Posterior Probability¶
Summary¶
The probability of a hypothesis after considering new evidence — the updated belief that Bayesian reasoning produces.
Definition¶
In Bayes' theorem: P(H|E) — the probability that hypothesis H is true given that evidence E is observed.
The Update Process¶
Prior (P(H)) + Evidence (P(E|H), P(E|¬H)) → Posterior (P(H|E))
The posterior becomes the prior for the next round of evidence — beliefs update iteratively.
Classic Examples¶
Steve the Librarian¶
| Stage | Probability |
|---|---|
| Prior (before description) | 1/21 = 4.8% |
| Posterior (after "meek and tidy" description) | 4/24 = 16.7% |
The description increased the probability from 4.8% to 16.7% — significant, but still unlikely.
Breast Cancer Screening¶
| Stage | Probability |
|---|---|
| Prior (before test) | 1% |
| Posterior (after positive mammogram) | 25% |
| Final (after follow-up tests) | 0% or 100% |
The Geometric View¶
In the 1×1 square, the posterior is the proportion of the hypothesis in the restricted shape — after evidence eliminates some possibilities, what fraction of the remaining space does the hypothesis occupy?
Key Insight¶
Irrelevant evidence doesn't change your beliefs. If the likelihood is the same whether the hypothesis is true or false (P(E|H) = P(E|¬H)), the posterior equals the prior.
Posterior as Actionable Belief¶
The posterior isn't just a number — it's the basis for decisions: - In medicine: whether to proceed with treatment - In science: whether a model is validated - In daily life: whether to update your opinion