Priors

Priors, in Bayesian statistics, refer to the prior distribution over a parameter before observing data. They encode beliefs or uncertainty about the parameter and can incorporate information from previous studies, expert judgment, or theoretical considerations.

Priors can be informative, reflecting specific knowledge; noninformative or objective priors aim to exert minimal influence when data are informative; and weakly informative priors provide regularization to stabilize inferences. They may be proper (integrate to one) or improper (do not). Conjugate priors are chosen for mathematical convenience because the posterior is of the same distribution family as the prior.

In Bayesian inference, the posterior distribution combines the likelihood with the prior via Bayes' theorem: posterior is proportional to likelihood times prior. With large data, the influence of the prior diminishes, but with sparse data it can dominate. Conjugate priors yield closed-form posteriors; common examples include Beta priors for binomial probabilities, Gamma priors for Poisson rates, normal priors for means with known variance, and Dirichlet priors for multinomial probabilities.

Choosing priors involves considerations of subjectivity and robustness. Analysts often perform prior sensitivity analysis, prior predictive checks, or use hierarchical priors to share information across related parameters. In model selection or averaging, priors over models or over parameter spaces affect results and must be stated.

In practical applications, priors appear in machine learning as regularization in Bayesian neural networks and in other probabilistic models. The prior structure influences learning, uncertainty quantification, and model complexity, and is a core component of Bayesian inference.