We assume that a problem domain is characterized by a set of random variables. Beliefs are represented by a Bayesian network (BN) --- an annotated directed acyclic graph, where nodes represent the random variables, and arcs represent probabilistic dependencies amongst the variables. We use the terms `node' and `variable' interchangeably. Associated with each node is a conditional probability of the variable given its parents.
In addition to the explicitly represented conditional probabilities, a BN also implicitly represents conditional independence assertions. Let , , ..., be an enumeration of all the nodes in a BN such that each node appears after its children, and let be the set of parents of a node . The Bayesian network represents the following independence assertion:
Each variable is conditionally independent of the variables in given values for its parents.The conditional independence assertions and the conditional probabilities together entail a joint probability over all the variables. By the chain rule, we have:
where the second equation is true because of the conditional independence assertions. The conditional probabilities are given in the specification of the BN. Consequently, one can, in theory, do arbitrary probabilistic reasoning in a BN.