Probabilistic Graphical Model

Synonyms

Bayesian network model

Definition

Probabilistic Graphical Models (or Bayesian networks) express jointly distributed probability distributions on a set of probabilistic variables, which is consistent with a set of graph relations in that the values of nodes depend directly only on immediately connected nodes.

Given a set of probabilistic variables X ₁, …, X _n , a probabilistic graphical model over these variables is defined by a set of directed edges (X _i , X _j ) (for which one writes more conveniently X _i → X _j ) such that there is no cycle (no closed loop X ₁ → X ₂ → X ₃ → … → X _l−2 → X _l−1 → X ₁).

This defines a directed acyclic graph over the probabilistic variables. Furthermore, consider for each probabilistic variable X _k its set of parents or predecessors Pa[X _k ] (the set of all X _i with X _i → X _k an edge of the graph. Then, for each such X _k , a conditional probability ${p\left( {{x_k}|{\bf Pa}\left[ {{x_k}} \right]} \right)}$is defined. This probability is called...