The problem with the Sleeping Beauty Problem (irony intended), is that it belongs more in the realm of philosophy and/or logic, than mathematics. The irony in that (double-irony intended), is that the supposed paradox is based on a fallacy of logic. So the people who perpetuate it should be best equipped to resolve it. Why they don’t, or can’t, I won’t speculate about.
Mathematicians, Philosophers, and Logicians all recognize how information introduced into a probability problem allows one to update the probabilities based on that information. The controversy in the Sleeping Beauty Problem is based on the fallacious conclusion that such “new” information is required to update probabilities this way. This is an example of the logical fallacy called affirming the consequent: concluding that “If A Then B” means “A is required to be true for B to be true” (an equivalent statement is “If B then A”).
All that is really needed for updating, is a change in the information. It almost always is an addition, but in the Sleeping Beauty Problem it is a removal. Sunday Sleeping Beauty (SSB) can recognize that “Tails & Awake on Monday” and “Tails & Awake on Tuesday” represent the same future (Manfred’s “AND”), both with prior probability 1⁄2. But Awakened Sleeping Beauty (ASB), who recognizes only the present, must distinguish these two outcomes as being distinct (Manfred’s “OR”). This change in information allows Bayes’ Rule to be applied in a seemingly unorthodox way: P(H&AonMO|A) = P(H&AonMO)/[P(H&AonMO) + P(T&AonMO) + P(T&AonTU)] = (1/2)/(1/2+1/2+1/2) = 1⁄3. The denominator in this expression is greater than 1 because the change (not addition) of information separates non-disjoint events into disjoint events.
The philosophical issue about SSA v. SIA (or whatever these people call them; I haven’t seen any two who define them agree), can be demonstrated by the “Cloned SB” variation. That’s where, if Tails is flipped, an independent copy of SB is created instead of two awakenings happening. Each instance of SB will experience only one “awakening,” so the separation of one prior event into two disjoint posterior events, as represented by “OR,” does not occur. But neither does “AND.” We need a new one called “ONE OF.” This way, Bayes’ Rule says P(H&Me on Mo) = P(H&Me on MO)/[P(H&Me on MO) + (ONE OF P(T&Me on MO), P(T&Me on TU))] = (1/2)/(1/2+1/2) = 1⁄2.
The only plausible controversy here is how SB should interpret herself: as one individual who might be awakened twice during the experiment, or as one of the two who might exist in it. The former leads to a credence of 1⁄3, and he latter leads to a credence of 1⁄2. But the latter does not follow from the usual problem statement.
The problem with the Sleeping Beauty Problem (irony intended), is that it belongs more in the realm of philosophy and/or logic, than mathematics. The irony in that (double-irony intended), is that the supposed paradox is based on a fallacy of logic. So the people who perpetuate it should be best equipped to resolve it. Why they don’t, or can’t, I won’t speculate about.
Mathematicians, Philosophers, and Logicians all recognize how information introduced into a probability problem allows one to update the probabilities based on that information. The controversy in the Sleeping Beauty Problem is based on the fallacious conclusion that such “new” information is required to update probabilities this way. This is an example of the logical fallacy called affirming the consequent: concluding that “If A Then B” means “A is required to be true for B to be true” (an equivalent statement is “If B then A”).
All that is really needed for updating, is a change in the information. It almost always is an addition, but in the Sleeping Beauty Problem it is a removal. Sunday Sleeping Beauty (SSB) can recognize that “Tails & Awake on Monday” and “Tails & Awake on Tuesday” represent the same future (Manfred’s “AND”), both with prior probability 1⁄2. But Awakened Sleeping Beauty (ASB), who recognizes only the present, must distinguish these two outcomes as being distinct (Manfred’s “OR”). This change in information allows Bayes’ Rule to be applied in a seemingly unorthodox way: P(H&AonMO|A) = P(H&AonMO)/[P(H&AonMO) + P(T&AonMO) + P(T&AonTU)] = (1/2)/(1/2+1/2+1/2) = 1⁄3. The denominator in this expression is greater than 1 because the change (not addition) of information separates non-disjoint events into disjoint events.
The philosophical issue about SSA v. SIA (or whatever these people call them; I haven’t seen any two who define them agree), can be demonstrated by the “Cloned SB” variation. That’s where, if Tails is flipped, an independent copy of SB is created instead of two awakenings happening. Each instance of SB will experience only one “awakening,” so the separation of one prior event into two disjoint posterior events, as represented by “OR,” does not occur. But neither does “AND.” We need a new one called “ONE OF.” This way, Bayes’ Rule says P(H&Me on Mo) = P(H&Me on MO)/[P(H&Me on MO) + (ONE OF P(T&Me on MO), P(T&Me on TU))] = (1/2)/(1/2+1/2) = 1⁄2.
The only plausible controversy here is how SB should interpret herself: as one individual who might be awakened twice during the experiment, or as one of the two who might exist in it. The former leads to a credence of 1⁄3, and he latter leads to a credence of 1⁄2. But the latter does not follow from the usual problem statement.