This is just the decomposition of P(A) into the sum over b of P(A|b)P(b). If she always cancels the bet though, she destroys this extra information, since the bet will end up cancelled regardless of whether the coin came up heads or tails.ġ) Yes, the prior is the weighted average of posteriors. And she can use this information to her benefit. She should lower her confidence in winning the bet, knowing the fact that she has been woken up. The fact that she is woken up on a not-Wednesday is extra information. I think this illuminates the apparent paradox too. So now she makes 28.125 cents for every dollar bet. But if she cancels the bet with 25% chance each time she wakes up, her expected win is 0.5 0.75 1.5 - 0.5 0.75 0.75 * 1 = 0.28125. That is she makes 25 cents per dollar bet, on average. If she always keeps the bet, her expected payout on Wednesday is 0.5 1.5 - 0.5 1 = 0.25. In other words she should keep the bet with 75% chance. instead she should she roll a 4-sided die and cancel the bet if she rolls a 1. Sleeping Beauty is right not to cancel the bet if she is woken up on not-Wednesday (so Monday or Tuesday, but she does not know which). What is this property actually called?Ģ) If it exists, is it a corollary that our prior distribution must remain unchanged unless we acquire new information? But now I can't find this reference anywhere, and I'm starting to doubt that I understood it correctly when I read it.ġ) Is there such a thing as this invariance requirement? Does anyone have a reference? It seems intuitive that the prior should be equivalent to the weighted average of posteriors, since it must contain all of our prior knowledge about a system. That is to say, the prior should be the weighted average of possible posteriors, where the weights are the likelihood that each posterior would be acheived after some measurement. In fact, I think I remember reading in one of Jaynes' papers about a requirement that any prior be invariant under the acquisition of new information. I suspect that these two distributions are always identical. Downey's resolution boils down to a conjecture that optimal bets on lotteries should be based on one's expected state of prior information just before the bet's resolution, as opposed to one's state of prior information at the time the bet is made. I recently read this blog post by Allen Downey in response to a reddit post in response to Julia Galef's video about the Sleeping Beauty problem.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |