A Bayesian inference network is a probabilistic graphical model that represents a set of random variables \(X = \{X_1, X_2, ..., X_n\}\) by a directed acyclic graph (DAG) \(G = (V, E)\), where each node \(v_i \in V\) corresponds to a variable \(X_i\).
The joint probability distribution can be expressed by factorization according to the structure of the DAG as:
\[ P(X_1, X_2, ..., X_n) = \prod_{i=1}^n P(X_i \mid ext{Pa}(X_i)) \]
where \( ext{Pa}(X_i)\) are the parent nodes of \(X_i\).
Inference in Bayesian networks involves computing conditional probabilities, e.g. \(P(X_i \mid E)\) with evidence \(E\), which is done using exact or approximate algorithms.
Bayesian networks are formally defined as a triple \((G, \Theta)\) with graph \(G\) and parameter set \(\Theta\) of the conditional probability distributions.
Definition:
“A Bayesian inference network is a directed acyclic graph that represents a factorization of the joint probability distribution of a set of variables according to their conditional independencies and enables probabilistic inference.”
Source:
Jensen, F. V., & Nielsen, T. D. (2007). Bayesian Networks and Decision Graphs. Springer.