The Sleeping Beauty Problem (SBP) is a puzzle in probability theory and on self-locating belief.

From the wiki on
The Sleeping Beauty Problem

The paradox imagines that Sleeping Beauty volunteers to undergo the following experiment. On Sunday she is given a drug that sends her to sleep. A fair coin is then tossed just once in the course of the experiment to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and then the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday, given a second dose of the sleeping drug, and awakened and interviewed again on Tuesday. The experiment then ends on Tuesday, without flipping the coin again. The sleeping drug induces a mild amnesia, so that she cannot remember any previous awakenings during the course of the experiment (if any). During the experiment, she has no access to anything that would give a clue as to the day of the week. However, she knows all the details of the experiment.

Each interview consists of one question, “What is your credence now for the proposition that our coin landed heads?”

Contradictory solutions:

This problem is considered paradoxical because the answer is often given as either 1/3 or 1/2.

It may appear that the different probabilities, as determined by Beauty and the experimenter, are due to their different levels of knowledge. This is not so. Beauty’s amnesia and ignorance of the day of the week are irrelevant. Even her waking up is irrelevant. The only factor that is relevant to the different probability calculations is sampling. If Beauty knew whether it was Monday or Tuesday then she would give the following odds. If it were Monday then the odds would be 1/2. If Tuesday then the odds would be 100% tails. Putting the two cases together we still get 1/3 chance of heads.

Which is the correct solution?

In this paper by Nick Bostrom , a hybrid solution is proposed.

From the abstract:

Opinion on this problem is split between two camps, those who defend the “1/2 view” and those who advocate the “1/3 view”. I argue that both these positions are mistaken. Instead, I propose a new “hybrid” model, which avoids the faults of the standard views while retaining their attractive properties.

The Self-Indication Assumption:

That assumption states that each observer should regard her own existence as evidence supporting hypotheses that imply the existence of a greater total population of observers in the world, the degree of support being proportional to the implied (expected) number of observers.

And the presumptuous philosopher:

It is the year 2100 and physicists have narrowed down the search for a theory of everything to only two remaining plausible candidate theories, T1 and T2 (using considerations from super-duper symmetry). According to T1 the world is very, very big but finite and there are a total of a trillion trillion observers in the cosmos. According to T2, the world is very, very, very big but finite and there are a trillion trillion trillion observers. The super-duper symmetry considerations are indifferent between these two theories. Physicists are preparing a simple experiment that will falsify one of the theories. Enter the presumptuous philosopher: “Hey guys, it is completely unnecessary for you to do the experiment, because I can already show to you that T2 is about a trillion times more likely to be true than T1!” (Whereupon the presumptuous philosopher explains the Self-Indication Assumption.)

Predicament in cosmology:

It is worth noting that the situation described in this modified version of Presumptuous Philosopher is by no means a farfetched possibility. Contemporary cosmologists face essentially that predicament. They are trying to determine whether the universe is finite or infinite.

If the universe is infinite then with probability one there are an infinite number of agent-moments in states subjectively indistinguishable to your current one. Therefore, if Elga’s 1/3 view is correct, we could conclude that we already have “infinitely strong” evidence that the universe is infinite.

Even though my field is statistics and not philosophy I was curious enough about the 1/3rd to look at the paper of Nick Bostrom summarizing the 1/3rd [erronous] proof.  On its surface it would seem to be intuitively wrong to claim beauty can do anything but say 1/2, but intuition has been known to fail people even myself.  Nevertheless,  this is an error: “Given that the Monday and the Tuesday awakening would be evidentially indistinguishable, we have P(T1) = P(T2) [By an indifference principle]”. 

This is not true, because to compare the probability of two events you are supposing in that notation that someone can distinguish between them, and just because the observer cannot does not mean that the observer should claim the two events are equally likely for the person that can.  That notation says I have an event: a set, T1.  And another event: a set, T2.  And someone is claiming that if you cannot tell these two sets apart, then you must say they are truly equally likely to occur.  This is wrong!

qutsemnie - 27 July 2010 02:44 PM

Even though my field is statistics and not philosophy I was curious enough about the 1/3rd to look at the paper of Nick Bostrom summarizing the 1/3rd [erronous] proof.  On its surface it would seem to be intuitively wrong to claim beauty can do anything but say 1/2, but intuition has been known to fail people even myself.  Nevertheless,  this is an error: “Given that the Monday and the Tuesday awakening would be evidentially indistinguishable, we have P(T1) = P(T2) [By an indifference principle]”. 

This is not true, because to compare the probability of two events you are supposing in that notation that someone can distinguish between them, and just because the observer cannot does not mean that the observer should claim the two events are equally likely for the person that can.  That notation says I have an event: a set, T1.  And another event: a set, T2.  And someone is claiming that if you cannot tell these two sets apart, then you must say they are truly equally likely to occur.  This is wrong!

That is the problem of self locating belief. From the paper by NIck Bostrom:

Sleeping Beauty is an example of a problem involving self-locating beliefs, i.e., beliefs that an agent, or a temporal part of an agent, might have about its own location. An agent-part that knew exactly which possible world is actual can still be ignorant about its own location in that world. That can happen if the world contains two or more agent-parts whose evidential states are subjectively indistinguishable. These agent-parts would then be unable to determine with certainty their own spatiotemporal location. (Even if Beauty knew that the outcome of the coin toss would be tails, she could not know whether it was currently Monday or Tuesday.)

Hence P(T1) = P(T2) [By an indifference principle]

Also, from the same paper

The Sleeping Beauty problem is but one piece of the larger puzzle of how to relate indexical to non-indexical information in our reasoning. If one studies the problem in isolation from this wider context, one risks coming up with answers and principles that do not fit with the other parts of the puzzle.

From the wiki on the SBP:

The core issue is whether Beauty is asked to estimate the probability of a certain outcome of the coin toss (which is assumed to be uniformly distributed) or the probability that a certain interview is a consequence of such a toss (which is skewed to the same extent as the number of samples).

This also applies to the experimenter, provided they are asked the question at the same time as Beauty. The reason why the experimenter’s answer is given as 1/2 above is because it is tacitly assumed that the experimenter is being asked the question before the experiment or after it is over, but not during. This is the key difference that decides whether the probability of heads is 1/2 or 1/3. It is purely a matter of sampling. The answer of 1/3 arises simply because we sample twice as much on the tails branch.

That is the philosophical conundrum. It appears that, from different locations, the probabilities are not the same. Is a hybrid solution the only way out of the conundrum?