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A Bayesian network (also known as a Bayes network, Bayes net, belief network, or decision network) is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). [1] While it is one of several forms of causal notation, causal networks are special cases of Bayesian ...
Bayesian inference using Gibbs sampling (BUGS) is a statistical software for performing Bayesian inference using Markov chain Monte Carlo (MCMC) methods. It was developed by David Spiegelhalter at the Medical Research Council Biostatistics Unit in Cambridge in 1989 and released as free software in 1991.
Originally intended to solve problems encountered in medical statistics, it soon became widely used in other disciplines, such as ecology, sociology, and geology. [2] The last version of WinBUGS was version 1.4.3, released in August 2007. Development is now focused on OpenBUGS, an open-source version of the package. WinBUGS 1.4.3 remains ...
Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well as other parameters describing the distribution of the regressand) and ultimately allowing the out-of-sample prediction of the regressand (often ...
The Markov boundary always exists. Under some mild conditions, the Markov boundary is unique. However, for most practical and theoretical scenarios multiple Markov boundaries may provide alternative solutions. [2] When there are multiple Markov boundaries, quantities measuring causal effect could fail. [3]
It can then be shown that the points of convergence of the sum-product algorithm represent the points where the free energy in such a system is minimized. Similarly, it can be shown that a fixed point of the iterative belief propagation algorithm in graphs with cycles is a stationary point of a free energy approximation.
Given a system transforming a set of inputs to output values, described by a mathematical function f, optimization refers to the generation and selection of the best solution from some set of available alternatives, [1] by systematically choosing input values from within an allowed set, computing the value of the function, and recording the best value found during the process.
where the first equality results from the marginalization rule, the second results from Bayes' theorem and the third corresponds to a second application of marginalization. The denominator appears to be a normalization term and can be replaced by a constant . Theoretically, this allows to solve any Bayesian inference problem.