## White’s coin puzzle for imprecise probabilities

[Caveat lector: I use a whole bunch of different labels for people who prefer sharp credences versus people who prefer imprecise credences. I hope the context makes it obvious which group I’m referring to in each instance. Also, this was all written rather quickly as a way for me to get my ideas straight. So I might well have overlooked something that defuses the problems I discuss. Please do tell me if this is the case.]

On two occasions now people have told me that there’s this paper by Roger White that gives a pretty strong argument against having imprecise degrees of belief. Now, I like imprecise credence, so I felt I needed to read and refute this paper. So I sat out on my tiny balcony in a rare spell of London sunshine and I read the paper. I feel slightly uneasy about it for two different reasons. Reasons that seem to pull in different directions. First, I do think the argument is pretty good, but I don’t like the conclusion. So that’s one reason to be uneasy. The other reason is that it feels like this argument can be turned against sharp probabilists as well…

The puzzle goes like this. You don’t know whether or not the proposition “P” is true or false. Indeed, you don’t even know what proposition “P” is, but you know that either P is true or ¬P is true. I write whichever of those propositions is true on the “Heads” side of a coin, after having painted over the coin such that you can’t tell which side is heads. I write the false proposition on the tails side. I am going to flip the coin, and show you whichever side lands upwards. You know the coin is fair. Now we want to know what sort of degrees of belief it is reasonable to have in various propositions.

It seems clear that your degree of belief in the proposition “The coin will land heads” should be a half. I’m not in the business of arguing why this is so. If you disagree with that, I take that to be a reductio of your view of probability. Whatever else your degrees of belief ought to do, they ought (ceteris paribus) to make your credence in a fair coin’s landing heads 1/2.

What ought you believe about P? Well, the set up is such that you have no idea whether P. So your belief regarding P should be maximally non-committal. That is, your representor should be such that C(P)=[0,1], the whole interval. This is, I think, the strength of imprecise probabilities over point probabilities: they do better at representing total ignorance. Your information regarding P and regarding ¬P is identical, symmetric. So, if you wanted sharp probabilities, the Principle of Indifference (PI, sometimes called the Principle of Insufficient Reason) suggests that you ought to consider those propositions equally likely. That is, if you have no more reason to favour one outcome over any other, all the outcomes ought to be considered equally likely. In this case C(P)=1/2=C(¬P). In sharp probabilities, you can’t distinguish total ignorance from strong statistical evidence that the two propositions are equally likely. Consider proposition M: “the 1000th child born in the UK since 2000 is male”. We have strong statistical evidence that supports assigning this proposition equal weight to the proposition F (that that child is female). I’ll come back to that later.

So what’s the problem with imprecise probabilities according to White? Imagine that I flip the coin and the “P” side is facing upward. What degrees of belief ought you have now in the coin’s being heads up? You can’t tell whether the heads or tails face is face up, so it seems like your degree of belief should remain unchanged: 1/2. Given that you can’t see whether it’s heads or tails, you’ve learned nothing that bears on whether P is the true proposition. So it seems that your degree of belief in P should remain the same full unit interval: [0,1].

But: you know that the coin landed heads IF AND ONLY IF P is true. This suggests that your degree of belief in heads should be *the same* as your belief in P. But they are thoroughly different: 1/2 and [0,1]. So what should you do? Dilate your degree of belief in heads to [0,1]? Squish your degree of belief in P to 1/2? Neither proposal seems particularly appetising. So this is a major problem, right?*

What I want to do now is modify the problem, and try and explore intuitions about what sharp credencers should do in similar situations. First I should note that the original problem is no problem for them, since PI tells them to have C(P)=1/2 anyway, so the credences match up. But I worry about this escape clause for sharp people, since it is still the case that *the reasons* for their having 1/2 in each case are quite different, and it seems almost an accident or a coincidence that they escape…

Consider a more general game. I have N cards that have a number 1..N on one side of each. The reverse sides are identical. Now, on those reverse sides I write a proposition as before. On card number 1 I write the true proposition out of P and ¬P. On the remaining 2..N I write the false one. Again you don’t know which is which etc. Now I shuffle the cards well, pick one and place it “proposition side up” on the table. For simplicity, let’s say it says “P”. I take it as obvious that your credence in the proposition “This is card number 1” should be 1/N. What credence should you have in P? Well, PI says it should be 1/2. But it should also be 1/N, since we know that P is true IF AND ONLY IF this is card number 1.

Imagine the case where N is large, 1 million, say. I get the feeling that in this case, you would want to say that it is overwhelmingly likely that P is false: 999,999 cards have the false proposition on, so it’s really likely that one of them has been picked. So my credence in P being true should be something like 1/1,000,000. Put it the other way round. Say there’s only 1 card. Then if you see the card says “P”, that as good as tells you that P is true, so your credence should move to 1.

On the other hand, if we’re thinking about a proposition we are very confident in, say A: “Audrey Hepburn was born in Belgium” (it’s true, look it up.). Let’s say C(A)=0.999 (not 1 because of residual uncertainty regarding wikipedia’s accuracy.). Now, if we have 2 cards and the one drawn says A, that’s good reason to believe that that card is the number 1 card. So in this case, it’s the belief regarding the card that moves, not the belief regarding the proposition.

What about the same game but with a million cards? Despite my strong conviction that Audrey Heburn was, if briefly an Ixelloise (?), the chance of the card drawn being number 1 is so small that maybe that should trump my original confidence and cause me to revise down my belief in A.

Here’s another trickier case. Now imagine playing the same game with some proposition I have strong reason to believe should have credence 1/2, like M defined above. And let’s say we’re playing with only 3 cards. For simplicity, let’s imagine that M is shown. How should your credences change in this situation? Again, it seems that C(M)=C(1) is required by the set up of the game. But I’m less sure which credence should move.

In any case, is there a principled way to decide whether it’s your belief about the card or your belief about the proposition that should change. And if there isn’t, doesn’t this tell against the sharp credentist as much as against the imprecise one?

On objection you might have is that this is all going to be cleared up by applying some Bayes’ theorem in these circumstances, since in these cases (as opposed to White’s original one) see which proposition is drawn really does count as learning something. I don’t buy this, since the set up requires that your degrees of belief be *identical* in the two propositions. Updating on one given the other is going to shift the two closer together, but I don’t think that’s going to solve the problem.

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* The third option, a little squish and a little dilate to make them match seems unappealing, and I ignore it for now, since it seems to have BOTH problems that the above approaches do…

Since I wrote this, I have been made aware of Jim Joyce’s paper which totally nails the point I was vaguely gesturing towards here.

http://www-personal.umich.edu/~jjoyce/papers/dic.pdf

SeamusOctober 14, 2010 at 5:16 pm