If she is paid for each correct guess, for example, she’ll say that she thinks the coin came up tails (this way she gets $2 half the time instead of $1 half the time for heads). If she’s paid only on Monday, she’s indifferent between the options, as she should be.
This clicked so hard it almost hurt. Indeed Bayesians should be willing to bet on their beliefs; so the rational belief depends on how specifically the bet is resolved. In other words, what specifically happens to the Sleeping Beauty based on her beliefs? (And if the beliefs have absolutely no consequence, what’s the point of getting them right?)
the rational belief depends on how specifically the bet is resolved
No. Bayesian prescribes believing things in proportion to their likelihood of being true, given the evidence observed; it has nothing to do with the consequences of those beliefs for the believer. Offering odds cannot change the way the coin landed. If I expect a net benefit of a million utilons for opining that the Republicans will win the next election, I will express that opinion, regardless of whether I believe it or not; I will not change my expectations about the electoral outcome.
There is probability 0.5 that she will be woken once and probability 0.5 that she will be woken twice. If the coin comes up tails she will be woken twice and will receive two payouts for correct guesses. It is therefore in her interests to guess that the coin came up tails when her true belief is that P(T)=0.5; it is equivalent to offering a larger payout for guessing tails correctly than for guessing heads correctly.
This clicked so hard it almost hurt. Indeed Bayesians should be willing to bet on their beliefs; so the rational belief depends on how specifically the bet is resolved. In other words, what specifically happens to the Sleeping Beauty based on her beliefs? (And if the beliefs have absolutely no consequence, what’s the point of getting them right?)
See also.
No. Bayesian prescribes believing things in proportion to their likelihood of being true, given the evidence observed; it has nothing to do with the consequences of those beliefs for the believer. Offering odds cannot change the way the coin landed. If I expect a net benefit of a million utilons for opining that the Republicans will win the next election, I will express that opinion, regardless of whether I believe it or not; I will not change my expectations about the electoral outcome.
There is probability 0.5 that she will be woken once and probability 0.5 that she will be woken twice. If the coin comes up tails she will be woken twice and will receive two payouts for correct guesses. It is therefore in her interests to guess that the coin came up tails when her true belief is that P(T)=0.5; it is equivalent to offering a larger payout for guessing tails correctly than for guessing heads correctly.