bayesian

Bayesian refers to methods, models, and a philosophy of probability that interprets uncertainty in terms of degrees of belief and updates those beliefs with evidence using Bayes' theorem. Bayes' theorem states that the posterior probability of a hypothesis H given data D is proportional to the likelihood of D given H times the prior probability of H: P(H|D) ∝ P(D|H) P(H). The constant of proportionality is the marginal likelihood, P(D).

This framework centers on three elements: the prior distribution, representing beliefs before observing data; the likelihood, describing how data would be generated under H; and the posterior distribution, the updated beliefs after observing D. From the posterior one can derive predictive distributions for future data.

Conjugate priors yield closed-form posteriors in simple models, but many practical problems require computational methods, including Markov chain Monte Carlo (MCMC) techniques such as Gibbs sampling and Metropolis-Hastings, and variational inference, which approximates the posterior with a tractable distribution.

Bayesian methods are used across disciplines, including medicine, ecology, finance, and machine learning, for parameter estimation, hypothesis testing, model comparison, and decision making under uncertainty. Advantages include a coherent probabilistic interpretation, principled incorporation of prior information, and the ability to update beliefs as new data arrive. Challenges include selecting priors, sensitivity to prior assumptions, and higher computational cost compared with some classical approaches.

Variants include hierarchical Bayes models, which share information across groups, and empirical Bayes, which estimates priors from the data. Bayesian reasoning contrasts with frequentist approaches, but both frameworks are used to analyze uncertainty in science and engineering.