(This post is elementary: it introduces a simple method of visualizing Bayesian calculations. In my defense, we've had other elementary posts before, and they've been found useful; plus, I'd really like this to be online somewhere, and it might as well be here.)

I'll admit, those Monty-Hall-type problems invariably trip me up. Or at least, they do if I'm not thinking very carefully -- doing quite a bit more work than other people seem to have to do.

What's more, people's explanations of how to get the right answer have almost never been satisfactory to me. If I concentrate hard enough, I can usually follow the reasoning, sort of; but I never quite "see it", and nor do I feel equipped to solve similar problems in the future: it's as if the solutions seem to work only in retrospect.

Minds work differently, illusion of transparency, and all that.

Fortunately, I eventually managed to identify the source of the problem, and I came up a way of thinking about -- visualizing -- such problems that suits my own intuition. Maybe there are others out there like me; this post is for them.

I've mentioned before that I like to think in very abstract terms. What this means in practice is that, if there's some simple, general, elegant point to be made, tell it to me right away. Don't start with some messy concrete example and attempt to "work upward", in the hope that difficult-to-grasp abstract concepts will be made more palatable by relating them to "real life". If you do that, I'm liable to get stuck in the trees and not see the forest. Chances are, I won't have much trouble understanding the abstract concepts; "real life", on the other hand...

...well, let's just say I prefer to start at the top and work downward, as a general rule. Tell me how the trees relate to the forest, rather than the other way around.

Many people have found Eliezer's Intuitive Explanation of Bayesian Reasoning to be an excellent introduction to Bayes' theorem, and so I don't usually hesitate to recommend it to others. But for me personally, if I didn't know Bayes' theorem and you were trying to explain it to me, pretty much the worst thing you could do would be to start with some detailed scenario involving breast-cancer screenings. (And not just because it tarnishes beautiful mathematics with images of sickness and death, either!)

So what's the right way to explain Bayes' theorem to me?

Like this:

W...

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