Bayes' Theorem Illustrated

The Venn diagram makes visualizing the update so easy, you start to wonder why the Bayes rule is a "thing" at all.

It works especially well with the examples, like Yudkowsy's breast cancer screeening example.

This works well for the cancer example. I'm not picturing how it would work for the Monty Hall problem. Guess it's time to open the drawing app.

Being able to prove the Total Prob. Theorem and from there the Bayes Theorem has helped me solve problems (granted from a textbook and not real-life ones).

-> In the problem, the sample space comes divided into mutually disjoint portions.

-> Using the definition of probability, we can compute joint distribution from the a-priori distributions.

-> Using the definition of conditional probability, we can flip around the inference direction.

I usually find that all problems (Naive-Bayes spam, cancer) etc. all show this pattern.

This

I'll give it a shot and get back to you. Should be doable.

Why?

The original diagram isn't important any more. It is replaced by the new values that are the results of the experiment. Kind of like:

  world = get_world()
  for experiment in experiments():
    world = combine(world, experiment())

The original world doesn't exist after the first experiment. The experiment has provided us a new world.