Posterior probability

Summary

The posterior probability of a hypothesis $H$ is its conditional probability $P(H \mid D)$ given the observed data $D$. By Bayes' theorem, $P(H \mid D) = P(D \mid H) \, P(H) / P(D)$. So the posterior is proportional to the likelihood $P(D \mid H)$ times the prior probability $P(H)$. For an unknown parameter $\theta$ with prior density $\pi(\theta)$ and likelihood $L(\theta \mid x)$, the posterior density is $\pi(\theta \mid x) = L(\theta \mid x) \, \pi(\theta) / \int L(t \mid x) \, \pi(t) \, dt$.

Remarks

The posterior of one experiment can be the prior of the next experiment. So Bayesian updates can be done one observation at a time. A credible interval is an interval that has a given posterior probability, for example $95\%$.

Explanations

L1 human

The posterior is what you believe after you see the data. It mixes what you believed before (the prior) with how well each hypothesis explains the data (the likelihood).