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  "body": "\n## Definition\n\nLet $G = (V, E)$ be a finite [[directed graph|0KFYGSHHF5]] with no directed cycles, that is, a directed acyclic graph.\nFor $v \\in V$ let $\\operatorname{pa}(v)$ be the set of parents of $v$, the vertices $u$ with an edge $u \\to v$.\nLet $X = (X_v)_{v \\in V}$ be random variables with a joint density or probability mass function $p$.\nThen $X$ is a *Bayesian network* with respect to $G$ if\n$$p(x) = \\prod_{v \\in V} p(x_v \\mid x_{\\operatorname{pa}(v)})$$\nfor all $x$, where $p(x_v \\mid x_{\\operatorname{pa}(v)})$ is the [[conditional density|0ZN9EZTEM6]] of $X_v$ given $X_{\\operatorname{pa}(v)}$.\n\n## Remarks\n\nThe factorisation is equivalent to the local Markov property.\nEach $X_v$ is [[conditionally independent|0THJ7416DX]] of its non-descendants given its parents.\nExample: for the chain $A \\to B \\to C$, the definition asks $p(a, b, c) = p(a) p(b \\mid a) p(c \\mid b)$.\nDividing by $p(b)$ gives $p(a, c \\mid b) = p(a \\mid b) p(c \\mid b)$, so $A \\perp C \\mid B$.\nNon-example: the graph $A \\to B \\to A$ has a directed cycle, so it does not define a Bayesian network; a product $p(a \\mid b) p(b \\mid a)$ is in general not a joint distribution.\nDifferent graphs can give the same model: $A \\to B \\to C$ and $A \\leftarrow B \\leftarrow C$ both encode exactly $A \\perp C \\mid B$.\n",
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