Talk:Sleeping Beauty problem

Editing needed

The article mentions Nick Bostrom three times as arguing for the thirder position. However from what I understood from his works he is actually a double halfer. He actually brings the "Extreme sleeping beauty" scenario to argue against both the halfer and thirder positions. Also, he argues against the Self indication assumption and argues that the halfer position is implied by the Self sampling assumption.

Here is a quote from his paper Sleeping Beauty and Self-Location: A Hybrid Model page 17:

"If the hybrid model is correct, it might explain the fact that both the 1/3- and the 1/2-views have some intuitive appeal. According to the hybrid model, both these views get something right. The 1/3-view is right that Beauty’s posterior credence in HEADS after being informed that it is Monday should be one-half. The 1/2-view is right that Beauty’s prior credence in HEADS, after awakening but before learning that it is Monday, should be one-half." 108.30.23.32 (talk) 21:04, 9 October 2022 (UTC)

I will attempt editing. 108.30.23.32 (talk) 16:59, 14 October 2022 (UTC)

Er... what ?

"the credence of HEADS after being informed it is monday is 50%" is obviously right.

But how on earth does it imply that the credence of HEADS before learning what day it is has to be the same ? SB has two different relevant pieces of information in one case ("I'm awake" and "it is monday") only one in the other ("I'm awake"), why should the credence be the same before and after learning the day ?

The math clearly shows it isn't at all.

It feels like debating the plausibility of different origins of the golden tooth in Fontenelle's story (mais on commença par faire des livres, et puis on consulta l'orfèvre.) 2A01:E0A:BED:6AA0:44C1:2CAB:37F2:B5EF (talk) 22:05, 21 October 2024 (UTC)

Looking for Reliable Source to this

Someone help me find the Reliable Source of this put-down to the so-called paradox. In my words but the reasoning is not originally mine:

The coin is fair. Sleeping Beauty is never awakened if Heads is tossed, and is wakened once if Tails is tossed. As before there is no hidden info and SB knows this. After being awoken SB is asked what probability is it that Heads was tossed. 0%, she says, correctly, as she flounces off in search of kinder people.

Paul Beardsell (talk) 11:32, 2 January 2024 (UTC)

I don't think there is a need for reliable source here. That's just a good demonstration of the difference between probability and credence, that shows very clearly why and how the discussion of Elga's article is a waste of time. 2A02:8440:7142:8425:A46E:F7C8:659A:EDB4 (talk) 08:06, 19 December 2024 (UTC)

... but if you want a source, that's pretty much the Snow White thing of referencz #12. 2A02:8440:7142:8425:A46E:F7C8:659A:EDB4 (talk) 11:10, 19 December 2024 (UTC)

It's just bad math and bad reasoning

Ok, first of all, sorry for the spelling and grammar mistakes: not a native English speaker. This sleeping beauty paradox thing seems to be a pointless discussion around faulty bayesianish demonstrations. Here is why and how. The experiment : there is one fair coin toss, resulting in H or T, and a sleeping subject of experiment, called B. If T, B is awaken on monday and left to sleep on wednesday. if H, B is awaken on monday and wednesday (there is no tuesday in this story, because both tuesday and tails begin with a t, which would throw off my notations). B knows all about the experiment, but forgets she has been awake as soon as she gets back to sleep, therefore does not know if she has been awaken before, does never know what day it is, nor if the result of the coin toss was H or T. Every time she is awake, she is asked what is the likelihood of H and the likelihood of T, knowing what she knows. What should she answer ?

Let's go for notations : uppercase are for parameters (M for monday, W for wednesday), states (A for being awake), results (H or T) ; lowercase are for the info B has on this objective reality (m if she knows it's monday and so on ; not m if she does not know that it is monday and so on) ; C is the probability function for the coin toss : C(T)=1/2 ; C(H)=1/2 ; c is for the knowledge of this function ; e is for the knowledge of the details of the experiment ; let's avoid frivolous distraction by assuming both "if A then a" and "if a then A" ; as we are talking to B, who is awake when we talk to her, there is no such thing as not a ; knowing e, not m and not w are obviously the same thing : B does not know what day it is ; L(X/y,z...) is the likelihood of the state/result/parameter X knowing the information y, z... It goes without saying that when B is awake, she has no context other than a, e, c, therefore should give the same answer in the three possible instances when she is awake : they are strictly identical from her point of view.

What we are looking for is L(H/a,e,c,not m) and L(T/a,e,c,not m). The 1/2 vs 1/2 intuition is obviously false : L(H/info)=C(H) only if the info is not relevant or linked to the result or the coin toss at all, but B knows she is awake (a) and that the experiment (e) provides a strong link between the result of the coin toss and the waking state (if H, B is awaken more often). In the frame of e, a is a crucial piece of information. There is therefore no reason to suppose L(./a,e,c)=C(.), quite the opposite. There should be a strong suspicion that L(T/a,e,c)<C(T) and L(H/a,e,c)>C(H). The 2/3 vs 1/3 intuition is quite lazy, does not use the available info and has no chance to be true : when B is awake, even if there are 3 situations when she will be, she has no reason to go for the L(M/a,c,not e) and L(W/a,c, not e) likelihoods with each of the three possibilities being equally plausible, because she knows e, and the experiment provides a strong link between the day it should be when she is awake and the result of the coin toss. Elga's "demonstration" in his 2000 article hinges solely on lousy and improper notations, where L(M and H), L(M/h) and L(H/m) are all written P(M,H) and used interchangeably, while they are different things with potentially different values.

Now for the calculus (we'll drop the notations e, c and not m from now on, for the sake of simplicity).

-first of all, we have to determine L(M/a) and L(W/a). if B knows the result of the toss was Tails, she also knows that if she is awake, it must be monday : L(M/a,t)=1 and L(W/a,t)=0 ; if B knows the result of the toss was Heads, she knows it's either monday or wednesday but does not know which, both being deemed equally likely for lack of relevant information : L(M/a,h)=L(W/a,h)=1/2 ; hence L(M/a)=C(H).L(M/a,h)+C(T).L(M/a,t)=1/2 . 1/2 + 1/2 . 1 =3/4 and L(W/a)=C(H).L(W/a,h)+C(T).L(M/a,t)=1/2 . 1/2 + 1/2 . 0 =1/4 (there we have it, we thirdists are definitely wrong).

-then we must calculate L(H/a,m), L(T/a,m), L(H/a,w) and L(T/a,w). obviously, if B knows it is wednesday, she knows the result of the toss must have been Heads, otherwise she would be sleeping : L(H/a,w)=1 and L(T/a,w)=0 ; if B knows it is monday, then she has no relevant info on the result of the coin toss : when it is monday, she is awaken anyway, whatever the result of the toss (in the context of e, m makes a irrelevant). Therefore : L(H/a,m)=C(H)=1/2 and L(T/a,m)=C(T)=1/2.

-this being done, we just have to calculate : L(H/a,not m)=L(H/a,m).L(M/a)+L(H/a,w).L(W/a)=1/2 . 3/4 + 1 . 1/4 =5/8 L(T/a,not m)=L(T/a,m).L(M/a)+L(T/a,w).L(W/a)=1/2 . 3/4 + 0 . 1/4 =3/8

So, there we are : neither 1/2 vs 1/2 nor 1/3 vs 2/3 and no paradox but an only mathematically rigourous solution 3/8 vs 5/8. If my reasoning is false, thank you for telling me where. If it isn't, I cannot fathom how philosophers have spent more than 20 years writing about this. 2A01:E0A:BED:6AA0:A5:3C67:B2E6:9DBB (talk) 19:29, 21 October 2024 (UTC)

At least one way that this is incorrect is the following equation: L(W/a)=C(H).L(W/a,h)+C(T).L(M/a,t). The correct formula for the Law of Total Probability with conditional probabilities would use P(h|a) where you have C(H), and similarly with C(T). Of course, that begs the question since P(h|a) is what you're trying to solve. Marfire (talk) 05:04, 13 April 2025 (UTC)

Well, that is why I took the precaution to differenciate between C(T) and L(T/...) : the fact that SB will be awake once or twice does not act retroactively on the coin toss : C(T) and C(H); although if obviously has an influence on the estimation she makes that the result of the coin toss was H or T : L(T/a,...) and L(H/a,...).

And the branching in the process of the experiment is determined by the actual result of the toss and its actual probabilities, ie C(T) and C(H), which are independent of a, not by the ex post estimation of the likelyhood of each result, which we are trying to solve and indeed depend on a.

It still might be wrong, but not in this way. ~2026-36810-18 (talk) 20:24, 26 June 2026 (UTC)

Sorry, too many implicit steps in that first response. Let's go for a step by step approach.

1- we of course start from the law of total probability with conditional probabilities :

P(W/a)=P(H/a).P(W/a,h)+P(T/a).P(W/a,t)

2- we've observed that P in fact covers two different things, C and L as described above, which are conceptually different and have no reason a priori to be equal. In this problem, we use both : C because there is a random event on which depend a chain of consequences ie the coin toss, C(T) and C(H) ; L because some information is hidden from SB (is it Monday ? was the result of the coin toss heads or tails ?) and we ask her to give her best estimate of the likelihood of each possible state given the info she has L(W...), L(T...) etc.

3- if e use P for both, there will be confusion and we stand no chance whatsoever to make a good demonstration. So we refrain from using P and we carefully determine, each time P would appear in a formula, whether we should use L or C. And L given what information exactly.

4- the day M or W is not a random event. There are just consecutive states on which SB as no info and we ask her to give her best guess. Therefore, there is no such thing as C(M) or C(W) and we use L(M/...) and L(W/...).

we thus have L(W/a)=P(H/a).L(W/a,h)+P(T/a).L(W/a,t)

5- now what do we use for P(T...) and P(H...) ?

L is the question we are asking SB. If we use it in the formula, that would be a circular reference, as aptly pointed above, and that would be a problem. But that's not an argument.

On a more fundamental level, the line of reasoning here is the following. Depending the result of the coin toss, there are two chains of event, ie two possible branches of the process : one where SB is woken once, one where she is woken twice. If we are on the branch "once", then the likelihoods of M and W are this and this. If we are on the branch "twice", then the likelihoods of M and W are that and that. We combine the two branches to determine the aggregate likelihoods of M and W. Now what determines whether we are on branch "once" or branch "twice" ? The actual coin toss and its actual result independent of any estimation that SB could come by afterwards. Then what P(T...) and P(H...) should we use ? The ones describing the actual coin toss, ie C (as opposed to the ones describing the estimations of SB, L, which do not determine anything in the chain of events).

We thus have L(W/a)=C(H/a).L(W/a,h)+C(T/a).L(W/a,t)

6- Now C describes the intrinsic probabilities of results of the actual coin toss, heads, or tails. Is the result of the actual coin toss in any way affected by the fact that we are going to do this or that afterwards ? Of course not. The causal chain goes the other way. In other terms, there is no way the actual events H or T depend from a, C(T/a)=C(T) and C(H/a)=C(H).

Thus we have L(W/a)=C(H).L(W/a,h)+C(T).L(W/a,t)

That's pretty much it. ~2026-36810-18 (talk) 22:54, 26 June 2026 (UTC)

I have argued about the SB "paradox" with someone who was upholding your comment as truth. It feels obvious to me it's 1/2 heads 1/2 tails. I didn't study probabilities but I spent a lot of time analyzing your comment and understood its logic. There might be more errors you have made but here's the error I found: "if B knows it is monday, then she has no relevant info on the result of the coin toss : when it is monday, she is awaken anyway, whatever the result of the toss (in the context of e, m makes a irrelevant). Therefore : L(H/a,m)=C(H)=1/2 and L(T/a,m)=C(T)=1/2.". it's not half chance it's Heads or Tails, it's twice as likely that it's Tails. I don't exactly know how to prove it but I can disprove your assumption using your results.

So if L(H/a,not m)=5/8, knowing that L(M/a,h)=L(W/a,h)=1/2 we get L(H,M/a)=L(H/a,not m)•L(M/a,h)=5/8•1/2=5/16

Similarly, L(T/a,not m)=3/8 and L(M/a,t)=1 means L(T,M/a)=L(T/a,not m)•L(M/a,t)=3/8•1=3/8=6/16≠5/16 so:

L(H,M/a)≠L(T,M/a) meaning L(H/a,m)≠L(T/a,m) which conflicts with "L(H/a,m)=C(H)=1/2 and L(T/a,m)=C(T)=1/2" where 1/2=1/2 so L(H/a,m)=L(T/a,m).

Again I didn't study probabilities so in case it wasn't clear, what I was trying to say is: assuming you're right, if the chance of the Heads world is 5/8 and the chances of it being Monday knowing we're in heads world is half, then the chances of Heads and also Monday are 2.5/8=5/16 whereas for Tails it's always Monday so it's 3/8=6/16. Then, clearly the assumption that "L(H/a,m)=C(H)=1/2 and L(T/a,m)=C(T)=1/2." is false, clearly since 5/16 and 6/16 differ there's a higher chance that it's Tails if we know it's Monday.

There's another inconsistency between the results and these functions but I don't disagree with these functions: "L(M/a)=C(H).L(M/a,h)+C(T).L(M/a,t)=1/2 . 1/2 + 1/2 . 1 =3/4 and L(W/a)=C(H).L(W/a,h)+C(T).L(M/a,t)=1/2 . 1/2 + 1/2 . 0 =1/4" so on waking up, it's three quarters chance it's Monday and one quarter chance it's Wednesday, I agree with that. However that means out of 5/8 chance of it being Heads, 3/8 is Monday and 2/8 is Wednesday which contradicts with the initial assumption that L(M/a,h)=L(W/a,h)=1/2.

Considering that odds for heads and tails worlds are in fact 50/50, what didn't check out for you checks out here: if L(H/a,not m)=1/2, knowing that L(M/a,h)=L(W/a,h)=1/2 we get L(H,M/a)=L(H/a,not m)•L(M/a,h)=1/2•1/2=1/4

Similarly, L(T/a,not m)=1/2 and L(M/a,t)=1 means L(T,M/a)=L(T/a,not m)•L(M/a,t)=1/2•1=1/2 so:

2•L(H,M/a)=L(T,M/a) meaning L(H/a,m)=L(T/a,m)/2 because 1/4=1/2/2

In (more) words, this means the Heads world having a 1/2 total chance, half of it being Monday so 1/4 chance, whereas Tails world has a 1/2 chance and it is always Monday, then it's twice less likely that if it's a Monday it's Heads because 1/4 is half of 1/2.

The other functions that were inconsistent with the results can be verified by adding up both chances for a total Monday chance L(M/a)=L(H,M/a)+L(T,M/a)=1/4+1/2=3/4 and L(W/a) is the rest it's 1-3/4=1/4 and it's also half of Heads chance L(W/a)=L(H/a,not m)/2=1/2/2=1/4.

Also to be honest I don't quite know why "not m" is used at all, if we don't say that we know it's Monday, it's implicitly unknown that it's Monday. But it doesn't affect the logic so it's not a critique it's a curiosity of mine. Ners14 (talk) 23:20, 26 June 2026 (UTC)

Zuboff's answer = 1/2 or 1/3?

So what's Zuboff's opinion? Is it 1/2 or 1/3? I feel like he never gave his answer, but I might be wrong. He just says that we're all the same person, but what's his answer then?

(the objective answer, I believe, is that it depends on the question you ask. but what's Zuboff's stance?) Niepodkoloryzowany (talk) 09:51, 29 October 2024 (UTC)

Shouldn't there be a refutation to the "no new information" argument in the article?

I don't know if it's because nobody bothered to write up a proper citeable article/paper about it, or if there's a different reason, but given that the main crux of the Halfer Position is "There's no new information so the result can't have changed" - easily refuted with "'How will/did the coin flip?' is a context-dependent question, it can have different answers depending on surrounding context". As in, "If you have yet to go to sleep, what do you think the coin flip will be" and "If you have just woken up, what do you think the coin flip was" are not the same question. You don't need new information to give different answers. ~2026-13436-24 (talk) 19:56, 1 March 2026 (UTC)