Sleeping Beauty Paradox

[PairTheBoard]

The is very apropo for an insight into the General Principle Sklansky has been preaching lately, that when you have two choices, A and B, and know nothing else about them than that there are two of them, you should consider them equally likely. The computation of 1/3 is the result of a Misapplication of that Principle. It's similiar in spirit to the misapplication of the Principle people commonly make in the Monty Hall puzzle when they don't think they should switch doors.

Solution in White:

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<font color="white">
When Beauty awakes she asks herelf, is it Monday or Tuesday, A or B? Thinking she has no other information about that, having amnesia, she decides to apply Sklansky's Principle and say to herself, there's a 50-50 chance it is Monday or Tuesday.

So she thinks to herself. 1/2 chance it's Monday which could equally well happen from Heads or Tails. 1/2 chance it's Tuesday which could only happen from Tails. So Heads has a Baysian weight of 1/2 while Tails has a weight of 1/2+1/2 = 1. So the relative Baysian weight for Heads is to Tails is 1:2. That is 1/3 chance the Coin was Heads.

What she fails to realize is that the two choices Monday,Tuesday are not equally likely. There is more information contained in the problem. But she corrects her poor thinking and recomputes. What are the real chances it is Monday or Tuesday?

She thinks to herself, am I waking up due to a Heads or a Tails? There's a 50% chance I'm waking up due to Heads in which case this must be Monday. There's a 50% chance I'm waking up due to Tails in which case it could equally likely be Monday or Tuesday. So, labeling the probabilities according to where they came from,

P(Monday) = (.5)H + (.5)(.5)T = 75%
P(Tuesday) = (.5)(.5)T = 25%

So having awoken she knows there is a 75% chance it is Monday, and a 25% chance it is Tuesday. 2/3 of the time that it's Monday will be because of Head. 1/3 of the time that it's Monday will be because of Tails. And all the time that it's Tuesday will be because of Tails. So the relative Baysian weights for Heads and Tails are,

Heads weighted 2/3(75%) = 50%
Tails weighted 1/3(75%) + (1)(25%) = 50%

50% chance heads was flipped.

No Paradox. Bad math by people who think there is one.</font>

PairTheBoard

[PairTheBoard]

Thinking more about my solution in white I'm not happy with it. I did not describe her thinking process accurately. And my purported valid method ended with the conclusion that if she is awake and it's Monday she's twice as likely to be there from Heads as Tails. That can't be right.

I was confused by your description which talked about 1.5 awakenings. An easier way for me to think about it is counting the awakenings. Over the long run - many repititions of this experiment - you will see twice as many awakenings from Tails as From Heads. Every 2 times through you will on average see 1 heads awakening and 2 tails awakenings. So if she is awake she must conclude it is twice as likely to be from tails being flipped as from heads. Thus from her reference frame of being awake she views P(heads having been flipped)=1/3

Here is a link to a good explanation for why 1/3 should be considered valid as a 1/3 measure of "credence" to her upon awakening, that a heads was flipped.

Sleeping Beauty Problem


Here is what they say about its relationship to,

From link:
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Bas Van Fraassen's `Reflection Principle' (1984:244, 1995:19), even an extremely qualified version of which entails the following:

Any agent who is certain that she will tomorrow have credence x in proposition R (though she will neither receive new information nor suffer any cognitive mishaps in the intervening time) ought now to have credence x in R.
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I'm reading the Wikipedia solution too. It seems like we need a really good description of the probability space here to clarify it. But I guess some smart people have already thought about everything we might come up with. Or maybe not. Who knows?

I'll think about it tomorrow. It's getting late.

PairTheBoard

[KipBond]

I think the problem, as stated, was ambiguous -- or rather, not as ambiguous as is required for the "paradox". This part here:

[ QUOTE ]
On Sunday evening we ask Beauty what she thinks the probability is of the coin landing heads. Unsurprisingly she says 1/2. Whenever she wakes up she will say 1/3.

[/ QUOTE ]

The probability of the coin landing heads??? 1/2. The probability of it having landed heads?? 1/3 is a valid answer.

The reason 1/3 is now valid is because it can be implied that the question is actually: "what is the probability that the coin landed heads and then we woke you and asked you this question?"

Since they are asking the question twice as often when the coin is tails, it's more probable that the question is being asked after a tails flip (2/3).

So, Sleeping Beauty can then answer the question either with or without the implied "and we woke you and asked you this question". That implication is assumed to be "her reference point", but I don't agree with that, actually. If she's completely rational, she understands the question is ambiguous.

[bigmonkey]

Siegmund:[ QUOTE ]
I'll be interested in seeing what good argument, if any, there is for 1/3... sure haven't heard one yet and have trouble imagining one yet.



[/ QUOTE ]

See PairTheBoard's link to Elga's paper, where his argument for 1/3 begins properly in the second section, although all these Sleeping Beauty papers are pretty short and accessible. It's easier to sympathise with 1/3 when you grant that the scientists may as well flip the coin after already waking Beauty up on Monday.

The weird process is how Elga's argument goes something like:

P(H) is obviously 1/2. (intuition)
*Do some calculations*
OK she should believe P(H) is 1/3.

David Lewis replies to Elga and uses this method.

Being rational she should believe P(H) is 1/2.
As far as Lewis is concerned this is the end of the story.
But if you apply Elga's method to Lewis' answer you get the following strange conclusion:
Upon waking on Monday if Beauty were told it was Monday she should believe the probability of a future coin landing heads is 2/3.

Both of their explanations seem to contradict themselves. I think the conclusion is that Elga reasons from Beauty's pre-credence (1/2) to her betting odds (1/3), and Lewis reasons from her post-credence (1/2) to her pre-credence (1/2) and in the process reveals her betting odds for a different game on Monday (2/3)

I'm not quite sure how to interpret what that game is that could make her believe or bet on heads with credence 2/3.

From Elga we know that if she's told it's Monday she should have credence 1/2 in H, because it hasn't been flipped yet. If she's told it's Tuesday then she should have credence 0 in heads. So waking up with no information she knows there's some chance it's Monday and some chance it's Tuesday, therefore some chance P(H)=1/2 and some chance P(H)=0. This is why it looks like her overall value for P(H) &lt; 1/2 and can be worked out as 1/3. Lewis responds by giving her an abnormal credence that H is true given it's Monday, to counterbalance the 1/3 claim.

I think the explnation might be: she knows there's some chance P(H)=1/2 and some chance P(H)=0. Interpreted this means there's some chance she is being offered a fair bet and some chance she is being offered a guaranteed-to-lose bet. She therefore demands abnormally good odds on the bet in case it is a guaranteed-to-lose bet.

Every time I re-read Elga and Lewis they seem pretty convincing, which suggests my mind is a bit fickle.

f97tosc:[ QUOTE ]
I can't say I am familiar with the reflection principle, but I suspect that the "resolution" will have to do with the fact during certain experimental outcomes, we supress Sleeping Beauty's ability to state (or even form) her belief.

[/ QUOTE ]

The reflection principle is that:

Px(A|Px+y[A]= c)=c

Where Px and Px+y are credence functions at time x and time x+y. So the principle is that given that at some later time your credence for A is c then your credence for A now should be C. If you believed now that tomorrow you would receive some important information about soemthing that will make you have credence c in A, then you should now already have credence c in A. I think it could be argued that Beauty isn't completely rational because she believes she will be given an amnesiac, and therefore cannot be in a rational cognitive state, but then people argue that she knows exactly what memories she will forget, and also the amnesiac is only given to her after going back to sleep to forget her awakening, so when she wakes up she is supposed to be fully rational.

PairTheBoard: [ QUOTE ]
Solution in White:

[/ QUOTE ]

I did originally think Beauty was using a principle of indifference over whether it's Monday or Tuesday where she shouldn't be.
Then in the re-computation you feed in the probabilities of heads and tails being 1/2 and effectively get them straight back out again, which is what Lewis does. But he also has the weird consequence that upon being told it is Monday Beauty believes a future coin has 2/3 chance of landing heads.
And then you get the Elga paper involved arguing for 1/3.

My current conclusion is that there are credences and betting odds, and these aren't always the same. However, it's quite hard to locate where Beauty is using credences and where she is using betting odds. The long-term argument whereby she knows that 2/3 of her total awakenings will have been preceded by a tails really seems like a credence and not a betting odds.Perhaps the reason this seems like a paradox is because as Bayesian thinkers we find it hard to tell the difference between credences and betting odds, and get punished in examples like this.

KipBond: [ QUOTE ]
The reason 1/3 is now valid is because it can be implied that the question is actually: "what is the probability that the coin landed heads and then we woke you and asked you this question?"

[/ QUOTE ]

They could ask her before the experiment "what is the probability of heads and that we will ask you again later?" As she knows for a fact that they will ask her again later she should still say 1/2 at this point. Is your point that the supposed paradox is really a counter-example and refutation of the Reflection Principle?

[KipBond]

[ QUOTE ]
KipBond: [ QUOTE ]
The reason 1/3 is now valid is because it can be implied that the question is actually: "what is the probability that the coin landed heads and then we woke you and asked you this question?"

[/ QUOTE ]

They could ask her before the experiment "what is the probability of heads and that we will ask you again later?" As she knows for a fact that they will ask her again later she should still say 1/2 at this point.

[/ QUOTE ]

Once you combine the "waking" event with the "coin flipping" event, you get a separate event: a "flipping+waking" event. Since she will be woken up twice as often after a "tails" outcome of the flipping event, the odds are 1/3 she will have a "waking+heads" event, and 2/3 she will have a "waking+tails" event. It doesn't matter if she's asked this before or after being woken. The probability that the coin will be heads is 1/2. The probability that the coin will be heads and she will be woken is 1/3.

After 1,000,000 coin flips, there will be about 500,000 heads+waking events, and 1,000,000 tails+waking events.

[bigmonkey]

I don't see how you are offering a solution that doesn't just concede that Beauty violates Reflection.

The scientists ask her before, "What is the probability that between t1 and t2 a heads will land and you will awake?"

When she wakes up they ask her "What is the probability that between t1 and t2 a heads landed and you woke up?"

She knows before that she will wake up, and she can already know exactly what the experience of waking up is like, just to make sure she gets no new information.

If you answer 1/2 to the first question and 1/3 to the second question, it sounds like they only wake her up 2/3 of the time after a heads lands, but we know that is false, so I don't see why this is not a contradiction.

[KipBond]

[ QUOTE ]
I don't see how you are offering a solution that doesn't just concede that Beauty violates Reflection.

[/ QUOTE ]

Well, she should be able to answer question in 2 different ways, while clarifying the ambiguity.

[ QUOTE ]
The scientists ask her before, "What is the probability that between t1 and t2 a heads will land and you will awake?"

[/ QUOTE ]

What is t1 &amp; t2?

It depends on the exact question being asked as to whether the answer is 1/2 or 1/3.

1/2 of the coin flips will be heads.
1/3 of the flip-wakings will be heads.

It's ambiguous as to what question is being asked, so it needs to be phrased precisely. If the question doesn't differentiate between the "coin flips" and the "flip-wakings", then it can be answered 2 ways.

[bigmonkey]

t1 and t2 were just two moments in time I made up and didn't define. It's not really important, and should be something like t1 as when Beauty fell asleep and t2 when she awoke.

[ QUOTE ]
1/2 of the coin flips will be heads.
1/3 of the flip-wakings will be heads.

[/ QUOTE ]

That does seem like the intuitive thing to say before she falls asleep. When she wakes up they seem to be the same thing.

Are you saying that when she is woken up and asked what credence she has that she just experienced a heads-awakening she answers 1/3? And if asked if the coin landed heads she answers 1/2?

But if a heads occurs then she will experience a heads-awakening, and if she experiences a heads-awakening, then a heads occurred. They are equivalent, so P(H) should equal P(heads-awakening)

[Siegmund]

[ QUOTE ]
But if a heads occurs then she will experience a heads-awakening, and if she experiences a heads-awakening, then a heads occurred. They are equivalent, so P(H) should equal P(heads-awakening)

[/ QUOTE ]

Well, there's your explanation. No paradox at all; your "they are equivalent, P(H) should equal P(heads-awakening)" is just plain wrong. P(H) is (heads)/(coinflip outcomes) while P(heads-awakening) is (heads-awakenings)/(all awakenings). Unless EVERY coin flip results in the same number of awakenings, P(H) and P(heads-awakening) are not equivalent.

It amazes me so much has been written about this.

[KipBond]

Here's a simpler scenario:

Same setup w/ the amnesiac Beauty -- only this time if the coin is heads, I don't wake her up at all. I flip the coin again. If it's tails, I wake her up &amp; ask what the P(H) is (*). What's her answer?

(*) The way I phrase the question is the ambiguous part. What is the P the coin will be heads when you flipped it? Or what is the P that the coin you last flipped before waking me (**) is heads?

(**) Note that this "waking" (&amp; the whole scenario) is actually additional information, so it's wrong to say Beauty doesn't have different information to change her mind regarding the P(H).