Bayesmodellek

Bayesmodellek are statistical models that apply Bayesian inference to update beliefs about unknown quantities as data are observed. They combine a prior distribution over parameters with a likelihood function derived from the data to yield a posterior distribution, according to Bayes' theorem. In simple terms, the posterior P(θ|D) is proportional to the likelihood P(D|θ) times the prior P(θ).

Key components include the prior, the likelihood, and the posterior. Models can be plain or hierarchical, where parameters themselves have distributions (hyperparameters). Conjugate priors produce analytic posteriors, while many models require numerical methods.

Inference methods: closed-form with conjugacy; otherwise Markov chain Monte Carlo methods such as Gibbs sampling and Metropolis-Hastings, or more modern Hamiltonian Monte Carlo. Variational inference provides faster approximations.

Model assessment and selection can use posterior predictive checks, Bayes factors via marginal likelihoods, or cross-validation with predictive criteria; model averaging combines several models weighted by posterior support.

Applications span linear and generalized linear regression, time series, Gaussian processes, Bayesian networks, and beyond. They are used in medicine, finance, environmental science, and machine learning, offering principled uncertainty quantification and the ability to incorporate prior information.

Strengths include coherent uncertainty, robustness in small samples, and flexible modeling with uncertainty propagation. Challenges involve computational demands, sensitivity to priors, and the need for careful specification and convergence diagnostics.

History: originated with Thomas Bayes and Pierre-Simon Laplace, formalized in the 20th century, and expanded with modern computational tools such as Stan and PyMC, which have broadened practical use.