Section 5.2

Bayes' Rule and Base Rates

Inverting conditional probabilities with priors: diagnostic tests, unreliable witnesses, guessing students, base-rate traps.

Toolkit

Bayes’ rule inverts a known conditional probability using a prior, which is essential whenever the question gives $\mathbb{P}(\text{evidence}\mid\text{state})$ but asks for $\mathbb{P}(\text{state}\mid\text{evidence})$ .

\mathbb{P}(H\mid E)=\frac{\mathbb{P}(E\mid H)\,\mathbb{P}(H)}{\mathbb{P}(E)} =\frac{\mathbb{P}(E\mid H)\,\mathbb{P}(H)}{\sum_i \mathbb{P}(E\mid H_i)\,\mathbb{P}(H_i)}.

For a binary state (e.g. disease present/absent, truth/lie, knows/guesses) with prior $p$ , sensitivity $\mathbb{P}(E\mid H)$ and false-positive rate $\mathbb{P}(E\mid H^c)$ :

\mathbb{P}(H\mid E)=\frac{p\,\mathbb{P}(E\mid H)}{p\,\mathbb{P}(E\mid H)+(1-p)\,\mathbb{P}(E\mid H^c)}.

When $p$ is small, the denominator is dominated by the $(1-p)\mathbb{P}(E\mid H^c)$ term, so even a reliable test yields a posterior far below naive intuition: always compute both terms explicitly rather than trusting the stated accuracy alone.

Problems

5.12 Diagnostic Test Base Rate+2 variations Easy Free