Sleeping Beauty as losing track of time

What makes the Sleeping Beauty problem non-trivial is Beauty's potential memory loss on Monday night. In my view, this means that Sleeping Beauty should be modeled as a case of potential epistemic fission: if the coin lands tails, any update Beauty makes to her beliefs in the transition from Sunday to Monday will also fix her beliefs on Tuesday, and so the Sunday state effectively has two epistemic successors, one on Monday one on Tuesday. All accounts of epistemic fission that I'm aware of then entail halfing.

But I seem to be alone in thinking that fission is the right paradigm for modeling Sleeping Beauty. A much more popular assumption is that Sleeping Beauty is essentially a problem about "losing track of time": as a result of the potential memory loss, it is claimed, Beauty can't tell upon awakening whether it is Monday or Tuesday, and that's what makes her case special. I don't agree that this adequately sums up Beauty's predicament. Surprisingly, though, I think this way of modeling Sleeping Beauty still supports halfing. (That's surprising because almost all authors who endorse the present interpretation are thirders).

Let's begin with an ordinary case where someone loses track of time.

Noisy Awakening I (incomplete). A loud noise wakes you up at night. You have a vague sense that you've slept for a few hours, but the sensation is equally compatible with it being 2 am or 3 am.

It's important to realize that this is not yet a case in which you're necessarily lost in time. For suppose you knew when you fell asleep that a loud noise was going to wake you up at 2am. Remembering this information, you should be confident upon awakening that it is 2am, despite your unspecific sensation of how much time has passed.

So your new credence in what time it is should be affected by two features. First there is your broadly sensory evidence of how long you've slept, as well as other pieces of "new evidence": your perception that it is still dark, etc. Second, there are your previous beliefs about when you might wake up. Since our topic is losing track of time and not forgetting or irrational priors, we can assume that these earlier beliefs were rational and you have no trouble recalling the reasons on which they were based.

It is not obvious how exactly these two factors determine the new beliefs. But the following special case should be uncontroversial.

(*) If before falling asleep you rationally gave credence x to waking up at t1 and 1-x to waking up at t2, and if upon awakening your new evidence is neutral between t1 and t2, then you should now give credence x to the time being t1 and 1-x to the time being t2.

So let's complete our first scenario.

Noisy Awakening I (complete). A loud noise wakes you up at night. You have a vague sense that you've slept for a few hours, but the sensation is equally compatible with it being 2 am or 3 am. Before going to sleep, you rationally gave credence 1/2 to the noise waking you at 2 am and 1/2 to the noise waking you at 3 am.

What should you believe about the time? Answer: you be 50% confident that it is 2 am and 50% confident that it is 3 am.

Now consider a simplified variant of the Sleeping Beauty problem in which Beauty is rationally certain that the coin lands tails. Before falling asleep, she then assigns credence 1/2 to waking up on Monday and 1/2 to waking up on Tuesday. Upon awakening, any sensations she may have about how much time has passed are presumably defeated by her knowledge of the setup, so we may as well assume that she has no relevant new evidence at all about the time.

If we model this as a "losing track of time" scenario, it is obviously analogous to Noisy Awakening.

The real Sleeping Beauty problem is a little more complicated because there is also the possiblity of heads. Let's make the corresponding adjustments to Noisy Awakening.

Noisy Awakening II. Before going to sleep on Sunday evening, you were given the following information. A fair coin will be tossed twice. If it comes up heads at least once, a loud noise will wake you up at 2 am, otherwise (if the coin lands tails twice) the noise will wake you up at 3 am. You fall asleep and are awakened by a loud noise. You have a vague sense that you've slept for a few hours, but the sensation is equally compatible with it being 2 am or 3 am.
In Noisy Awakening II, 50% of your Sunday credence goes to the hypothesis that the first toss lands heads and you will wake up at 2am. This mirrors Beauty's 50% credence that her coin will land heads and she will be woken on Monday. The other 50% of your credence goes to the previously discussed scenario, corresponding to the tails scenario in Sleeping Beauty, in which you might wake up either at 2am or at 3am.

What should you believe in Noisy Awakening II when you are woken up by the noise?

Fortunately, we don't need any new principles. For the upshot of the information about the coin tosses is that on Sunday you give credence 3/4 to waking at 2am and 1/4 to waking at 3am. By (*), absent relevant new evidence, you should give credence 3/4 to the hypothesis that it is 2am.

If Sleeping Beauty is a problem about losing track of time, we should give the parallel answer: Beauty should give credence 3/4 to the hypothesis that it is Monday. That's what halfers say. They hold that Beauty's Monday credence should be divided 1/2 - 1/4 - 1/4 between the possibilities Heads & Monday - Tails & Monday - Tails & Tuesday; thirding says it should be divided 1/3 - 1/3 - 1/3.

Just to be clear: this is an argument by analogy. We can't directly apply (*) to Sleeping Beauty, precisely because Sleeping Beauty is not a straightforward case in which someone is otherwise perfectly rational but loses track of time. My point is that if we model Sleeping Beauty as nonetheless analogous to such cases (as I think we shouldn't: we should rather model it as a case of epistemic fission), we still get an argument for halfing.

(Sleeping Beauty as a case of fission is interesting because it arguably reveals the tension between evidentialism and conservatism. Case of merely losing track of time don't: in Noisy Awakening II, your evidence upon awakening plausibly supports the 2 am hypothesis to degree 3/4.)


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