Bayes' theorem: updating a belief given evidence

The problem, from first principles

A medical test for a rare disease comes back positive. Should you be worried? Your gut says "the test is 99% accurate, so I'm almost certainly sick," but that's the wrong question, it ignores how rare the disease was in the first place. If the disease affects 1 in 100 people, and even a small fraction of HEALTHY people also test positive (a false positive rate), the healthy false-positives can vastly outnumber the sick true-positives in raw counts, even with a 99%-accurate test. The right question is: "of everyone who tests positive, healthy and sick combined, what fraction are actually sick?"

Bayes' theorem is the formula that answers exactly this: it takes a prior belief (how common is the disease, before any test), combines it with how the evidence behaves under each hypothesis (how likely is a positive test if you're sick, vs if you're not), and produces the correct, updated belief (how likely are you sick, given the positive test).

From theory to code

Theory gives the raw formula (posterior = likelihood * prior / evidence) and, for the common binary-hypothesis case, shows how to compute the evidence term yourself from the two conditional likelihoods and the prior.

Implement bayes_theorem(prior, likelihood, evidence) first, the direct formula, then posterior_binary(prior_h, likelihood_e_given_h, likelihood_e_given_not_h), which computes the evidence term and calls the first function.

Constraints

• All probabilities are plain floats in [0, 1].
• posterior_binary must compute the evidence term itself (P(evidence), the total probability of seeing the evidence at all, summed over both the H-true and H-false cases), not take it as a separate argument.
• posterior_binary should call bayes_theorem rather than reimplementing the same division.

Hints

Open one at a time. Each gives away a little more than the last.