The Actual Sleeping beauty Problem

[JeffJo]

I am also not convinced by the “no new information” claims. I would like to see an actual calculation from information theory to support that claim. I did such a calculation previously and got 1 bit of information, which seems too high. So I am not confident in my result, but I really don’t think that the 0 information claim is convincing.

The problem with the Sleeping Beauty Problem, as it is usually presented, is that the breaks the paradigm under which other probability problems are solved. The point is, you have to develop new rules to cover it, and the controversy always revolves around those rules. That's what I tried to address with my version (it doesn't add new information, btw, it adds the same information Elga added to it but in a more conventional way).

I haven't looked at your proof (I find applying information theory to be a bit of a kludge anyway), but based on past (and present, unfortunately) experience I suspect that those who don't like your answer will invent a different way to apply it that gets their answer.

 

[Dale]

The point is, you have to develop new rules to cover it

I don’t think that is the case. The standard rules applied in a straightforward manner lead to a well defined result. It isn’t that new rules are needed, it is just that some people trust their intuition more than the rules.

 

[JeffJo]

That's not the relevant "prior" because Sleeping Beauty already knows on Sunday Night that she will not be awakened

It's not the prior that you claim is relevant. It is a prior, and has relevancy somewhere. You just won't say what you think makes it relevant, or not, to this problem. You just insist on this one.

Do you have such criteria? What are they? They seem to be based on what she knows on Sunday night, which will prove why you are wrong. (I'll include the days, which actually are not relevant, to aid exposition).

On Sunday Night, SB learns:

1. She will be put to sleep soon.
2. Then two coins, a Quarter and a Dime, will be flipped and then left untouched for 24 hours.
   1. At this point, Sunday SB knows what the probability distribution will be. Each of the outcomes in {(H,H), (H,T), (T,H), (T,T)} will have a probability of 1/4. Call this distribution A.
   2. This is a valid prior for something.
3. On Monday Morning, the coins will be examined.
   1. Sunday SB knows that distribution A will still apply at this time
4. If either coin is showing Tails, SB will be wakened. If that happens:
   1. Sunday SB knows this result depends on the state of the coins. She knows that distribution A will be used to determine if it happens, and knows that it will not happen if the state is (H,H).
   2. Monday SB will be asked for her credence in Quarter=Heads. Let's delay talking about her answer, because she doesn't know she is Monday SB.
   3. Monday SB will be put back to sleep, with amnesia.
5. The Dime will be turned over on Monday Night, and then left untouched for 24 hours.
   1. Even though the state changes, distribution A still applies to the new state.
6. On Tuesday Morning, the coins will be examined.
   1. Sunday SB knows that distribution A will still apply at this time.
7. If either coin is showing Tails, SB will be wakened. If that happens:
   1. Sunday SB knows this result depends on the state of the coins. She knows that distribution A will be used to determine if it happens, and knows that it will not happen if the state is (H,H).
   2. Tuesday SB will be asked for her credence in Quarter=Heads. Again, delay talking about her answer.
   3. Tuesday SB probably should be put back to sleep, with amnesia, but this does not affect anything except symmetry.
8. On Wednesday, SB is wakened and the experiment ends.

Based on this, Sunday SB knows that:

• Both Awake Monday SB and Awake Tuesday SB will have the same knowledge.
• In fact, it is the same knowledge Sunday SB has about their situation.
• So I can just call her Awake SB, and base any answer on this shared knowledge.
• Awake SB knows that her existence, as an Awake SB, depended on distribution A.
• But Awake SB also knows that (HH) can no longer be part of the distribution. THIS IS NEW INFORMATION.
• Awake SB updates her distribution to {(H,T), (T,H), (T,T)}, with each outcome having probability 1/3.

Of course, all of this was pretty trivial from the beginning. But I'd be interested in knowing why you think Awake SB does not know both distributions, A and B, or why her existence does not depend on the change from A to B.

 

[PeterDonis]

You are using circular reasoning, by claiming I can only use the same space for both prior and posterior.

No, I'm not. I'm simply saying that the information you claim is "new" is not, since Sleeping Beauty already has it on Sunday, before the experiment is started. All the reasoning you go through to purportedly claim that "the sample space changes" or "new information is gained", or however you want to phrase it, is reasoning Sleeping Beauty herself can go through on Sunday. Basically you are arbitrarily saying Sleeping Beauty cannot reason from all the information she has on Sunday, but has to wait until she is awakened. Sorry, not buying it.

Awake SB also knows that (HH) can no longer be part of the distribution. THIS IS NEW INFORMATION.

No, it isn't, because Sunday SB already knows that Awake SB will only occur if (H, H) does not occur, and Sunday SB already knows that she will only be asked questions as Awake SB. So Sunday SB is already in the same logical position as Awake SB.

 

[JeffJo]

I don’t think that is the case. The standard rules applied in a straightforward manner lead to a well defined result. It isn’t that new rules are needed, it is just that some people trust their intuition more than the rules.

There actually is a way to use conventional probability for Sleeping Beauty. It does require using a modified a definition that, while it applies to conventional problems, adds an unorthodoxy that is needed to include the separate the possible awakenings, as Elga did. And it is what halfers specifically deny is valid.

This isn't going to be rigorous, so leave me some slack. An outcome in a discrete, finite sample space is "trivial" if it (well, the event containing just it) has zero probability. I mean, you can include 3.5 as an outcome for rolling a six-sided die, but it is trivial. Outcomes with positive probability are non-trivial. A sample space is non-trivial if it is a valid sample space (mutually exclusive and collectively exhaustive) that contains only non-trivial outcomes.

"New information" traditionally occurs when you learn something that makes some outcomes in a non-trivial sample space become trivial. To update the probabilities, you make a new ("posterior") sample space out non-trivial outcomes, and divide each prior probability by the sum of all of them. While worded in an unorthodox way, I don't think anybody will disagree so far.

In the traditional Sleeping Beauty Problem, a non-trivial sample space on Sunday Night is {H,T}, with probabilities 1/2 for both. But Sunday SB also see that there are two days, which are needed to describe the state at any point in time (this is regardless of whether she is awake). So there are four possible outcome descriptions {H&Mon, H&Tue, T&Mon, T&Tue}. These are used in the Sunday Night sample space, since H&Mon and H&Tue represent the same future, and so are not mutually exclusive.

But I could use {H&Mon, T&Mon} as a non-trivial sample space, each with having probability 1/2.

But when SB is awake during the experiment, she needs an outcome description that includes the day. H&Tue is not needed, because it is trivial. But T&Tue, which in the prior sample space was the same outcome as T&Mon and had probability 1/2, is now disjoint. The space {H&Mon, T&Mon} is no longer a valid sample space, so it is also not a non-trivial sample space. This was exactly my definition for new information. We need to use {H&Mon, T&Mon, T&Tue}. Each had probability 1/2 in the prior. Their sum is 3/2, and we update each to (1/2)/(3/2)=1/3.

The objection halfers have - that Peter insisted I identify earlier while he wouldn't identify his reasons - is that T&Mon and T&Tue still represent the same outcome to Awake SB. Their logic is that the day of the week is an "indexical" quality that can't be used as an outcome description. I still don't understand this logic, which is why I suggested he ask a halfer. Better yet, google for "SLEEPING BEAUTY INDEXICAL."
But when SB is awake,

 

[PeterDonis]

all of this was pretty trivial from the beginning.

I agree that getting the correct answer is trivial, but we disagree on what the correct explanation of the correct answer is, so I think you need to re-think this claim.

I'd be interested in knowing why you think Awake SB does not know both distributions, A and B, or why her existence does not depend on the change from A to B.

I have never made either of these claims. You are attacking a straw man.

Awake SB knows the same things that Sunday SB knows. Both of their knowledge includes both "distributions". But both of their knowledge also includes the fact that only one distribution is relevant, because both of their knowledge includes the fact that awakening only occurs if (H, H) is not the state of the coins.

 

[PeterDonis]

This isn't going to be rigorous, so leave me some slack.

For someone who claims to have a wonderful new way of resolving all controversy about the Sleeping Beauty problem, this request is, to say the least, dubious.

 

[PeterDonis]

H&Mon and H&Tue represent the same future

No, they don't, because as the problem is specified, H&Mon is possible but H&Tue is impossible, and is already known to be so on Sunday. So including H&Tue in the sample space is already wrong on Sunday; it's already known to be "trivial" (in your idiosyncratic terminology) then.

 

[PeterDonis]

The objection halfers have - that Peter insisted I identify earlier while he wouldn't identify his reasons - is that T&Mon and T&Tue still represent the same outcome to Awake SB.

Please give a reference for this claim. Telling me to Google is not enough. If you are going to make a claim about what halfers say, you need to actually give a reference where a halfer is saying it. Those are the rules you signed up to when you joined this forum. Either follow them or your thread will be closed.

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

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