Bayesian probability

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Bayesian statistics
Posterior = Likelihood × Prior ÷ Evidence
Background
Bayesian inference Bayesian probability Bayes' theorem Bernstein–von Mises theorem Coherence Cox's theorem Cromwell's rule Likelihood principle Principle of indifference Principle of maximum entropy
Model building
Conjugate prior Linear regression Empirical Bayes Hierarchical model
Posterior approximation
Markov chain Monte Carlo Laplace's approximation Integrated nested Laplace approximations Variational inference Approximate Bayesian computation
Estimators
Bayesian estimator Credible interval Maximum a posteriori estimation
Evidence approximation
Evidence lower bound Nested sampling
Model evaluation
Bayes factor (Schwarz criterion) Model averaging Posterior predictive
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Bayesian probability (/ˈbeɪziən/ BAY-zee-ən or /ˈbeɪʒən/ BAY-zhən)^[1] is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation^[2] representing a state of knowledge^[3] or as quantification of a personal belief.^[4]

The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses;^[5][6] that is, with propositions whose truth or falsity is unknown. In the Bayesian view, a probability is assigned to a hypothesis, whereas under frequentist inference, a hypothesis is typically tested without being assigned a probability.

Bayesian probability belongs to the category of evidential probabilities; to evaluate the probability of a hypothesis, the Bayesian probabilist specifies a prior probability. This, in turn, is then updated to a posterior probability in the light of new, relevant data (evidence).^[7] The Bayesian interpretation provides a standard set of procedures and formulae to perform this calculation.

The term Bayesian derives from the 18th-century mathematician and theologian Thomas Bayes, who provided the first mathematical treatment of a non-trivial problem of statistical data analysis using what is now known as Bayesian inference.^[8]: 131 Mathematician Pierre-Simon Laplace pioneered and popularized what is now called Bayesian probability.^{[8]: 97–98}