Wolfgang Schwarz

Terry Horgan on Sleeping Beauty

I've been participating in a couple of workshops here at ANU lately, and I thought I'd share some notes. First, we had a little Sleeping Beauty workshop where Terry Horgan and Mike Titlebaum defended thirding, and me halfing. Unfortunately, I think we didn't quite get to the heart of our disagreement. Each of us said their own thing, without saying enough about what's wrong with the reasoning of the other sides. So I'll do that here. I start with Terry's account.

Terry lets Beauty reason as follows. Consider the evidence she has on Monday morning. It includes a) her memories of the setup, and b) the information that she was just awakened. From (a), Beauty can tell that a fair coin was tossed on Sunday night, but not how much time has passed since then: whether it is Monday or Tuesday or Wednesday etc. At this stage, Terry suggests, Beauty should assign the same credence to Heads & Monday, Heads & Tuesday, Tails & Monday, and Tails & Tuesday. (b) then rules out all Heads possibilities except Monday and all Tails possibilities except Monday and Tuesday. She ends up with probability 1/3 for Heads.

I have two objections. First, Terry has left out another piece of evidence, (c): Beauty remembers what she believed on Sunday evening. This is potentially relevant because rational agents should trust their previous self. Suppose you know that you believed that if the next thing you learn is E, then A will be true; and suppose now you learn E, and you know that you have not learned anything in between; then (ceteris paribus) you should think that A is true. More generally, you should follow the conditional expert principle

EP: P(A) = P(A | P'(+A|+E)=x & E) = x

where P' is your previous credence function and +A is the proposition that at the next time you gain any evidence, A will obtain (similalry for +E).

The principle is only applicable to Beauty's situation if she (assumes that she) has not learned anything relevant between Sunday evening and now. So we need a version of the story where in case of Tails, both Beauty's Monday state and her Tuesday state are naturally regarded as direct successors of her Sunday state. I think many versions of Sleeping Beauty fit that description, including the original. In these versions, EP applies, for Beauty can be certain that she has learned nothing relevant between Sunday and now. By EP, Beauty's new credence in Heads given (a) & (b) should equal her (known) previous credence in Heads given +(a) & +(b). But on Sunday, Beauty was certain of +(a) and +(b): she was certain that the next time she will learn anything, she will learn (a) and (b). Hence her new credence in Heads, which should equal her previous conditional credence in Heads given +(a) & +(b), should equal her previous unconditional credence in Heads -- which is 1/2.

My second objection doesn't depend on the details of the story, nor on the assumption that Beauty has any relevant evidence besides (a) and (b). Instead, I object to the 'prior' probabilities Terry takes to be revised by (a) and (b). In effect, Terry lets Beauty start out with a probability function that has forgotten everything Beauty has ever learned. For instance, even though Beauty already knew of the setup on Sunday, she has to re-learn it again on Monday, based on her evidence (a). We can imagine that instead she keeps knowing what she knew before, without having to reconstrue it every time from her memories. A natural generalisation of this is an 'external' version to the expert principle:

EEP: P(A) = P'(+A|+E),

where E is the total new evidence Beauty receives at the time of P.

Once again, on Sunday, Beauty's credence in Heads conditional on +(a) & +(b) was 1/2. Hence it should still be 1/2 on Monday.

(I have described Terry's position as if Beauty was conditioning on (a). Terry himself seemed closer to the view that the information about the setup isn't re-learned, but somehow carried over from Sunday. If so, he is even closer to my view.)

The downside of this second argument is that it assumes that there are constraints on rational credence that are in a sense inaccessible to the subject: Beauty should stick to her previous credence in Heads even if she has no direct evidence telling her what her previous credence was. (She does have indirect evidence if she can introspect her new credence, knows her evidence and also that she follows the external principle: these three facts substantially narrow down possibilities regarding her previous credence function.)

As David Chalmers pointed out, at this point our disagreement depends on what exactly we are asking. If we ask what Beauty's credence in Heads should be if she follows a sensible method of updating her previous beliefs in the light of new evidence, I would answer 1/2. If instead we ask what probability an ideal agent should assign to Heads given only the evidence Beauty has on Monday and nothing else to go by, then the answer might be 1/3, at least if we're dealing with versions of the story where the 'internal' EP principle doesn't apply.

The upshot is that the answer to the Sleeping Beauty problem depends on which of the two questions we ask, and what version of the story we consider. On one of the four combinations, Terry's analysis is correct, and I agree that Beauty should assign credence 1/3 to Heads. On the other three combinations, Terry's analysis leaves out a relevant fact, and the credence should be 1/2.

What's left is to explain why EP is a plausible constraint on rational credence, and why EEP captures a reasonable way to update one's beliefs. In my paper, I have tried to do this by showing that just about any argument ever offered for conditioning turns into an argument for EEP when self-locating propositions are taken into account. Another way to motivate the principles is to contrast them with some variations, such as Mike's. On which more hopefully soon.