Follow up to: Probability is fake, frequency is real
There is something wrong with the normal formulation of the Sleeping Beauty problem. More precisely, there is something wrong about postulating a single "fair" random coin flip. So here is an improved version of the Sleeping Beauty problem. After explaining the setup, I will recover the normal Sleeping Beauty problem, but in a more well defined way.
There are no truly random coins. There are only pseudo random coins which has the property that you don't have the capacity to calculate the outcome. A fair pseudo random coin have the additional property that when flipped enough times, the ratio of Heads v.s. Tails will approach one. Note that fairness is only defined if you actually flip the coin a sufficient number of times. Because of this, the Sleeping Beauty problem should be a repeated game.
(Alternatively, you could solve this by using counterfactuals. However, we don't yet know how to deal with counterfactuals. Also, I suspect that any method of handling counterfactuals will be, at best, useful but wrong.)
Repeated Sleeping Beauty setup: Every Sunday a mysterious person flips a pseudo random fair coin. If the coin comes up Heads, Sleeping Beauty will wake up on Monday, and then sleep for the rest of the week. If the coin comes up Tails, she will wake up on Monday and Tuesday and then sleep for the rest of the week. No-one is telling Sleeping Beauty what is going on, she gets to rely on her own past experiences.
Every morning when Sleeping Beauty wakes up she does not know what day it is. However there is an easy experiment she can do to find out, namely asking anyone she meets on the street. Because Sleeping is a curious person, she is keeping a science journal. Every day she finds out what day it is and writes it down. She soon notices some patterns.
1) There are two kinds of days, Monday and Tuesday.
2) Every Tuesday is followed by a Monday.
3) A Monday can be followed by either a Tuesday or a Monday.
After some more time she starts to notice the frequencies of which different days occur.
1/3 of days are Mondays that are followed by Monday (corresponds to Heads & Monday)
1/3 of days are Mondays that are followed by Tuesday (corresponds to Tails & Monday)
1/3 of days are Tuesdays (corresponds to Tails & Tuesday)
Sleeping Beauty tries to find more patterns in the data, but none of the more complicated hypothesizes she can come up with survives further observation.
Recovering the original Sleeping Beauty problem: Sleeping Beauty have been slacking off for a few days, and not asking for what day it was. What likelihood should she assign to the current day being a Monday followed by Monday?
The obvious answer based on Sleeping Beauty's own experience is 1/3.
Conclusion and after-though: If you take your probability from how things have played out in the past, you will learn the Thirder position / Self-indication assumption (SIA). Also, doing what has worked well in the past leads to Evidential decision theory (EDT). This is a sad fact of the universe, because EDT combined with SIA leads to a sort of double counting of actions which add up to the wrong policy [citation].