From my point of view, the problem statement mixes probabilities with credences. The coin flip is stated in terms of equal unconditional probability of outcomes. An awakening on Monday is not probabilistic in the same sense: it always happens. Sleeping Beauty has some nonzero subjective credence upon awakening that it is Monday now.
Let’s use P(event) to represent the probabilities, and C(event) to represent subjective credences.
We know that P(Heads) = P(Tails) = 1⁄2. We also know that P(awake on Monday | Heads) = P(awake on Monday | Tails) = P(awake on Tuesday | Tails) = 1, and P(awake on Tuesday | Heads) = 0. That fully resolves the probability space. Note that “awake on Tuesday” is not disjoint from “awake on Monday”.
The credence problem is underdetermined, in the same way that Bertrand’s Paradox is underdetermined. In the absence of a fully specified problem we have symmetry principles to choose from to fill in the blanks, and they are mutually inconsistent. If we specify the problem further such as by operationalizing a specific bet that Beauty must make on awakening, then this resolves the issue entirely.
I think you are on the right path. Indeed, awakenings on Monday is somehow different than the coin coming up a particular side. But the separation between probabilities and credences is not helpful. They have to be one and the same, otherwise something unlawful is going on.
If we specify the problem further such as by operationalizing a specific bet that Beauty must make on awakening, then this resolves the issue entirely.
No, this doesn’t work like that. Halfer scoring rule counts per experiment, while thirder per awakening, but regardless of the bet proposed they produce correct betting scores (unless we are talking about halfers who subscribe to Lewis’ model which is just wrong).
If there is only one bet per experiment a thirder would think that while probability of Tails is twice as large as Heads, it is compensated by the utilities of bets: only one of the Tails outcome is rewarded, not both. So they use equal betting odds, just as a halfer, who would think that both probability and utility are completely fair.
Likewise, if there is a bet on every awakening, halfer will think that while the probabilities of Heads and Tails are the same, Tails outcome is rewarded twice as much, so they will have betting odds favouring Tails, just as a thirder, for whom utilities are fair but probability is in favour of Tails.
Betting just adds another variable to the problem, it doesn’t make the existent variables more preciese.
We know that P(Heads) = P(Tails) = 1⁄2. We also know that P(awake on Monday | Heads) = P(awake on Monday | Tails) = P(awake on Tuesday | Tails) = 1, and P(awake on Tuesday | Heads) = 0. That fully resolves the probability space. Note that “awake on Tuesday” is not disjoint from “awake on Monday”.
I’m going to present a correct model in the next post with all the detailed explanations, but I suspect you should be able to deduce it on your own. Suppose that all that you’ve written here is correct. What’s stopping you from filling the blanks? What is P(Heads|awake on Monday)? If you know it, you should be able to calculate P(awake on Monday) and so on.
In situations when you don’t know about (potential) memory wipe or, more generally, in situations where you were lied to/not revealed all the necessary information about a setting, your probability estimate differs from a probability estimate of a person who knows all the relevant information. But I don’t see a reason to call one “subjective credence” and the other “objective probability”. Just different estimates based on different available information.
In any way, it’s not what happening in the Sleeping Beauty problem, where the Beauty is fully aware that her memories are to be erased, so the point is moot
Yes, I have filled in all the blanks, which is why I wrote “fully resolves the probability space”. I didn’t bother to list every combination of conditional probabilities in my comment, because they’re all trivially obvious. P(awake on Monday) = 1, P(awake on Tuesday) = 1⁄2, which is both obvious and directly related to the similarly named subjective credences of which day Beauty thinks it is at a time of awakening.
By the way, I’m not saying that credences are not probabilities. They obey probability space axioms, at least in rational principle. I’m saying that there are two different probability spaces here, that it is necessary to distinguish them, and the problem makes statements about one (calling them probabilities) and asks about Beauty’s beliefs (credences) so I just carried that terminology and related symbols through. Call them P_O and P_L for objective and local spaces, if you prefer.
From my point of view, the problem statement mixes probabilities with credences. The coin flip is stated in terms of equal unconditional probability of outcomes. An awakening on Monday is not probabilistic in the same sense: it always happens. Sleeping Beauty has some nonzero subjective credence upon awakening that it is Monday now.
Let’s use P(event) to represent the probabilities, and C(event) to represent subjective credences.
We know that P(Heads) = P(Tails) = 1⁄2. We also know that P(awake on Monday | Heads) = P(awake on Monday | Tails) = P(awake on Tuesday | Tails) = 1, and P(awake on Tuesday | Heads) = 0. That fully resolves the probability space. Note that “awake on Tuesday” is not disjoint from “awake on Monday”.
The credence problem is underdetermined, in the same way that Bertrand’s Paradox is underdetermined. In the absence of a fully specified problem we have symmetry principles to choose from to fill in the blanks, and they are mutually inconsistent. If we specify the problem further such as by operationalizing a specific bet that Beauty must make on awakening, then this resolves the issue entirely.
I think you are on the right path. Indeed, awakenings on Monday is somehow different than the coin coming up a particular side. But the separation between probabilities and credences is not helpful. They have to be one and the same, otherwise something unlawful is going on.
No, this doesn’t work like that. Halfer scoring rule counts per experiment, while thirder per awakening, but regardless of the bet proposed they produce correct betting scores (unless we are talking about halfers who subscribe to Lewis’ model which is just wrong).
If there is only one bet per experiment a thirder would think that while probability of Tails is twice as large as Heads, it is compensated by the utilities of bets: only one of the Tails outcome is rewarded, not both. So they use equal betting odds, just as a halfer, who would think that both probability and utility are completely fair.
Likewise, if there is a bet on every awakening, halfer will think that while the probabilities of Heads and Tails are the same, Tails outcome is rewarded twice as much, so they will have betting odds favouring Tails, just as a thirder, for whom utilities are fair but probability is in favour of Tails.
Betting just adds another variable to the problem, it doesn’t make the existent variables more preciese.
I’m going to present a correct model in the next post with all the detailed explanations, but I suspect you should be able to deduce it on your own. Suppose that all that you’ve written here is correct. What’s stopping you from filling the blanks? What is P(Heads|awake on Monday)? If you know it, you should be able to calculate P(awake on Monday) and so on.
I don’t see why. If someone is messing with you, eg. by wiping your memory, then your subjective credences could depart from objective probabilities.
In situations when you don’t know about (potential) memory wipe or, more generally, in situations where you were lied to/not revealed all the necessary information about a setting, your probability estimate differs from a probability estimate of a person who knows all the relevant information. But I don’t see a reason to call one “subjective credence” and the other “objective probability”. Just different estimates based on different available information.
In any way, it’s not what happening in the Sleeping Beauty problem, where the Beauty is fully aware that her memories are to be erased, so the point is moot
Yes, I have filled in all the blanks, which is why I wrote “fully resolves the probability space”. I didn’t bother to list every combination of conditional probabilities in my comment, because they’re all trivially obvious. P(awake on Monday) = 1, P(awake on Tuesday) = 1⁄2, which is both obvious and directly related to the similarly named subjective credences of which day Beauty thinks it is at a time of awakening.
By the way, I’m not saying that credences are not probabilities. They obey probability space axioms, at least in rational principle. I’m saying that there are two different probability spaces here, that it is necessary to distinguish them, and the problem makes statements about one (calling them probabilities) and asks about Beauty’s beliefs (credences) so I just carried that terminology and related symbols through. Call them P_O and P_L for objective and local spaces, if you prefer.
Your frequentism is showing. Bayesian probabilities are subjective credences, not objective features of the universe.