$P(H \mid E) = \frac{P(E \mid H)\,P(H)}{P(E)} = \frac{P(E \mid H)\,P(H)}{\displaystyle\sum_{i} P(E \mid H_i)\,P(H_i)}$ $\text{where } H = \text{hypothesis},\quad E = \text{observed evidence}$ Bayes' Theorem gives a formula for reversing conditional probabilities: it updates the prior probability $P(H)$ of a hypothesis to the posterior probability $P(H \mid E)$ after observing evidence $E$. The denominator $P(E)$ is computed via the Law of Total Probability across a partition $\{H_i\}$ of the sample space. Bayes' Theorem is central to credibility theory, risk classification, and predictive modeling in actuarial science. > [!example]- Identifying a High-Risk Policyholder After a Claim {💡 Example} > 20% of policyholders are high-risk ($H$) and 80% are low-risk ($L$). A high-risk policyholder files a claim in year 1 with probability 0.40; a low-risk one with probability 0.10. A randomly selected policyholder files a claim. What is the probability they are high-risk? > > > [!answer]- Answer > > First find $P(\text{claim})$ by the Law of Total Probability: > > $P(C) = P(C \mid H)P(H) + P(C \mid L)P(L) = (0.40)(0.20) + (0.10)(0.80) = 0.08 + 0.08 = 0.16$ > > Now apply Bayes' Theorem: > > $P(H \mid C) = \frac{P(C \mid H)\,P(H)}{P(C)} = \frac{0.40 \times 0.20}{0.16} = \frac{0.08}{0.16} = 0.50$ > > After observing a claim, the probability the policyholder is high-risk rises from 20% to 50%.