Sleeping Beauty Paradox

[f97tosc]

I agree that the central difficulty here is making the right defnition and probability model, not solving it, and that this has some philosophical flavor too it. I would say though, that I don't think there is any fundamental ambiguity in this problem. In the "bent coin" problem it is not so clear how we rigorously model a coin that is "bent in an unknown way", because different people may have different ideas of what this may mean. In this problem, however, the problem is quite well-specified, anybody reading the problem knows exactly what Sleeping Beauty experiences and knows and does not know in a given situtation. The problem has some unorthodox assumptions that make things difficult, but I don't think it is ill-specified. Any disagreement is of our own making!

And I am just surprised that you are willing to throw out causality in favor of an intuition or a simpler model. I hope that you agree that there is no proof that "waking up at least once" captures all the information in waking up. Without causality, however, it is unclear how anybody could ever form a rational belief about anything. Other than that I agree that it is time to close the debate and I will summarize my position in a different post.

Cheers,

f97tosc

[f97tosc]

This has been an interesting discussion but I feel it is starting to take up too much time for me so I am going to wrap up my position.

The central difficulty of this problem is that under certain conditions, sleeping beauty sleeps and cannot observe what happens; the question is then how she should rationally form beliefs in the cases when she is awake. As I understand it, discussants on "both sides" of the issue are in agreement that if the test protocol is "uncensored" (i.e., she is woken up and told the coin flip under Tue-Heads), then the probability of heads (when she wakes up and not told this) is 1/3, and the reflection principle holds as well. It is not particularly strange that the reflection principle may go out the window if we censor the observer's ability to state or form beliefs under certain conditions. When this happens we must keep separate probabilities of what may happen, and probabilities of what may be experienced (of course, the problem is about what is experienced).

I think the best way to deal with the unusual censorship phenomenon (in this problem and potentially in others, when the observer may be kept asleep, killed, etc) is to build a formal, uncensored probability model. Then from there we argue, based on basic principles of causality, symmetry and information, that under those outcomes when the observer actually does observe, it doesn't matter whether there would or would not be censorship under some other outcomes. This point is difficult to prove rigorously, but as Jason1990 pointed out, when we translate an informally stated word problem to a mathematical one there must always be some element of the translation that cannot be completely formalized. I would just say that the principle of causality is a stronger guide than is the assumption that there is no information in waking up, which is just that: an assumption. I concede that the informationless event "wake up more than once" makes for nice and easy models, and even that it may sound somewhat appealing, but there is simply no proof that this event captures all the information of the situation. Frankly, I think that many posts in the thread are just rephrasing, restating and assuming the no-information claim over and over - but we can't assume away that which is being disputed. To me, it is just as appealing that there is information in waking up, based on the biases associated with the different wake-up frequencies. I wish that everybody could at least be open to the possibilities that there may or may not be information in waking up, and then reach a conclusion based on other properties of the problem. As I see it, none of the points below have been given an adequate explanation by those who favor 1/2.

1. Comparing to the uncensored problem and invoking causality, as discussed above, suggest P(Heads|Wake up unknown day) = 1/3.

2. Long-term frequency. Repeating the experiment many times we have 1/3 of the wake-ups in the heads case. PTB wrote "Just because there are three types of awakenings doesn't mean they are equally likely to be the one she is experiencing right now." No, but by what information does she determine that the present one is more or less likely? Having no distinguishing feature she must assign the same probability to them all, and for this to be consistent with the frequency (which she knows) it must be 1/3.


3. Wagering. Similarly, Sleeping Beauty should make wagers she should bet on 1/3. Any other probability assignment would enable somebody else with no more information than her to exploit her with wagers.


4. Modified problems/Bayes' theorem (this is related to 1.). If we change the problem, by adding or removing certain pieces of information, only 1/3 gives sensible results. 1/2 would make the belief contingent on future and/or unrealized events and therefore violate causality. This can also be seen by writing the following equation - it is impossible to make sensible right hand side assignments that lead to 1/2.

P(Heads|Wake up unknown day) = P(Heads|Tuesday)xP(Tuesday) + P(Heads|Monday)xP(Monday)


5. Information content of answer. How can it be that two reasonable approaches lead to different answers? One explanation is that neither path does anything wrong, but that one path makes less use of the available information. IMO, conditioning on "waking up at least once" is not wrong, just incomplete. But how do we know that it is not the 1/3 answer that is incomplete? Because 1/3 - 2/3 answer has higher information content (lower enthropy), and is closer (lower average error) to the actual flip outcome, than is 1/2-1/2.

[PairTheBoard]

I too would like to end my participation in this discussion and leave the problem to the professional philosoper-mathematicians who it appears are still working on it.

I'll give a summary of my current thinking on it. I think the key element of difficulty are these "Agent-Parts" of Beauty. How do we handle these things? I think our difficulty was illustrated in my example where a coin is flipped; If Tails, 100 White Balls are put in a Bag. If Heads, 1 Black Ball is put in a Bag. Now I say, Imagine you are a Ball in the Bag, but you don't know what color you are. You ask yourself, "What is the probability I am a Black Ball?". Certainly, in half the experiments, when you ask yourself that question you will be a Black Ball.

The Difficulty was illustrated when I got the response from one poster, "I don't know what it means for me to be a Ball". Exactly. We don't know what it means to be a Ball in the Bag. We are not outside looking in and computing probabilities of Balls in the Bag. We ARE a ball in the Bag. Being a Ball in the Bag is what I mean by Beauty being one of three "Agent-Parts" of herself. What does it mean for her to be such things?

The best analysis I've seen of what this means and how we might reasonably handle such "Agent-Parts" was given by Bostrom in the link:

Bostrom's Hybrid Model for Sleeping Beauty

He shows how either of the Pure 1/3 and 1/2 models produce very Counter-Intuitive results when the principles they apply to produce them are carried into more extreme examples. He shows how they ignore the real difficulty of handling the peculiarities of these "Agent-Parts" by not adopting a language with which they can be clearly discussed. He introduces such a language and in doing so finds implications that support his Hybrid Model.

So in summary, I think Bostrom has the best handle on the problem of any I've seen so far.

PairTheBoard

[jman3232]

there are two seperate questions here.

What is the probability that the coin landed heads? that is 1/2.

What is the probability that her awakening resulted from a heads flip? The answer to that question is 1/3.

[jason1990]

[ QUOTE ]
I hope that you agree that there is no proof that "waking up at least once" captures all the information in waking up.

[/ QUOTE ]
I already said there is no proof either way. But in the absence of proof, my default assumption is that waking up with amnesia tells her nothing about the coin. I personally demand evidence to give up that belief. I have not seen any yet.

Regarding proof, the only thing we can mathematically prove is that the statements

(1) P(Mon or Tue) = 1
(2) P(heads and Tue) = 0
(3) P(heads | Mon or Tue) = 1/2
(4) P(heads | Mon) = 1/2
(5) P(Mon | tails) = 1/2

are inconsistent. We cannot remove (1). The problem says that when she wakes up, she knows it is Mon or Tue. We cannot remove (2). The problem says that when she wakes up she knows it is not a Tue-heads awakening. So we must remove one of (3)-(5).

You want to reject (3) and replace it with 1/3. As I said, I find that extremely counterintuitive, since I cannot see how the amnesiac SB gets any useful info. You cannot prove she gets any, so I have no desire to reject (3).

Some people want to reject (4). You say they "reject causality." Maybe they do, maybe they do not. I leave that question for the philosophers. But you are wrong if you think I am one of them. I was merely playing devil's advocate in order to emphasize your lack of proof. Personally, I want to keep (4). To me, changing (4) to 2/3 is as disgusting as changing (3) to 1/3. I will not do it without evidence.

So my personal preference is to remove (5). I do not want to change it to something else. I simply remove it entirely. The price I pay is that if I wake up and they tell me the coin is tails and they ask me the probability that today is Monday, then I must say I do not have enough information to make that assessment. What I gain is that I am able to keep both of the very intuitive statements (3) and (4). I do not need to lay claim to some mysterious unidentifiable "information." And I do not need to claim that I can predict the future based on the day of the week. I realize that "not enough information" is an unpopular probabilistic model both on and off this forum. But I think it has its place. And it is my personal favorite for this problem.

[oe39]

[ QUOTE ]
there are two seperate questions here.

What is the probability that the coin landed heads? that is 1/2.

What is the probability that her awakening resulted from a heads flip? The answer to that question is 1/3.

[/ QUOTE ]

the only thing wrong here is that you are using common sense to trivialize the problem. in order to be taken seriously in this thread, at least a page of agent-event-ball-Bayesian-awakening crap is required.