The Sleeping Beauty paradox

Great Debates
1

I’ve been spending some time lately thinking about the Sleeping Beauty paradox, which is one of the well-known paradoxes in epistemology and Bayesian probability theory.

Here’s the description from Wikipedia:

There are two common answers to this paradox. Some argue that she should give a credence of 1/2. Others argue that she should give a credence of 1/3. The former are called “halfers”, and the latter are called “thirders”.

I have a response of my own, but I’m curious to see what other people here think of the paradox. Of course, I’m especially interested to hear reactions to my answer.

Here’s how I think of the paradox.

[spoiler]My response to the paradox depends on whether, upon awakening, Sleeping Beauty (SB) has an objective means of indicating the current day. If she does, then I’m a thirder. But if she doesn’t, then I’m a halfer.

I have a perhaps idiosyncratic view about when SB can indicate the current day. I hold that SB can’t indicate that day merely by saying “today” or “the day of this very utterance”. She at least needs to have (what I call) a “random calendar” for the duration of the experiment, or something equivalent.

A random calendar is a device with a display that shows one symbol per day in random order from a known finite set of symbols, never repeating a symbol. Hence, such a calendar has a finite span of days during which it is active. SB’s random calendar must be known to be active throughout the longest-possible duration of the experiment. (It doesn’t matter, but, for simplicity, let’s suppose that her random calendar is active for precisely that duration.)

With a random calendar, SB can now indicate the current day by referring to it as “the day on which my calender displays <blah>”, where <blah> is the name of the symbol that she’s looking at. We assume that the first thing that SB does upon awakening is to look at the display on her random calendar. Only then does she determine her credence that the coin-flip came up heads.

Briefly, I’m a thirder in this case because SB’s credence that the coin came up heads should be the probability that the coin came up heads conditioned on the fact that the experiment’s duration included the day corresponding to the symbol on the calendar. This makes it more probable that the experiment lasted longer, and hence less probable that the coin came up heads.

On the other hand, if she doesn’t have a random calendar (or equivalent), she is not, on my view, able to condition on something like “the experiment’s duration included today”, because she has no way to make any particular day the referent of “today”. Hence, her credence that the coin came up heads remains the unconditioned probability of 1/2.

Incidentally, it follows from my view that, in the calendar-less case, she’s not able to assign a credence to the statement “It’s Monday” at all. If she uttered this statement, it would fail to refer to any particular day. Even though her utterance happens on a particular day, she has no way of specifying that day, so she can’t imbue her utterance with that meaning.[/spoiler]

(Self-plagarism alert: I posted much of the above earlier in this thread.)

2

Well, I’ll continue the self-plagiarism:

[spoiler]Interesting. So she wouldn’t even be able to imbue the statement “It’s currently the case that everyone outside is going around calling the day ‘Monday’ (even though I can’t access them)” with meaning, even though she can imbue the statement “The coin flip came up heads (even though I can’t access any record of its results)” with meaning?

Well…

I imagine what you are saying is that SB knows there are two days in consideration, one where people outside say Monday and one where people outside say Tuesday, but she can’t manage to assign to these the labels “now” and “not now” in any fashion, because they’re totally symmetric to her; there’s no way to distinguish the one from the other.

I am amenable to speaking this way. However, while doing so, I would feel compelled to also similarly argue that SB knows there are two histories in consideration, one where the coin came up heads and one where it came up tails, but she can’t manage to assign to these the labels “the actual history” and “an alternative history” in any fashion, because they’re totally symmetric to her; there’s no way to distinguish the one from the other.

Why should we be able to say “there was only one past coin-flip [i.e., there is only one coin flip result in the ‘actual’ history]” if we cannot equally as well say “there is only one outside day-value [i.e., there is only one day-value for the ‘current’ outside]”? Seems to me we can accept both or reject both, but we oughtn’t treat one differently from the other.
[/spoiler]

3

Why would someone say 1/3 instead of 1/2?
I’m not sure I quite get this. It seems like it should always be 1/2 if there’s no way of figuring out what Day it is (Like being held in an isolation chamber for 24 hours or 48 hours), if she can’t recall being previously awakened, then she’s only got the coinflip to base her guess on. So it’s 50-50 that she’s been either out for a day or out for two days.

Where does the 1/3 come into this in that scenario?

EDIT: Ah… I read the article now. Huh. Interesting. I understand the 1/3 thing. Didn’t think about it like that.

4

Here’s one standard thirder argument as I understand it:

When Sleeping Beauty wakes up, she has no way of knowing whether
[ul]
[li]the coin came up heads and it’s Monday,[/li][li]the coin came up tails and it’s Monday,[/li][li]the coin came up tails and it’s Tuesday.[/li][/ul]
Her experience of any one of these scenarios would be indistinguishable from her experience of either of the others. She has no evidence to lead her to prefer one over the others. Hence, they should all be given equal credence. But in only one of these scenarios did the coin come up heads, so she should give credence of 1/3 to that possibility.

Another argument is to suppose that the whole experiment (coin flip and all) were repeated many times. Then, among all interviews, the coin flip preceding a given interview will have been “heads” only one-third of the time. The argument is then that, in any given interview, SB should reason that there is only a one-third probability that she is in one of the interviews following a heads flip.

5

Though, let us point out, unfortunately, symmetry arguments for equiprobability often clash. After all, just focusing on the day of the week, she has no way of knowing whether it’s Monday or it’s Tuesday; her experience would be indistinguishable. So, that means P(Monday) = P(Tuesday) = 1/2? But that conflicts with the 1/3 result just deduced [and similarly with heads-tails symmetry].

Granted, one can then give a more sophisticated response [sure, she has no way of knowing which day of the week it is, but she still has reason to prefer Monday to Tuesday, because of the setup, which involves symmetry-breaking fact…], but then, to do this is to give a different, more sophisticated one-thirder argument as well.

6

I disagree with this paragraph from the wikipedia article:

The author is trying to argue that P[Monday] = 2/3 and P[Tuesday] = 1/3 (by analogy to the fact Monday is the first day of the test, and Tuesday is the second). But the experimenter has no such probability distribution for the experimenter because he knows the day with certainty. The last sentence about “putting the two cases together” makes no sense if Sleeping Beauty (or the experimenter) knows the day of the week with certainty, since you cannot simultaneously hold that P[Monday] = 2/3 and P[Monday] = 1.

7

The paradox is poorly posed, or maybe thats just me. How hungry is she? How thirsty? If she’s pretty thirsty, its probably Monday. If she’s so thirsty she doesn’t give a rats ass if its Tuesday or Doomsday, its Tuesday.

8

It’s just you. Don’t fight the hypothetical.

9

Those three outcomes do not have equal probability of occurring. Think of it this way: the coin has even odds for coming up heads and tails. However, if it comes up tails, then Sleeping Beauty has even odds that her interview is on either Monday or Tuesday. Thus,
-The coin coming up tails AND it being Monday has a 1/4 chance of being true.
-The coin coming up tails AND it being Tuesday has a 1/4 chance of being true.
-The coin coming up heads AND it being Monday has a 1/2 chance of being true.

Thus, during her interview, she would respond 1/3, but that’s only because she’s a spoiled princess and never studied math. The correct answer is 1/2.

10

She is twice as likely to be woken on monday than tuesday. She has a 2/3 chance of being woken up monday, of which 1/3 is the result of a heads flip and 1/3 is the result of a tails flip. She also has a 1/3 chance of being woken up tuesday, which is the result of a tails flip.

She can’t say 1/2, because she is woken up twice for every tails result, meaning that when she wakes up, it is twice as likely to be due to a tails result than a heads one.

To me this is a thousand times easier to comprehend than the game show paradox.

11

She isn’t being woken up 100% of the time, and that’s why the numbers are off. Since she’s being woken up 150% of the time (either once or twice divided between two equally likely possibilities), and each of the three possible interviews occurs 50% of the time, the chance that she was woken up after a heads is only 1/3.

12

Never mind.

If it’s tails, she gets interviewed twice. So 1/3.

13

Upon reflection, I appear to have been wrong. I apologize for my failure.

14

Let’s look at it another way. I decide to go on vacation, and flip a coin to decide between:
(1) One night in London;
(2) Two nights in Paris.
(It’s more expensive in London).

Next, I wake up in a strange hotel bed, and remember the decision-making process, but don’t remember which way the coin flip went. And (if it’s the second day) I can’t remember the first day – my short term memory is bad that way.

So what are the odds between London and Paris? I think it’s obvious: 1/3 I’m in London, and 2/3 I’m in Paris. And I think the odds work the same for the question as originally posed: 1/3 heads, and 2/3 tails.

15

Alright, but just in case some of you fail to grasp what is (considered by some to be) paradoxical:

Just before you learn the coin-flip results, what probability would you give to it being heads or tails? 1/2, right? Fine.

Ok, now things play out, you wake up, you remember nothing. Now what probability would you give to the coin flip being heads? Some of you are saying 1/3. Also fine.

But something very interesting has happened here, hasn’t it? You haven’t learnt any new information from the former to the latter time, yet you’ve changed the probabilities you assign. Isn’t that a little weird? If you haven’t learnt anything new, why would the probabilities (thought of as representing your levels of credence in various propositions) shift? Therein lies (part of) the paradox.

16

She should just go by how badly she needs to use the bathroom. . .

17

You have learnt new information: you know that you woke up, an event that was twice as likely to occur if the coin came up tails.

18

You already knew you were going to wake up at some point; you just didn’t know how many times. And all you’ve learnt is that you’ve woken up at some point; you don’t know how many times. So that isn’t really new information.

19

You haven’t learned anything new, per se, but it’s not the same circumstances. As someone touched on upthread, this is very similar to the Monty Hall Paradox, where in you learn nothing new, but the probabilities still change because the circumstances are not the same as they would appear to be intuitively. That is, it is intuitive that the coin flip (let’s call it P(C)) and waking up (let’s say P(W)) are independent events, but they clearly are not because the circumstances change based upon the outcome of the coin flip. What we’re actually calculating is P(C|W). The simplest way to look at it is to look at all the possibilites, each having equal probability (0.25)

A: It’s Monday and it was Tails > She wakes up
B: It’s Monday and it was Heads > She wakes up
C: It’s Tuesday and it was Tails > She wakes up
D: It’s Tuesday and it was Heads > She was already awake

We’re only looking at 3 of potential situations because case D doesn’t fall under the given condition that she wakes up. Thus, of those 3 potential cases, 1 is heads, which is obviously the probability of 1/3.

It’s only paradoxical until you realize that you’re not actually in the same circumstances and you’re calculating a different probability.

20

That’s not true. Before the experiment, you knew that you were going to wake up once or twice not remembering the recent past. Now you have woken up: that’s a different fact. (Unless you think that the sentences “Tomorrow will be Tuesday” and “Today is Tuesday” mean the same thing).