Bayes’ Theorem

Bayes' theorem states P(H|E) = P(E|H)P(H)/P(E) when P(E) is positive. Equivalently, posterior odds equal prior odds times the likelihood ratio for two alternatives.

Bayes' theorem reverses a conditional probability using the base rate: P(H|E) = P(E|H)P(H)/P(E), provided P(E) is positive. The denominator accounts for all ways the evidence can occur. A high probability of a signal given a hypothesis is therefore not the same as a high probability of the hypothesis given that signal.

In two-hypothesis odds form, posterior odds equal prior odds multiplied by the likelihood ratio. It is odds, not probability, that multiply this way; probability is recovered by dividing the odds by one plus the odds. Natural-frequency tables often make the same calculation easier to inspect.

When to use it

When analyzing data; when building predictive models; when evaluating statistical claims; when quantifying uncertainty.

How it can help

Define base rates and signal likelihoods, include all ways the evidence can arise, and calculate with normalized probabilities or odds. A natural-frequency table can make the denominator visible.