Friday, March 13, 2015

Cage Match: Delong vs. Motl

Brad and Lubos are insulting each other. The subject is the sleeping beauty problem, my stripped down version of which goes like this: Sleeping Beauty knows probability, but can't remember being awakened. A fair coin is tossed, and if it comes up heads, SB is awakened only on Monday. If tails, SB is awakened on both Monday and Tuesday. Each time she is asked to say what the probability is that the coin was head. She doesn't know what day it is, so what should she say the probability is that the coin was heads?

To cut to the chase, Brad is right and Lumo wrong. The point is that even though the two branches are equally probable, the tails branch is sampled twice as often as the heads branch. Consequently, two out of three times she is asked, she will be on the tails branch. Heads, 1/3, tails 2/3.

One could make the point a bit starker by reducing the number of awakenings in the heads branch to zero or increasing the number of awakenings in the tails branch to ten. SB is really reporting on the joint probability that (1)heads was thrown and (2)she was awakened. Since P(heads) is 1/2, P(awakening) is in control.

UPDATE: I see that Lumo has now updated to consider sampling, but unfortunately still doesn't seem to get it. Here is another way to think of it. Suppose the experiment is done a large number of times, say N, such that heads has come up approximately N/2 times and tails N/2 times - we will just assume the equality is exact. Then SB will have been awakened N/2 times on the heads branch and N/2 + N/2 = N times on the tails branch. Given that she doesn't know what branch she is on, what's her best estimate that the current awakening occurred on the heads branch? Since 1/3 of the awakenings occur on the heads branch, it's obviously 1/3.

BTW, for those who wonder why we worry about such things, the answer is that it's a purely intellectual matter, forcing us to think deeply about what we mean by probability. Even better, a lot of smart guys, Princeton and Harvard physics profs, get it wrong.