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  "body": "\n## Definition\n\nLet $G = (V, E)$ be a finite [[graph|00C0BR7D7H]], directed or undirected, and let $X = (X_v)_{v \\in V}$ be a family of [[random variables|0PNRKMRC55]] indexed by the vertices.\nA *Markov property* for $G$ is a list of [[conditional independence|0THJ7416DX]] statements $X_A \\perp X_B \\mid X_S$, for subsets $A, B, S \\subseteq V$, that is read off the graph.\nThe *graphical model* of $G$ is the set of all [[joint distributions|0DR81R1F2B]] of $X$ that satisfy the Markov property of $G$.\nFor a directed acyclic graph it is called a Bayesian network, and for an undirected graph a Markov random field.\n\n## Remarks\n\nHere $X_A = (X_v)_{v \\in A}$.\nExample: the undirected path $1 - 2 - 3$ gives the model of all distributions with $X_1 \\perp X_3 \\mid X_2$.\nA Markov chain $X_1, X_2, X_3$ lies in this model.\nNon-example: let $X_1, X_2$ be independent fair coins and $X_3 = X_1$.\nThen $X_1$ and $X_3$ are dependent given $X_2$, so this distribution is not in the model of the path.\nA complete graph imposes no condition, so its model contains all joint distributions.\nOften the model is restricted further to a parametric family, such as the Gaussian graphical models.\n",
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