Bayesian

Study guides: Exam ASTAM, Exam ATPA

Bayesian probability interprets probability as a degree of belief about uncertain propositions, updated as new evidence arrives. Starting from a prior P(H) for a hypothesis H and evidence E with likelihood P(E \mid H), Bayes Theorem yields the posterior P(H \mid E).

P(H \mid E) = \frac{P(E \mid H)\, P(H)}{P(E)}

• The denominator P(E) = \sum_i P(E \mid H_i)\,P(H_i) is the marginal probability of the evidence, from the law of total probability over all competing hypotheses.
• The prior encodes initial belief; the likelihood reweights it by how well each hypothesis explains the evidence.
• This contrasts with the frequentist view, which treats probability as a long-run frequency rather than a belief to be updated.

Updating Disease Probability After a Positive Test A disease affects 1% of the population. A diagnostic test has 95% sensitivity (true positive rate) and a 5% false positive rate. A randomly selected person tests positive. What is the probability they actually have the disease?

Answer Let H = has the disease and E = tests positive, so P(H) = 0.01, P(E \mid H) = 0.95, P(E \mid H^c) = 0.05. First find the marginal probability of a positive test:

\begin{align*} P(E) &= P(E \mid H)P(H) + P(E \mid H^c)P(H^c) \\ &= 0.95(0.01) + 0.05(0.99) \\ &= 0.059 \end{align*}

Then apply Bayes' theorem:

\begin{align*} P(H \mid E) &= \frac{P(E \mid H)\,P(H)}{P(E)} \\ &= \frac{0.95(0.01)}{0.059} \\ &\approx 0.161 \end{align*}

Despite the positive test, there is only about a 16.1% chance the person has the disease, because the disease is rare.