Priorijakauman

Priorijakauma (Finnish for prior distribution) is a probability distribution that encodes beliefs about a parameter before observing data in Bayesian statistics. In Bayes' theorem, the posterior distribution is proportional to the product of the likelihood and the prior: p(theta|data) ∝ p(data|theta) p(theta). A prior can be informative, encoding substantive beliefs, or noninformative (sometimes called vague or objective), aiming to exert minimal influence when data are plentiful. Priors may be proper (integrates to one) or improper (e.g., p(theta) ∝ 1), the latter requiring the likelihood to yield a proper posterior.

Conjugate priors are chosen for mathematical convenience because the posterior is in the same family as the prior, e.g., Beta prior for a binomial proportion, Gamma prior for a Poisson rate, Normal prior for a known-variance mean. More flexible priors include normal or hierarchical structures, such as hyperpriors on group-level effects in hierarchical models.

Prior elicitation involves translating expert knowledge or historical data into a prior distribution. Empirical Bayes uses the data to estimate prior hyperparameters, while objective or reference priors seek to minimize influence in a formal sense (e.g., Jeffreys prior, reference priors).

Priors play a central role in regularization, shrinking estimates toward prior beliefs when data are sparse. They also affect model comparison and posterior uncertainty, making sensitivity analysis important. Inference is usually performed by sampling from the posterior (e.g., MCMC) or via variational methods, with the prior sometimes impacting convergence and efficiency.

Critics note subjectivity and potential bias in priors, but proponents argue that transparent articulation of prior knowledge improves inference and interpretability. Priorijakauman is a foundational concept in Bayesian data analysis and appears across statistics, machine learning, and applied sciences.