In a recent post, it has been said the Sleeping Beauty Problem is resoundingly considered resolved in academia. As someone who has been following the literature extensively for the past few years, I can positively say that is not true. The post also used a frequentist argument (in the form of a computer simulation) as support for thirdism. I want to point out the shortcoming of this argument. Most importantly, I want to show what I consider to be the correct frequentist model. And why perspectives shall be regarded as fundamental to solve anthropic paradoxes. as I have argued in this previous post.
Per Toss or Per Awakening
If the SB experiment is performed 100 times then roughly speaking there will be 150 awakenings. Out of these 150 awakenings, about 50 would be after Heads. Neither camp is contending this. A computer simulation reiterating it cannot settle the debate.
The problem is Halfers, in general, suggest the relative frequency should be divided over the total number of tosses: 50/100. Thirders on the other hand want to divide it over the total number of awakenings: 50/150. No consensus has been reached so far.
I argue the above interpretations are both wrong. I purpose an alternative model that would give the same relative frequency over toss or awakening. It requires treating perspectives as fundamental.
For ease of expression, I will first use the following thought experiment I call Cloning Beauty. It presents the same anthropic uncertainty as SB. If you have doubt that SB and CB are equivalent, do not worry. I will go back to SB and present its frequentist model momentarily.
CB is set up the same way as the original Sleeping Beauty. Instead of memory wipes, the subjectively similar instances are obtained by cloning. Omega would toss a coin after Beauty falls asleep. If it lands Heads, nothing happens. If Tails, Omega would create a highly accurate clone of Beauty and put her in the empty identical room. The cloning is highly accurate even memories are preserved. So a Beauty would not be able to tell if she is physically new. Beauty wakes up the next day, what's the probability of Heads?
For clarity let's call the pre-existing agent the original, call the newly created agent the clone, and call any agent a copy.
The same debate of Halferism vs Thirderism also exists in CB. After waking up Beauty could say there is no new information so it is 1/2. Or she could say I exist, so Tail is more likely therefore it is 1/3. SSA and SIA would give the same answers as SB.
Let's repeat this experiment so we can get some frequencies. Here the effect of perspective is prominent. Imagine yourself in Beauty's shoes. I fall asleep, wake up the next day not knowing what happened. I can enter another iteration of the same experiment hereon and repeat it again and again. Granted, through these repetitions I may not refer to the same physical copy (just like in the first experiment I might not be the same physical copy after waking up). Yet it is always clear to me which copy the indexical I refers to. It means the first-person which is primitively clear from my perspective. Through these repetitions I would experience Heads and Tails with roughly equal numbers.
If we imagine repeating the experiment from an outsider's perspective (as a third-person, or from a god's eye perspective, or with "a view from nowhere", or however you like to call it), the picture would be completely different. Indexicals such as I can no-longer be used to pinpoint a copy. Discussing a particular Beauty requires selecting it out first. Depending on how the copy is selected it will give different probabilities/ relative frequencies. E.g. a copy is selected from all actually exist copies, the probability, and relative frequency would be 1/2. Alternatively if a copy is selected from potentially existing copies, given the selection actually exists, the probability and relative frequency would be 1/3.
But none of the selection methods is relevant for Beauty to get the relative frequency from her own viewpoint. In anthropic problems, there are two sets of seemingly impeccable logic. On one hand, from an observer's first-person perspective, the analysis naturally focuses on indexicals such as I or now. The indexical references are also primitively clear to the first-person. On the other hand, from an outsider's perspective, it is correct to say no one is innately special. Every observer/moment should be treated with indifference. But the two sets of logic stem from different perspectives, they should be kept parallel to each other. Anthropic paradoxes are caused by trying to forcibly mix them. It will lead to assuming the primitively identified indexicals, like I or now, as the result of some imaginary sampling process, such as SSA or SIA.
Back to SB's Frequentist Model
The SB should have the same frequentist model as CB. The only nick is that the memory wipe of a previous iteration could interrupt a later iteration. But that can be easily avoided. It should be obvious that the exact duration of the SB experiment is inconsequential. E.g. The experiment can take two days or two hours. It doesn't matter. As long as a memory wipe happens after Tails, making the two awakenings subjectively similar, it is the same problem.
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. Following iterations are progressively shorter. This way the later experiments will not be interrupted by the memory resets of previous iterations. So from Beauty’s perspective, she can have a continuous first-person experience of the repetitions. From an outsider’s perspective, this structure is a bifurcating supertask. Theoretically, the repetition could go to infinity, yet it will all happen within two days.
I argue all frequentist arguments, including arguments using bets and monetary rewards, should employ this model. And from beauty's perspective, the relative frequency would approach 1/2 as the repetition goes on.