Mathematics and statistics

Bayes risk

Bayes risk is the expected loss of a statistical decision rule averaged over a prior on the unknown parameter; the rule minimizing it is the Bayes rule.

Bayes' theorem

Bayes' theorem, also called Bayes' law or Bayes' rule, is a rule of probability theory that computes the probability of a cause given its effect from the reverse conditional probability.

Bayesian additive regression trees

Bayesian additive regression trees (BART) is a Bayesian nonparametric model for regression and classification that sums many small trees, introduced by Chipman, George, and McCulloch in 2010.

Bayesian calibration is a statistical method that estimates a computational model's uncertain parameters by combining model outputs with observed data through Bayes' theorem, yielding posterior distributions rather than point estimates.

Bayesian clustering

Bayesian clustering is a statistical method that groups observations by fitting mixture or hierarchical models with priors, producing a full posterior distribution over cluster assignments and their uncertainty.

Bayesian decision theory

Bayesian decision theory is the branch of statistical decision theory that ranks actions by posterior expected loss, the criterion behind estimators like the posterior mean, median, and MAP.

Bayesian deep learning libraries

Bayesian deep learning libraries are software packages that place probability distributions over neural-network weights, producing calibrated uncertainty estimates through variational inference, MC dropout, SGLD, and Laplace methods.

Bayesian design

Bayesian design is a statistical method of experimental design that chooses the design maximizing expected utility, typically expected information gain, computed under a prior distribution and probabilistic model.

Bayesian design of computer experiments

Bayesian design of computer experiments is the use of Bayesian decision theory to choose input points for evaluating a deterministic computer simulator, using criteria like expected improvement.

Bayesian estimator

A Bayesian estimator is a statistical method that combines a prior distribution with observed data via Bayes' theorem, yielding a posterior distribution from which point estimates and credible intervals are drawn.

Bayesian Gaussian process regression

Bayesian Gaussian process regression, also known as kriging, is a Bayesian method that places a Gaussian process prior over the unknown function, giving closed-form predictions with full uncertainty.

Bayesian hierarchical model

A Bayesian hierarchical model, also called a multilevel or mixed effects model, is a statistical model whose prior parameters are themselves given distributions and estimated from data.

Bayesian hierarchical modeling

Bayesian hierarchical modeling, also known as hierarchical Bayesian modeling, is a statistical approach for grouped data in which partial pooling shrinks each group's estimate toward the overall mean.

Bayesian history matching

Bayesian history matching is a method for calibrating complex computer simulators by ruling out parameter regions inconsistent with observed data, using fast statistical emulators in iterative waves.

Bayesian inference

Bayesian inference, or Bayesian updating, is a method of statistical inference that combines a prior distribution with the likelihood of observed data to produce a posterior distribution, treating probability as strength of belief.

Bayesian inference in phylogeny

Bayesian inference in phylogeny, also known as Bayesian phylogenetics, is a statistical method for reconstructing evolutionary trees by combining priors with molecular data likelihoods to yield posterior tree probabilities, popularized by MrBayes.

Bayesian information criterion

The Bayesian information criterion (BIC), also called the Schwarz information criterion, is a statistic for comparing statistical models fitted to the same data, derived by Gideon Schwarz in 1978.

Bayesian inversion

Bayesian inversion is a statistical approach to inverse problems whose solution is a posterior probability distribution over model parameters, combining data and prior knowledge via Bayes' theorem.

Bayesian lasso

The Bayesian lasso is a Bayesian variable selection method for regression, introduced by Park and Casella in 2008, that places a Laplace prior on coefficients.

Bayesian logistic regression

Bayesian logistic regression is a statistical model for binary outcomes that keeps the likelihood of ordinary logistic regression but treats the coefficients as random variables with prior distributions.

Bayesian meta-analysis

Bayesian meta-analysis is a statistical method for combining the results of multiple studies using a hierarchical model, computing full posterior distributions for the pooled effect and between-study heterogeneity.

Bayesian model selection

Bayesian model selection is a statistical method that compares competing probability models by scoring each with its evidence, reported as Bayes factors or posterior model probabilities.

Bayesian network

A Bayesian network, or belief network, is a probabilistic graphical model that encodes a joint probability distribution as a directed acyclic graph, supporting reasoning under uncertainty.

Bayesian network software

Bayesian network software covers the full pipeline of learning a network structure from data, fitting its parameters, and answering probabilistic and causal queries, in tools like bnlearn and pgmpy.

Bayesian nonparametric regression and classification

Bayesian nonparametric regression and classification places priors on infinite-dimensional function spaces, such as splines, trees, and Gaussian processes, letting the fitted function's complexity grow with the data.

Bayesian nonparametric survival analysis

Bayesian nonparametric survival analysis places stochastic-process priors, such as Dirichlet process and beta process priors, on an unknown survival function or cumulative hazard without parametric shape assumptions.

Bayesian optimal design and decision-analysis software

Bayesian optimal design and decision-analysis software picks experimental settings or decision strategies by maximizing expected utility or information gain, spanning tools like acebayes, Pyro BOED, GeNIe, and HUGIN.

Bayesian optimal interval design

The Bayesian optimal interval (BOIN) design is a statistical design for phase I oncology dose-finding trials, proposed by Liu and Yuan at MD Anderson in 2015.

Bayesian optimization

Bayesian optimization (BO) is a sequential method for optimizing expensive black-box functions, typically under 20 dimensions, using a Gaussian process surrogate and acquisition functions to choose each evaluation.