Sleeping Beauty and the Red Dice

In response to my previous article on the Sleeping Beauty Problem, I got this comment from a reader:

The late great philosopher David Lewis was a halfer. I'd be interested in any reactions to his paper on it: http://fitelson.org/probability/lewis_sb.pdf

The context of the paper is a disagreement between Lewis and Adam Elga; specifically, Lewis's paper is a response to Elga's paper "Self-locating belief and the Sleeping Beauty Problem".

Elga presents the Sleeping Beauty problem like this:

Some researchers are going to put you to sleep. During the two days that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you to back to sleep with a drug that makes you forget that waking. [Just after you are] awakened, to what degree ought you believe that the outcome of the coin toss is Heads?

And then he states the two most common responses to the problem

First answer: 1/2, of course! Initially you were certain that the coin was fair, and so initially your credence in the coin’s landing Heads was 1/2. Upon being awakened, you receive no new information (you knew all along that you would be awakened). So your credence in the coin’s landing Heads ought to remain 1/2. 

Second answer: 1/3, of course! Imagine the experiment repeated many times. Then in the long run, about 1/3 of the wakings would be Heads-wakings — wakings that happen on trials in which the coin lands Heads. So on any particular waking, you should have credence 1/3 that that waking is a Heads-waking, and hence have credence 1/3 in the coin’s landing Heads on that trial. This consideration remains in force in the present circumstance, in which the experiment is performed just once.

In his Section 2, Elga then proves that the correct answer is 1/3.  His proof is correct (although there are a few spots where it would be helpful to fill in some intermediate steps).  So Lewis is wrong to reject this proof.

But Elga's Section 3 introduces some confusion around the meaning of "information".  Elga says:

Let H be the proposition that the outcome of the coin toss is Heads. Before being put to sleep, your credence in H was 1/2. I’ve just argued that when you are awakened on Monday, that credence ought to change to 1/3. This belief change is unusual. It is not the result of your receiving new information — you were already certain that you would be awakened on Monday.

And then in a footnote:

To say that an agent receives new information (as I shall use that expression) is to say that the agent receives evidence that rules out possible worlds not already ruled out by her previous evidence.

This is where Elga and I disagree.  I would say that an agent receives information if they receive evidence that is not equally likely in all possible worlds.  In that case, the evidence should cause the agent to change their credences (subjective beliefs) about at least some possible worlds.

In particular (as I explained in my previous article), when Sleeping Beauty is awakened, she observes an event, awakening, that is twice as likely under T (the proposition that the coin toss is Heads) than under H, and she should change her credences accordingly.

So in my solution, her belief change is not unusual; it is an application of Bayes's Theorem that is only remarkable because it is not immediately obvious what the evidence is and what its likelihood is under the two hypotheses.  In that sense, it is similar to the Elvis Problem.

In the rest of Section 3, Elga tries to reconcile the seemingly contradictory conclusions that Beauty receives no new information and Beauty should change her credences.  I think this argument addresses a non-problem, because Beauty does receive information that justifies her change in credences.  So I agree with Lewis that Elga is wrong to conclude that the Sleeping Beauty problem raises, "a new question about how a rational agent ought to update her beliefs over time".

In summary:

1) Lewis is wrong about the answer to the problem and wrong to reject Elga's proof,

2) Also, his claim that Beauty does not receive information is wrong.

3) However, he is right to reject the argument in Elga's Section 3.


The Red Dice

At this point, we have three arguments to support the "thirder" position:

1) The argument based on long-run frequencies (I quoted Elga's version above).

2) The argument based on the principle of indifference (Elga's section 2).

3) The argument based on Bayes's theorem (in my previous article).

But if you still find it hard to believe that Beauty gets information when she wakes up, the Red Dice problem might help.  I wrote about several versions of it in this previous article:

Suppose I have a six-sided die that is mostly red -- that is, red on 4 sides and blue on 2 -- and another that is mostly blue -- that is, blue on 4 sides and red on 2. 

I choose a die at random (with equal probability) and roll it.  If it comes up red, I tell you "it came up red".  Otherwise, I put the die back, choose again, and roll again.  I repeat until the outcome is red.

If I follow this procedure and eventually report that the die came up red, what is the probability that the last die I rolled is mostly red?

A halfer might claim (incorrectly) that you have received no relevant information about the die because the outcome was inevitable, eventually.  The evidence you receive when I tell you the outcome is red is identical regardless of which die it was, so it should not change your credences.

A thirder would respond (correctly) that the outcome you observed is twice as likely if the die is mostly red, and therefore it provides evidence in favor of the hypothesis that it is mostly red.  Specifically, the posterior probability is 2/3.

If you don't believe this answer, you can see a more careful explanation and a demonstration by simulation in this Jupyter notebook (see Scenario C).

The Red Dice problem suggests that we should be skeptical of an argument with the form "The observation was inevitable under all hypotheses, and therefore we received no information."  If an event happens once under H and twice under T, it is inevitable under both; nevertheless, a random observation of the event is twice as likely under T, and therefore provides evidence in favor of T.