A probability distribution over unknown parameters or latent quantities obtained by updating a prior distribution with observed data via the likelihood; it assigns to each parameter value its plausibility conditional on the observed dataset.

Principle

Update prior beliefs by weighting parameter values according to how well the observed data are predicted by those values (Bayes' rule): posterior ∝ prior × likelihood, normalized to integrate to one.

Demonstration

With a Beta(α,β) prior for a Bernoulli success probability and observing k successes in n trials, the posterior is Beta(α+k, β+n−k), giving updated credible mass for the probability.

Misapplication

Reporting only a point summary such as the mode as if it represented the whole posterior, or interpreting the posterior as a frequency statement about repeated sampling without conditioning on the same observed data and prior.

Consequence

A posterior distribution provides a coherent quantified uncertainty for parameters that permits propagation through predictions, decision rules under loss functions, and model comparison when combined with marginal likelihoods.

Reversal

The prior is the distribution before conditioning on data; ignoring data yields the prior, whereas ignoring prior information yields only the likelihood, which is not a probability distribution over parameters.

Boundary

Requires a specified prior and likelihood; improper priors or misspecified likelihoods can produce improper or misleading posteriors; results depend on both data and the chosen prior model class.

Semantic Tension

Often contrasted with frequentist confidence intervals or point estimators: a posterior is a conditional probability distribution expressing degree of belief, whereas frequentist constructs are sampling-procedure guarantees over hypothetical repetitions.

Synthesis

A posterior distribution is the normalized update of a prior by the likelihood that quantifies the conditional plausibility of parameter values given the observed data.