or because he couldn't find it? Ref "What evidence bad arguments".

So the prior that you're updating for each point the clever arguer makes starts out low. It crosses 0.5 at the point where his argument is about as strong as you would expect given a 50/50 chance of A or B.

I don't believe this is exactly correct. After all, when you're just about to start listening to the clever arguer, do you really believe that box B is almost certain not to contain the diamond? Why would you listen to him, then? Rather, when you start out, you have a spectrum of expectations for how long the clever arguer might go on - to the extent you believe box A contains the diamond, you expect box B not to have many positive portents, so you expect the clever arguer to shut up soon; to the extent you believe box B contains the diamond, you expect him to go on for a while.

The key event is when the clever arguer stops talking; until then you have a probability distribution over how long he might go on.

The quantity that slowly goes from 0.1 to 0.9 is the estimate you would have if the clever arguer suddenly stopped talking at that moment; it is not your actual probability that box B contains the diamond.

Your actual probability starts out at 0.5, rises steadily as the clever arguer talks (starting with his very first point, because that excludes the possibility he has 0 points), and then suddenly drops precipitously as soon as he says "Therefore..." (because that excludes the possibility he has more points).

I mostly concur, but I think you can (and commonly do) get some "negative" information before he stops. If CA comes out with a succession of bad arguments, then even before you know "these are all he has" you know "these are the ones he has chosen to present first".

I know that you know this, because you made a very similar point recently about creationists.

(Of course someone *might* choose to present their worst arguments first and delay the decent ones until much later. But people usually don't, which suffices.)

g, agreed.

I don't believe this is exactly correct. After all, when you're just about to start listening to the clever arguer, do you really believe that box B is almost certain not to contain the diamond?

Where do you get that A is "almost certain" from? I just said the prior probability of B was "low". I don't think that's a reasonable restatement of what I said.

Your actual probability starts out at 0.5, rises steadily as the clever arguer talks (starting with his very first point, because that excludes the possibility he has 0 points), and then suddenly drops precipitously as soon as he says "Therefore..." (because that excludes the possibility he has more points).

It doesn't seem to me that excluding the possibility that he has more points should have that effect.

Consider the case where CA is artificially restricted to raising a given number of points. By common sense, for a generous allotment this is nearly equivalent to the original situation, yet you never learn anything new about how many points he has remaining.

You can argue that CA might still stop early when his argument is feeble, and thus you learn something. However, since you've stipulated that every point raises your probability estimate, he won't stop early. To make an argument without that assumption, we can ask about a situation where he is required to raise exactly N points and assume he can easily raise "filler" points.

ISTM at every juncture in the unrestricted and the generously restricted arguments, your probability estimate should be nearly the same, excepting only that you need compensate slightly less in the restricted case.

Now, there is a certain sense of two ways of saying the same thing, raising the probability per point (presumably cogent) but lowering it as a whole in compensation.

But once you begin hearing CA's argument, you know tautologically that you are hearing his argument, barring unusual circumstances that might still cause it not to be fully presented. I see no reason to delay accounting that information.

Tom, if CA's allotment of points is generous enough that the limit makes little difference then it's no longer true that "you never learn anything new about how many points he has remaining" because he'll still stop if he runs out.

If he knows that he's addressing Eliezer and that Eliezer will lower his probability estimate when CA stops, then indeed he'll carry on until reaching the limit (if he can), but in that case what happens is that as he approaches the limit without having made any really strong arguments Eliezer will reason "if the diamond really were in box B then he'd probably be doing better than this" and lower his probability.

Suppose you meet CA, and he says "I think you should think the diamond is in box B, and here's why", and at that instant he's struck by lightning and dies. Ignoring for the sake of argument any belief you might have that liars are more likely to be smitten by the gods, it seems to me that your estimate of the probability that the diamond is in box B should be almost exactly 1/2. (Very slightly higher, perhaps, because you've ruled out the case where there's no evidence for that at all and CA is at least minimally honest.)

Therefore, your suggestion that you lower your probability estimate as soon as you know CA is going to argue his case must be wrong.

What actually happens is: after he's presented evidence A1, A2, ..., Ak, you know not only that A1, ..., Ak are true but also that those are the bits of evidence CA chose to present. And you have some idea of what he'd choose to present if the actually available evidence were of any given strength. If A1, ..., Ak are exactly as good as you'd expect given CA's prowess and perfectly balanced evidence for the diamond's location, then your probability estimate should remain at 1/2. If they're better, it should go up; if they're worse, it should go down.

Note that if you expect a profusion of evidence on each side regardless, k will have to be quite large before good evidence A1 ... Ak increases your estimate much. If that's the case, and if the evidence really does strongly favour box B, then a really clever CA will try to find a way to aggregate the evidence rather than presenting it piecemeal; so in such situations the presentation of piecemeal evidence is itself evidence against CA's claim.

G, you're raising points that I already answered.

Only in the sense that you've said things that contradict one another. You said that knowing that you're listening to CA modifies your prior estimate of P(his preferred conclusion) from the outset, and then you said that actually if you stop him speaking immediately then your prior shouldn't be modified. These can't both be right.

I don't see any way to make the "modified prior" approach work that doesn't amount to doing the same calculations you'd do with the "modified estimation of evidence provided by each point made" approach and then hacking the results back into your prior to get the right answer, and I don't see any reason for preferring the latter.

Of course, as a *practical* matter, and given the limitations of our reasoning abilities, prior-tweaking may be a useful heuristic even though it sometimes misleads. But, er, "useful heuristics that sometimes mislead" is a pretty good characterization of what's typically just called "bias" around here :-).

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