Bayes and Total Probability

Bayes’ rule is easiest to understand as re-restricting the sample space after evidence arrives.

If an event A can produce evidence B, then the path A followed by B has joint probability

P(A ∩ B) = P(A)P(B | A).

If several mutually exclusive starting cases can all produce B, add their joint paths to obtain P(B).

Then

P(A | B) = P(A ∩ B) / P(B).

The numerator is the part of the observed evidence that came through A. The denominator is all observed evidence. The Bayes lab keeps the tree, joint paths, evidence denominator, and posterior synchronized so the formula never becomes a symbol swap.