Graphical model

Builds on Graph Joint probability distribution Conditional independence Random variable

Definition

Let $G = (V, E)$ be a finite graph, directed or undirected, and let $X = (X_v)_{v \in V}$ be a family of random variables indexed by the vertices. A Markov property for $G$ is a list of conditional independence statements $X_A \perp X_B \mid X_S$ , for subsets $A, B, S \subseteq V$ , that is read off the graph. The graphical model of $G$ is the set of all joint distributions of $X$ that satisfy the Markov property of $G$ . For a directed acyclic graph it is called a Bayesian network, and for an undirected graph a Markov random field.

Remarks

Here $X_A = (X_v)_{v \in A}$ . Example: the undirected path $1 - 2 - 3$ gives the model of all distributions with $X_1 \perp X_3 \mid X_2$ . A Markov chain $X_1, X_2, X_3$ lies in this model. Non-example: let $X_1, X_2$ be independent fair coins and $X_3 = X_1$ . Then $X_1$ and $X_3$ are dependent given $X_2$ , so this distribution is not in the model of the path. A complete graph imposes no condition, so its model contains all joint distributions. Often the model is restricted further to a parametric family, such as the Gaussian graphical models.

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Tag	`J10YF5CXX9`
URI	`m2://J10YF5CXX9`
Kind	definition
Writing	W0 stub
MSC	62H22
Defines	Graphical model
Uses	Graph, Joint probability distribution, Conditional independence, Random variable

Graphical model

Definition

Remarks

Reading path

Further reading

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