Other meanings of Posterior distribution
BAYESIAN INFERENCE
A posterior distribution represents updated uncertainty about an unknown quantity after combining prior information with observed data. It is the central probability distribution in Bayesian inference, turning assumptions and evidence into a quantified basis for prediction and decision-making.
The posterior distribution is the conditional distribution of an unknown parameter or hypothesis after data have been observed. If θ denotes the quantity of interest and y the data, Bayes’ rule gives p(θ|y) = p(y|θ)p(θ) / p(y). The prior distribution p(θ) expresses uncertainty before seeing y, while the likelihood p(y|θ) measures how compatible the observed data are with each possible value of θ. The denominator p(y), called the marginal likelihood or evidence, normalizes the result so that the posterior is a valid probability distribution.1 In proportional form, the rule is often written posterior ∝ likelihood × prior, because the normalizing constant does not depend on θ.
A posterior distribution combines evidence with a declared model rather than replacing judgment with a purely data-driven calculation. A prior may be informative, weakly informative, or deliberately broad; its effect is usually strongest when data are sparse and diminishes when the likelihood is highly concentrated. The posterior can describe a continuous parameter, a discrete hypothesis, a latent variable, or an entire collection of unknowns. Summaries include posterior means, medians, modes, variances, credible intervals, and posterior probabilities such as Pr(θ > 0|y). A 95% credible interval is a probability statement about θ conditional on the model, prior, and data; it is not the same interpretation as a frequentist confidence interval.2
Posterior distributions support estimation, prediction, model comparison, and decision analysis. In simple conjugate models, the posterior has a familiar closed form: a beta prior combined with binomial data produces a beta posterior, and a normal prior with a normal sampling model produces another normal distribution under suitable assumptions. More complex models generally require numerical methods. Markov chain Monte Carlo, including the Metropolis algorithm and Hamiltonian Monte Carlo, generates dependent draws whose distribution approximates the posterior; importance sampling, variational inference, and sequential methods offer other approximations.3 Posterior predictive distributions extend inference to future observations by averaging predictions over posterior uncertainty rather than plugging in a single estimate.
The posterior is conditional on the model as well as the data, so a sharply concentrated posterior can still be misleading when the likelihood is misspecified or important variables are omitted. Prior sensitivity analysis examines how conclusions change under plausible alternative priors, while prior predictive checks ask whether a model generates data resembling the domain before observations are used.4 In hierarchical models, posterior distributions can produce partial pooling: estimates for small groups are drawn toward a shared population distribution, often improving predictive performance while retaining group-level differences. Bayesian model averaging goes further by averaging across models according to their posterior probabilities. In high-dimensional problems, posterior geometry can create strong correlations, funnels, or multiple modes that make computation and convergence assessment difficult even when the mathematical posterior is well defined.
Posterior conclusions are conditional on the chosen prior, likelihood, and model; transparent analysis commonly reports assumptions, diagnostics, and sensitivity to reasonable alternatives.
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