Repeated (and improved) Sleeping Beauty problem

by Linda Linsefors1 min read10th Jul 20185 comments


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Sleeping Beauty Paradox
Crossposted from the AI Alignment Forum. May contain more technical jargon than usual.

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].


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This argument seems to depend on the fact that Sleeping Beauty is not actually copied, but just dissociated from her past self and so that from her perspective it seems like she is copied. If you deal with actual copies then it is not clear what is the sensible way for them to all pass around a science journal to record their experiences, or all keep their own science journals, or all keep their own but then recombine somehow, or whatever. Though if this thought experiment gives you SIA intuitions on the Sleeping Beauty problem then maybe those intuitions will still carry over to other scenarios.

This statement of the problem concedes that SB is calculating subjective probability. It should be obvious that subjective probabilities can diverge from each and objective probability -- that is what subjective means. It seems to me that the SB paradox is only a paradox if y ou try to do justice to objective and subjective probability in the same calculation.

I'm confused, isn't the "objective probability" of heads 1/2 because that is the probability of heads in the definition of the setup? The halver versus thirder debate is about subjective probability, not objective probability, as far as I can tell. I'm not sure why you are mentioning objective probability at all, it does not appear to be relevant. (Though it is also possible that I do not know what you mean by "objective probability".)

Keeping with my comment to your previous post, the probability of the coin landing heads is a model for the frequency measurements that the Sleeping Beauty undertakes. (Also, you have some typos in the text, my favorite being "panthers in the data"). This frequency depends on how the measurement is made, and so is not an invariant. Specifically, if the SB in question checks the state of the dice each time it is woken up in your repeated SB setup, she measures the frequency approaching 1/3 for the heads, because of the double counting the same throw on Monday and Tuesday. While an outsider measures 1/2, since they don't double count. So there is no conflict or contradiction, just different measurements and hence different hypotheses.

If you need a long-run frequency for 'fairness', is it sufficient to have a "fair coin" (rather than a "fair coin flip")? By a fair coin I mean a coin that has been flipped many times in the past to establish non-predictability. Then the game itself doesn't need to be repeated for it to be well-defined, but it seems to retain more of the original problem's tension.