A quick, example-driven note on Bayes’ theorem that computes prior/posterior probabilities step by step.
1) Bayes’ Theorem
Bayes’ theorem:
\[P(H\mid E) \;=\; \frac{P(E\mid H)\,P(H)}{P(E)} \tag{1}\]Here (H) is a hypothesis and (E) is the evidence (new information).
- (P(H)): prior — belief before seeing the evidence
- (P(H\mid E)): posterior — updated belief after seeing the evidence
Interpreting probability as a degree of belief is the Bayesian view (as opposed to the traditional frequentist view).
2) Why it matters (in one line)
Bayes’ theorem updates belief about (H) using new information (E).
Frequentist workflows are more deductive (given a fixed model/space), while Bayesian workflows are inductive—they start with a prior and move closer to the truth by iteratively updating with data.
3) Quick glossary
- (H): Hypothesis — the claim/event of interest
- (E): Evidence — newly observed information
- (P(H)): prior, (P(H\mid E)): posterior
“Prior/Posterior” simply mean before/after observing the evidence.
4) Example 1 — Posterior after one positive test
- Prevalence (prior): (P(H)=0.001) (0.1%)
- Sensitivity: (P(E\mid H)=0.99)
- Specificity: (P(E^c\mid H^c)=0.98 \Rightarrow P(E\mid H^c)=0.02)
Using the law of total probability for (P(E)):
\[P(H\mid E) =\frac{P(E\mid H)P(H)}{P(E\mid H)P(H)+P(E\mid H^c)P(H^c)} =\frac{0.99\times 0.001}{0.99\times 0.001+0.02\times 0.999} \approx 0.047. \tag{2}\]Interpretation: With one positive test only, the posterior is about 4.7% (very low prevalence makes false positives matter a lot).
5) Example 2 — Posterior after two consecutive positives
Use the posterior from Example 1 (0.047) as the new prior, then update again:
- New (P(H)=0.047), still (P(E\mid H)=0.99), (P(E\mid H^c)=0.02)
Interpretation: Two positives in a row raise the posterior to ≈70.9%—a classic illustration of Bayesian updating.
6) Practical notes
- (P(A\mid B)=1-P(A^c\mid B)) does hold, but (P(A\mid B)) is not related to (1-P(A\mid B^c)).
- With very small priors (low prevalence), the PPV can be low even for accurate tests → consider repeat testing or combining with a more specific test.
- Decision-making should also weigh cost/utility (costs, invasiveness, treatment benefits/risks), not just probabilities.
References
- Source (in Korean): https://angeloyeo.github.io/2020/01/09/Bayes_rule.html