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.
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 probability 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.
Suppose we write out SB as a world-program:
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.
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 probability 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.