Nobody Understands Probability

For those who didn't make it all the way down:

People often intuitively think of probabilities as a fact about the world, when in reality probabilities are a fact about our model of the world.

This article is not particularly clear. It doesn't have a clear discussion of Bayesian versus frequentist interpretations of probability or inferential statements that are conditioned on the unobserved true parameter versus the observed data. It's hard to understand the subtlety of probability without understanding p(theta), p(x), p(theta|x) and p(x|theta).

> However, the answer is not, in fact, 1/3. Why is this?

This seems like a canard to me.

Here is my defense of 1/3 as a correct answer: http://gist.github.com/578386

> Is Bayes’ theorem wrong?

> No, the answer comes from an unfortunate namespace collision in the word “given”. The man “gave” us the information that he has at least one male child. By this we mean that he asserted the statement “I have at least one male child.” Now our issue is when we confuse this with being “given” that the man has at least one male child, in the sense that we should restrict to the set of universes in which the man has at least one male child. This is a very different statement than the previous one. For instance, it rules out universes where the man has two girls, but is lying to us.

No, we are assuming that the givens are facts that are true.

> Even if we decide to ignore the possibility that the man is lying, we should note that most universes where the man has at least one son don’t even involve him informing us of this fact, and so it may be the case that proportionally more universes where the man has two boys involve him telling us “I have at least one male child”, relative to the proportion of such universes where the man has one boy and one girl. In this case the probability that he has two boys would end up being greater than 1/3.

No, we don't have to consider universes where the man has at least one male child but does not inform of us of this fact. We have a set of givens that are assumed to be true, and based on those givens and the rules of logic, we can make justifiable statements of probabilities.

It's a canard, but still informative. I do wish he hadn't claimed that 1/3rd is wrong by Bayesian statistics because the frequentist approach, with the same interpretation of the problem, yields exactly the same results.

It's still a valuable example of how to represent unreliable measurement processes in your model, and the importance of doing so.