Bayesian Network

graph-theory statistics machine-learning

Definition

Bayesian Network

A Bayesian network is a probabilistic graphical model that represents a set of random variables and their conditional dependencies via a Directed Acyclic Graph (DAG). Formally, for a set of variables $\{X_1,\dots ,X_n\}$, the joint probability distribution is factorised according to the graph structure:

$P(X_1,\dots ,X_n)={\prod }_{i=1}^{n}P(X_i|\text{parents}(X_i))$

$P(X_1,X_2,X_3,X_4)=P(X_1)P(X_2|X_1)P(X_3|X_1)P(X_4|X_2)$

Properties

Independence: The graph encodes local Markov properties; a node is conditionally independent of its non-descendants given its parents. This allows for a significantly more succinct representation of the joint distribution compared to full probability tables.

Inference Efficiency: Factorisation reduces the computational complexity of marginalisation and posterior updates by allowing the summation/integration to be performed over local subsets of variables.