Bayesian network

Builds on Graphical model Directed graph Conditional probability distribution Conditional independence

Definition

Let $G = (V, E)$ be a finite directed graph with no directed cycles, that is, a directed acyclic graph. For $v \in V$ let $\operatorname{pa}(v)$ be the set of parents of $v$ , the vertices $u$ with an edge $u \to v$ . Let $X = (X_v)_{v \in V}$ be random variables with a joint density or probability mass function $p$ . Then $X$ is a Bayesian network with respect to $G$ if $p(x) = \prod_{v \in V} p(x_v \mid x_{\operatorname{pa}(v)})$ for all $x$ , where $p(x_v \mid x_{\operatorname{pa}(v)})$ is the conditional density of $X_v$ given $X_{\operatorname{pa}(v)}$ .

Remarks

The factorisation is equivalent to the local Markov property. Each $X_v$ is conditionally independent of its non-descendants given its parents. Example: for the chain $A \to B \to C$ , the definition asks $p(a, b, c) = p(a) p(b \mid a) p(c \mid b)$ . Dividing by $p(b)$ gives $p(a, c \mid b) = p(a \mid b) p(c \mid b)$ , so $A \perp C \mid B$ . Non-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. Different graphs can give the same model: $A \to B \to C$ and $A \leftarrow B \leftarrow C$ both encode exactly $A \perp C \mid B$ .

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Conditional independence concept
Conditional probability distribution concept
Directed graph concept
Graphical model concept
Bayesian network definition

Tag	`G2XY7J2Z5Q`
URI	`m2://G2XY7J2Z5Q`
Kind	definition
Writing	W0 stub
MSC	62H22, 68T37
Defines	Bayesian network
Uses	Graphical model, Directed graph, Conditional probability distribution, Conditional independence

Bayesian network

Definition

Remarks

Reading path

Further reading

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