Bayes Theorem

Study guide: Exam P-1

Bayes' Theorem is a formula for reversing conditional probabilities, updating 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:

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)}

• H is the hypothesis and E is the observed evidence
• If H and E are independent (with P(E) > 0), observing E leaves the prior unchanged: P(H \mid E) = P(H)
• See also Bayesian Credibility

Identifying a High-Risk Policyholder After a Claim 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 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%.