Graphical Model

Graphical model is a probabilistic representation that uses a graph to encode conditional independence structure among random variables. Nodes represent variables; edges represent direct probabilistic dependencies. The graph structure determines which factorizations of the joint distribution are valid, transforming high-dimensional inference problems into tractable local computations.

The two dominant families are Bayesian networks (directed acyclic graphs representing causal or temporal dependencies) and Markov random fields (undirected graphs representing symmetric interactions). Both frameworks exploit the factorization implied by the graph to make inference tractable, though the computational complexity depends sharply on the graph's treewidth.

Graphical models are the backbone of Bayesian statistics, machine learning, and computational biology. They provide the structure on which variational inference and expectation propagation operate. The graph is not merely a visualization; it is a computational contract, specifying which conditional probabilities must be estimated and which can be derived.