I remember (dimly, as human memories go) the first time I self-identified as a "Bayesian". Someone had just asked a malformed version of an old probability puzzle, saying:

If I meet a mathematician on the street, and she says, "I have two children, and at least one of them is a boy," what is the probability that they are both boys?

In the correct version of this story, the mathematician says "I have two children", and you ask, "Is at least one a boy?", and she answers "Yes". Then the probability is 1/3 that they are both boys.

But in the malformed version of the story—as I pointed out—one would common-sensically reason:

If the mathematician has one boy and one girl, then my prior probability for her saying 'at least one of them is a boy' is 1/2 and my prior probability for her saying 'at least one of them is a girl' is 1/2. There's no reason to believe, a priori, that the mathematician will only mention a girl if there is no possible alternative.

So I pointed this out, and worked the answer using Bayes's Rule, arriving at a probability of 1/2 that the children were both boys. I'm not sure whether or not I knew, at this point, that Bayes's rule was called that, but it's what I used.

And lo, someone said to me, "Well, what you just gave is the Bayesian answer, but in orthodox statistics the answer is 1/3. We just exclude the possibilities that are ruled out, and count the ones that are left, without trying to guess the probability that the mathematician will say this or that, since we have no way of really knowing that probability—it's too subjective."

I responded—note that this was completely spontaneous—"What on Earth do you mean? You can't avoid assigning a probability to the mathematician making one statement or another. You're just assuming the probability is 1, and that's unjustified."

To which the one replied, "Yes, that's what the Bayesians say. But frequentists don't believe that."

And I said, astounded: "How can there possibly be such a thing as non-Bayesian statistics?"

That was when I discovered that I was of the type called 'Bayesian'. As far as I can tell, I was born that way. My mathematical intuitions were such that everything Bayesians said seemed perfectly straightforward and simple, the obvious way I would do it myself; whereas the things frequentists sai...

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