The fairness of the Sleeping Beauty

post by MrMind · 2015-07-07T08:25:56.863Z · LW · GW · Legacy · 2 comments

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This post will attempt a (yet another) analysis of the problem of the Sleeping Beauty, in terms of Jaynes' framework "probability as extended logic" (aka objective Bayesianism).

TL,DR: The problem of the sleeping beauty reduces to interpreting the sentence “a fair coin is tossed”: it can mean either that no results of the toss is favourite, or that the coin toss is not influenced by anthropic information, but not both at the same time. Fairness is a property in the mind of the observer that must be further clarified: the two meanings cannot be confused.

What I hope to show is that the two standard solutions, 1/3 and 1/2 (the 'thirder' and the 'halfer' solutions), are both consistent and correct, and the confusion lies only in the incorrect specification of the sentence "a fair coin is tossed".

The setup is given both in the Lesswrong's wiki and in Wikipedia, so I will not repeat it here. 

I'm going to symbolize the events in the following way: 

- It's Monday = Mon
- It's Tuesday = Tue
- The coin landed head = H
- The coin landed tail = T
- statement "A and B" = A & B
- statement "not A" = ~A

The problem setup leads to an uncontroversial attributions of logical structure:

1)    H = ~T (the coin can land only on head or tail)

2)    Mon = ~Tue (if it's Tuesday, it cannot be Monday, and viceversa) 

And of probability:

3)    P(Mon|H) = 1 (upon learning that the coin landed head, the sleeping beauty knows that it’s Monday)

4)    P(T|Tue) = 1 (upon learning that it’s Tuesday, the sleeping beauty knows that the coin landed tail)

Using the indifference principle, we can also derive another equation.

Let's say that the Sleeping Beauty is awaken and told that the coin landed tail, but nothing else. Since she has no information useful to distinguish between Monday and Tuesday, she should assign both events equal probability. That is:

5)    P(Mon|T) = P(Tue|T)

Which gives

6)    P(Mon & T) = P(Mon|T)P(T) = P(Tue|T)P(T) = P(Tue & T)

It's here that the analysis between "thirder" and "halfer" starts to diverge.

The wikipedia article says "Guided by the objective chance of heads landing being equal to the chance of tails landing, it should therefore hold that". We know however that there's no such thing as 'the objective chance'.

Thus, "a fair coin will be tossed", in this context, will mean different things for different people.

The thirders interpret the sentence to mean that beauty learns no new facts about the coin upon learning that it is Monday.

They thus make the assumption:

(TA) P(T|Mon) = P(H|Mon)

So:

7)    P(Mon & H) = P(H|Mon)P(Mon) = P(T|Mon)P(Mon) = P(Mon & T)

From 6) and 7) we have:

8)    P(Mon & H) = P(Mon & T) = P(Tue & T)

And since those events are a partition of unity, P(Mon & H) = 1/3.

And indeed from 8) and 3):

9)    1/3 =  P(Mon & H) = P(Mon|H)P(H) = P(H)

So that, under TA, P(H) = 1/3 and P(T) = 2/3.

Notice that also, since if it’s Monday the coin landed either on head or tail, P(H|Mon) = 1/2.

The thirder analysis of the Sleeping Beauty problem is thus one in which "a fair coin is tossed" means "Sleeping Beauty receives no information about the coin from anthropic information".

There is however another way to interpret the sentence, that is the halfer analysis:

(HA) P(T) = P(H)

Here, a fair coin is tossed means simply that we assign no preference to either side of the coin.

Obviously from 1:

10)  P(T) + P(H) = 1

So that, from 10) and HA)

11) P(H) = 1/2, P(T) = 1/2

But let’s not stop here, let’s calculate P(H|Mon).

First of all, from 3) and 11)

12) P(H & Mon) = P(H|Mon)P(Mon) = P(Mon|H)P(H) = 1/2

From 5) and 11) also

13) P(Mon & T) = 1/4

But from 12) and 13) we get

14) P(Mon) = P(Mon & T) + P(Mon & H) = 1/2 + 1/4 = 3/4

So that, from 12) and 14)

15) P(H|Mon) = P(H & Mon) / P(Mon) = 1/2 / 3/4 = 2/3

We have seen that either P(H) = 1/2 and P(H|Mon) = 2/3, or P(H) = 2/3 and P(H|Mon) = 1/2.

Nick Bostrom is correct in saying that self-locating information changes the probability distribution, but this is true in both interpretations.

The problem of the sleeping beauty reduces to interpreting the sentence “a fair coin is tossed”: it can mean either that no results of the toss is favourite, or that the coin toss is not influenced by anthropic information, that is, you can attribute the fairness of the coin to prior or posterior distribution.

Either P(H)=P(T) or P(H|Mon)=P(T|Mon), but both at the same time is not possible.

If probability were a physical property of the coin, then so would be its fairness. But since the causal interactions of the coin possess both kind of indifference (balance and independency from the future), that would make the two probability equivalent. 

That such is not the case just means that fairness is a property in the mind of the observer that must be further clarified, since the two meanings cannot be confused.

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