jimrandomh comments on Conditioning on Observers - Less Wrong

6 Post author: Jonathan_Lee 11 May 2010 05:15AM

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Comment author: jimrandomh 12 May 2010 04:11:32PM 0 points [-]

I already addressed this elsewhere. The problem is that W is not a boolean, it's a probability distribution over observer moments, so P(W) and P(~W) are undefined (type errors).

Comment author: neq1 12 May 2010 04:24:13PM 0 points [-]

At one point in your post you said "For convenience let us say that the event W is being woken" and then later on you suggest W is something else, but I don't see where you really defined it.

You're saying W itself is a probability distribution. What probability distribution? Can you be specific?

P(H) and P(H|W) are probabilities. It's unclear to me how those can be well defined, but the law of total probability doesn't apply.

Comment author: jimrandomh 12 May 2010 05:03:34PM *  2 points [-]

Suppose we write out SB as a world-program:

SleepingBeauty(S(I)) =
{
coin = rnd({"H","T"})
S("starting the experiment now")
if(coin=="H"):
S("you just woke up")
S("you just woke up")
else:
S("you just woke up")
S("the experiment's over now")
return 0
}

This notation is from decision theory; S is sleeping beauty's chosen strategy, a function which takes as arguments all the observations, including memories, which sleeping beauty has access to at that point, and returns the value of any decision SB makes. (In this case, the scenario doesn't actually do anything with SB's answers, so the program ignores them.)

An observer-moment is a complete state of the program at a point where S is executed, including the arguments to S. Now, take all the possible observer-moments, weighted by the expected number of times that a given run of SleepingBeauty contains that observer moment. To condition on some information, take the subset of those observer-moments which match that information. So, P(coin=heads|I="you just woke up") means, of all the calls to S where I="you just woke up", weighted by probability of occurance, what fraction of them are on the heads branch? This is 1/3. On the other hand, P(coin=heads|I="the experiment's over now")=1/2.

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