Bayesian Network

BayesianMachineLearningMachineLearningProbabilityTheoryStatistics A Bayesian Network or belief network is a Probabilistic Graphical Model over a collection of random variables defined by a Directed Acyclic Graph (DAG) which determines a set of (local) conditional distributions that express conditional independence/causality assumptions. The full joint distribution of is given by Thus, the direction of the arrows indicates that the local distribution of some random variable is defined in terms of the other random variables. This is useful for expressing causality and a generative model of data. Sampling can be done by sampling from the local conditionals according to the topological ordering of the DAG.

Modeling the conditional distributions requires parameters, where is some set of discrete values that each can take. For high-dimensional generative models, as long as then the graphical model will be feasible.

Modeling the joint distribution for the Linear Gaussian Model for Bayesian linear regression: A Bayesian network can be converted into an undirected graphical model by replacing directed edges by undirected edges and connecting the edges between parents in Head-Head conditional independence relationships. This is called the moralization of the graph, and results in the loss of the conditional independence relationship between the three variables in the Head-Head configuration. This type of conversion is common in order to apply Belief Propagation algorithms for efficient inference.