From a friend: “1. What’s the probability of a HS athlete going pro? 2. Suppose we know a pro athlete. What’s the probability she was a college athlete?”
So I was thinking about my favorite intuitive illustrations and explanations of conditional probability and Bayes’ theorem, e.g.
I asked for help here, there’s some good discussion: https://math.stackexchange.com/questions/2407913/is-it-possible-to-divide-a-square-into-four-parts-of-arbitrary-size-with-two-lin
To-do:
One cool thing about this is that you can feel out which things are linear and which are not. The slope of the diagonal line is independent of P(A), which I would not have intuited. And P(A|B) and P(A|¬B) are nonlinear functions of P(A), P(B|A), and P(B|¬A), which I don’t think I had any intuition about, but feels central to a lot of counterintuitive results of conditional probability questions.
As nice as it is to “feel out”, I want to be able to SEE any of those things I feel — spatialize the state space. Plot everything as a function of everything else, see the steepest slopes, marginal sensitivities, etc. A kind of phase space, idk. Ideally with the same visualization. Explode it along a third axis of all possible values of the current parameter… yessssssss that’d be so good, so doable. Whichever parameter you’re currently holding, explode out all possible values along the z-axis, so you can see the nonlinear effects of dragging by dx.
I still want to make something that captures some feeling I have of weighing prior and posterior confidences, and the updating flowing one way or the other accordingly, almost hydraulically.