In "Sleeping Beauty and Self-Location: A Hybrid Model", Bostrom comes up with a way to have Beauty believe that P(Tails) = 1/2, both before and after learning that it's Monday. The idea is to treat the situation as five separate possible cases: three cases before learning it's Monday (Heads^Mon_1, Tails^Mon_1, Tails^Tue_1) and then two cases after learning it's Monday (Heads^Mon_2, Tails^Mon_2). Once Beauty learns that it's Monday she's now in a new reference class, and since each world contributes exactly one observer moment, she ends up back at P(Tails) = 1/2. Here, "learning" doesn't mean narrowing the set of possible cases we conditionalize upon, as in standard Bayesian conditionalization, but dropping the model and replacing it with a new one having different measures. Note that there's no situation in which she'd be unaware of whether she's in the before or after learning cases, since she'd always know whether she has or has not learned that today is Monday, so treating them as being in the same probability space or not makes no difference.
This move is similar to the introduction of reference classes: a new variable is injected into the equation whose value depends on philosophical interpretation rather than well defined rules, and this variable fully determines the probability of each possible world. For instance, including her Heads^Tue (aspeep) state in the reference class makes her a thirder, while excluding it makes her a halfer.
So on the one hand we can fully determine the probabilities of each possible world by playing with reference classes, and on the other, we can decide whether to use conditionalization or instead replace the model with a new one by deciding when to embed learning steps into the model. Admitting these devices in probability theory would end up rendering the whole thing pretty meaningless insofar as it is a system of rules.
In "Sleeping Beauty and Self-Location: A Hybrid Model", Bostrom comes up with a way to have Beauty believe that P(Tails) = 1/2, both before and after learning that it's Monday. The idea is to treat the situation as five separate possible cases: three cases before learning it's Monday (Heads^Mon_1, Tails^Mon_1, Tails^Tue_1) and then two cases after learning it's Monday (Heads^Mon_2, Tails^Mon_2). Once Beauty learns that it's Monday she's now in a new reference class, and since each world contributes exactly one observer moment, she ends up back at P(Tails) = 1/2. Here, "learning" doesn't mean narrowing the set of possible cases we conditionalize upon, as in standard Bayesian conditionalization, but dropping the model and replacing it with a new one having different measures. Note that there's no situation in which she'd be unaware of whether she's in the before or after learning cases, since she'd always know whether she has or has not learned that today is Monday, so treating them as being in the same probability space or not makes no difference.
This move is similar to the introduction of reference classes: a new variable is injected into the equation whose value depends on philosophical interpretation rather than well defined rules, and this variable fully determines the probability of each possible world. For instance, including her Heads^Tue (aspeep) state in the reference class makes her a thirder, while excluding it makes her a halfer.
So on the one hand we can fully determine the probabilities of each possible world by playing with reference classes, and on the other, we can decide whether to use conditionalization or instead replace the model with a new one by deciding when to embed learning steps into the model. Admitting these devices in probability theory would end up rendering the whole thing pretty meaningless insofar as it is a system of rules.