Bayesian statistics in the context of "Equilibrium distribution"

A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables. Graphical models are commonly used in probability theory, statistics—particularly Bayesian statistics—and machine learning.

In statistics, point estimation involves the use of sample data to calculate a single value (known as a point estimate since it identifies a point in some parameter space) which is to serve as a "best guess" or "best estimate" of an unknown population parameter (for example, the population mean). More formally, it is the application of a point estimator to the data to obtain a point estimate.

Point estimation can be contrasted with interval estimation: such interval estimates are typically either confidence intervals, in the case of frequentist inference, or credible intervals, in the case of Bayesian inference. More generally, a point estimator can be contrasted with a set estimator. Examples are given by confidence sets or credible sets. A point estimator can also be contrasted with a distribution estimator. Examples are given by confidence distributions, randomized estimators, and Bayesian posteriors.

A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election. The unknown quantity may be a parameter of the model or a latent variable rather than an observable variable.

In Bayesian statistics, Bayes' rule prescribes how to update the prior with new information to obtain the posterior probability distribution, which is the conditional distribution of the uncertain quantity given new data. Historically, the choice of priors was often constrained to a conjugate family of a given likelihood function, so that it would result in a tractable posterior of the same family. The widespread availability of Markov chain Monte Carlo methods, however, has made this less of a concern.

In Bayesian statistics, a credible interval is an interval used to characterize a probability distribution. It is defined such that an unobserved parameter value has a particular probability ${\displaystyle \gamma }$ to fall within it. For example, in an experiment that determines the distribution of possible values of the parameter ${\displaystyle \mu }$, if the probability that ${\displaystyle \mu }$ lies between 35 and 45 is ${\displaystyle \gamma =0.95}$, then ${\displaystyle 35\leq \mu \leq 45}$ is a 95% credible interval.

Credible intervals are typically used to characterize posterior probability distributions or predictive probability distributions. Their generalization to disconnected or multivariate sets is called credible set or credible region.

Richard McElreath (born 18 April 1973) is an American professor of anthropology and a director of the Max Planck Institute for Evolutionary Anthropology in Leipzig, Germany. He is an author of the Statistical Rethinking applied Bayesian statistics textbook, among the first to largely rely on the Stan statistical environment, and the accompanying rethinking R language package.

He earned his B.S. at Emory University in 1995 and a Ph.D. in anthropology under Robert Boyd at the University of California, Los Angeles in 2001 with field research in Tanzania.