Let be a variable in for the true hypothesis, and let be the possible values of such that is mutually exclusive and exhaustive. Then, Bayes' theorem states:
with a proof that runs as follows. By the definition of conditional probability,
By the law of marginal probability:
By the definition of conditional probability again:
Done.
Note that this proof of Bayes' rule is less general than the proof of the odds form of Bayes' rule.
Using the Diseasitis example problem, this proof runs as follows:
Numerically:
Using red for sick, blue for healthy, and + signs for positive test results, the proof above can be visually depicted as follows:
if we replace the other Venn diagram for the proof of Bayes' rule, we should probably update this one too.