## The Sleeping Beauty Problem

I just came back from the DICE2014 conference and as I recover from the jet lag and prepare posts about the conference I'll present the last topic in the statistics mini-series, the sleeping beauty problem. Unlike the Monty Hall problem, there is no consensus on the right solution even among the experts which makes this problem that much more interesting.

So here is the setting: Sleeping Beauty participates in an experiment. Every day she is explained the process and she is asked about the the credence (degree of belief) that a certain fair coin landed heads or tails. So what is the big deal you may ask? The coin is fair and this means that it lands heads 50% of the time, and tails 50% of the time. However, there is a clever catch.

Whenever the sleeping beauty is put to sleep she takes an amnesia drug which erase all her prior memory. If the coins lands tails, she will be woken up Monday and Tuesday but if the coins lands heads, she will be woken up only Monday. On Wednesday the experiment ends.

So now for the majority opinion: the thirders:

To make this very clear, let's change the experiment and keep waking up the Sleeping Beauty a million time if the coin lands tails, and only once if the coin lands heads. If she is woken up on a day at random, the chances are really small that she hit the jackpot and was woken up in the one and only Monday. So being woken up more times when the coin lands tails, means that the in the original problem the credence the coin landed heads should be one third. If you play this game many times and attach a payout for the correct guess, you maximize the payout overall if your credence is one third.

Now for the opposing minority point of view: the halfers

On Sunday, the credence is obviously 50-50 because the coin is fair. Even the thirders agree with this. However, From Sunday to Monday, no new information is gained, therefore the credence should be unchanged and the overall credence should remain 50% throughout the experiment. If you adopt the thirder position you should explain how does the credence change if no new information is injected into the problem.

So which position would you take? There were all sorts of approaches to convince the other side, but no-one had succeeded so far.

1. The sleeping beauty problem is ambiguous because it does not say what sample space she is using. Probabilities are defined on a per sample space basis. The sample space of the coin toss is {H,T} and the sample space for the questions about the coin state is {MH,MT,UT} where H=heads, T=tails, M=Mondays and U=Tuesdays. The probability of heads for the first sample space is 1/2 and the probability of heads for the second sample space is 1/3, since they are both equiprobable sample spaces. To see equiprobability, just notice that out of every 1000 coin tosses about 500 will be heads, 500 will be tails, and about 1500 questions will be asked about 500 which will occur when it is Monday and heads, another 500 which will occur when it is Monday and tails, and the remaining 500 which will occur when it is Tuesday and tails.

Now she knows the proposition is true one out of every three times she is asked and she is not going to mistake that for the fact that the coin comes up heads one out of every two times during coin tossing. So, for the proposition "the coin landed heads" the frequency of this proposition being true during repeated questioning in many repetitions of the experiment is different than the frequency of it being true during repeated coin tosses. If the coin toss actually came up heads then the proposition "the coin landed heads" is true but if the coin toss actually came up tails then the proposition "the coin landed heads" is false. How often the proposition is true or not depends on the circumstances. So adding two different prepositional phrases onto the original question highlights the ambiguity of that question:

(case 1)
What is your belief now for the proposition that "the coin landed heads" in the case of repeated questioning in repetitions of the experiment?

(case 2)
What is your belief now for the proposition that "the coin landed heads" in the case of repeated tosses of the coin?

The conclusion: the sleeping beauty problem is ambiguous because case 1 and case 2 use different sample spaces and if one removes the phrase "questioning in repetitions of the experiment" from case 1 and removes the phrase "tosses of the coin" from case 2 then the ambiguity of the original question is exposed.

1. Interesting argument, but I don't think it works. The problem asks for credences, not probabilities. If the problem would ask for probabilities then the answer would clearly be 50% because the coin is fair. But by asking for degree of belief, the setting is Bayesian. And in this setting, the context is unique: this is what "me, myself, and I" believe, regardless of the sample space because there is only one me.

By the way as you probably know, Lubos did a Sleeping beauty post recently. I disagree with his position on Gobbles' quote "a lie repeated 100 times becomes the truth". I agree with Lubos that repeating a lie 100 times does not make it become truth, but that was not the point of Gobbles. Mass brainwashing does work as shown in countless example. People in the Russian gulag cried and were genuinely saddened when Stalin died, although he sent them there in the first place.