Posterior
Consider an unknown quantity, such as the chance that a coin lands heads, and a prior belief about the quantity. The posterior is a description of what you believe about the quantity after you see the data, found by updating the prior with the data. In the case of the coin, you may have a prior belief that the coin is fair: It lands on heads 50% of the time. But if you flip the coin 1000 times and it comes up heads 900 times, your posterior belief may become that the probability of the coin landing on heads is around 90%.
Consider a statistical model $p(x \mid \theta)$ with parameter $\theta$, a prior $p(\theta)$, and observed data $x$. The posterior is the conditional probability distribution of the parameter given the data, which is given by Bayes’ rule:
\[p(\theta \mid x) = \frac{p(x \mid \theta) \, p(\theta)}{p(x)}.\]Properties
- The posterior is proportional to the likelihood times the prior: $p(\theta \mid x) \propto L(\theta) \, p(\theta)$.
- The maximum a posteriori (MAP) estimate of $\theta$ is the value that maximizes $p(\theta \mid x)$. If the prior is uniform, then the MAP estimate equals the maximum likelihood estimate.