In this section, I will explain the answer to the Sleeping Beauty Problem based on my core argument. It assumes readers are familiar with the invalidity of self-locating probabilities from Part 3 and the rationale of perspective disagreements from Part 4.
A quick review of the current camps for this (in)famous paradox: Lewisian halfers (after David Lewis) think the probability of Heads should be 1/2 then changes to 2/3 after learning it is Monday. Thirders suggest the probability of Heads should be 1/3 then changes to 1/2 after learning it is Monday. This is the position with majority support in the literature. Then there are the double halfers, who suggest the probability of Heads should be 1/2 at waking up and remains at 1/2 after learning it is Monday. Spoiler: I am a double halfer.
Just Another Way of Duplication
Astute readers may already realize the Toss&Fission from Part 4 is conceptually equivalent to the Sleeping Beauty Problem. Their difference is in the mechanics of creating subjectively similar instances. In Toss&Fission they are created by cloning. Similar states are experienced by two distinct physical agents. In the Sleeping Beauty Problem, they are created by memory manipulation. So the similar states are experienced by the same physical person but at two different times. Subsequently, the relevant indexical in Toss&Fission is “I” while in the Sleeping Beauty Problem it is “now” (or by extension “today”). The analysis would be otherwise identical.
It should be noted that almost all anthropic camps treat agent-based and time-based problems the same way. For example, supporters of SIA would apply it to counter the doomsday argument, and in the same manner, apply it to solve the sleeping beauty problem. So the above claim of equivalency is not unique to my argument.
However, my argument does give a novel theoretical explanation for this parallelism. These problems all become paradoxical because special attention is given to the first-person center while claiming to be objective and impartial. On a side note, for the same reason, the Strong Self-Sampling Assumption due to Nick Bostrom, which uses all observer-moment pairs as the proposed reference class of “I-now” is not different from other anthropic camps. It would also produce paradoxical results.
Right After Waking Up
Upon waking up Beauty has not received any new information regarding the coin toss. So the probability of Heads ought to remain at 1/2. Thirders may argue Beauty has learned the new information that she is awake “now” or “today”. However “I exist (am conscious), now (and here)” is a tautology of first-person reasoning. It is formally true because all indexicals involved in that statement are defined by the same first-person center. It cannot be regarded as new evidence.
Like in Toss&Fission, perspective disagreement can also happen in the Sleeping Beauty Problem. For example, let the experimenter randomly label one day red and the other day blue in his mind. Let one of Beauty’s friends take part in the experiment by choosing between a red and a blue pill. If the friend chooses the red pill he will be awake on the red day and sleep through the blue day. The reverse is true for the blue pill. Say the friend chooses the red pill then wakes up finding Beauty also awake. Here Beauty and the friend would have different answers to the probability of Heads.
From the friend’s perspective, he learned Beauty is awake on the randomly chosen day. It is new evidence favoring more awakenings. A simple calculation using Baye’s Theorem would reduce the probability of Heads to 1/3. However, Beauty would keep her answer as 1/2 as discussed in Part 4.
The friend’s reasoning above resembles SIA. The experiment can be modified to make his reasoning SSA like. For example, keeping everything else unchanged, the experimenter could always wake the friend up on Monday in the case of Heads. This way seeing Beauty awake would no longer be a surprise for him. His probability of Heads shall remain at 1/2. However, using this as a halfer argument would still be wrong. Beauty’s probability has to be calculated from her own perspective, not from the viewpoint of some outsider. SSA and SIA are equally incorrect because they switched to an outsider view.
After Learning It’s Monday
The probability of Heads after Beauty learns it is Monday has been an ongoing predicament for halfers. While all thirders agrees Beauty should update her belief so that the probability changes from 1/3 to 1/2, halfers are divided on this issue. Lewisian halfers update the probability according to Bayes’ theorem from 1/2 to 2/3. However, the coin can be tossed after the first awakening. In this case, Beauty would be giving Heads a probability of 2/3 to a fair coin yet to be tossed (this argument is due to Adam Elga). It is a consequence hard to accept.
Another camp argues the probability of Heads should remain at 1/2. Yet, this seems to violate Bayes’ Theorem. So double halfers have been theorizing different ways to validate Beauty’s unchanging probability (E.g. the arguments by Christopher Meacham, Nick Bostrom, and Rachael Briggs respectively). However, these arguments are ultimately unsuccessful. Michael Titelbaum pointed out that as long as Beauty assign a nonzero probability to “today being Tuesday” right after waking up, her probability of Heads would be greater than the initial 1/2 once learning it is Monday.
However, the probabilities of “today” being Monday/Tuesday (or “this awakening” being the first/second) are self-locating probabilities. As explained in Part 3, they are invalid concepts due to perspective inconsistencies. So there is no correct value for them. That is why after learning it is Monday Beauty’s probability of Heads can still be 1/2 without violating Baye’s theorem. There can be no Bayesian update since no corresponding probability exists.
In the core argument, I explained perspective centers are primitively identified logical starting points. It is analogous to axioms in axiomatic systems. New information regarding the perspective center, such as “now” is Monday, or “I” am the clone, is a modification to the axiom. It changes the starting point of the entire perspective reasoning. Therefore it requires a complete remodel of the probability calculation. It is distinctly different from common information which can be incorporated by Bayesian updates.
The Frequentist Model
Unsurprisingly many arguments, from both camps, uses the frequentist approach. In these arguments the experiment is repeated many times, sometimes with bets and monetary rewards involved, then the conclusion is drawn based on the long-run outcomes. However, these arguments rarely pay any attention to how the experiments are repeated. Most would simply arrange them chronologically, e.g. the second experiment is performed after Wednesday and so forth. This is not a problem from the perspective of an outsider observer. To them, the subject and time of the experiment are irrelevant. As long as an experiment has the same procedure, it can be counted as another iteration.
In contrast, more structure is required if the experiment is to be repeated from the subject’s perspective. As shown in Part 3 and 4, this means imagine being Beauty, after waking up from the first experiment, take part in another iteration of the same experiment. However, if the second experiment also takes 2 days it would be interrupted by the memory wipe from the first. So the second iteration’s overall duration needs to be shortened to fit in the time slot divided by the previous iteration’s memory wipe. Subsequent iterations’ durations need to be progressively shorter.
For simplicity let’s assume the time awake in each iteration is negligible and let the first awakenings happen immediately after the experiments start. After waking up from the first experiment, Beauty can take part in another iteration where the overall duration is only one day instead of the initial two days. After waking up from the second experiment she can take part in another iteration with the duration of half a day, and so on. This way the new experiments will not be interrupted by the memory reset of the previous iteration. So from Beauty’s perspective, she can have a continuous first-person experience of the repetitions. From an outsider’s perspective, this repetition structure is a bifurcating supertask, analogous to the continuous fission in agent-based problems.
All frequentist arguments, including arguments using long-running monetary rewards, should employ this repetition model. Otherwise, they are calculating the probability from the outsiders’ perspectives, not Beauty’s. And as expected, the relative frequency for beauty is 1/2 in this model.