3.1 Introduction

The concept of conditional probability is a useful one. There are countless real world examples where the probability of one event is conditional on the probability of a previous one. While the sum and product rules of probability theory can anticipate this factor of conditionality, in many cases such calculations are NP-hard. The prospect of managing a scenario with 5 discrete random variables (2⁵-1=31 discrete parameters) might be manageable. An expert system for monitoring patients with 37 variables resulting in a joint distribution of over 2³⁷ parameters would not be manageable[6].

3.2 Definition

Consider a domain U of n variables, x₁,...x_n. Each variable may be discrete having a finite or countable number of states, or continuous. Given a subset X of variables x_i where x_i U, if one can observe the state of every variable in X, then this observation is called an instance of X and is denoted as X= for the observations . The "joint space" of U is the set of all instances of U. denotes the "generalized probability density" that X= given Y= for a person with current state information . p(X|Y, ) then denotes the "Generalized Probability Density Function" (gpdf) for X, given all possible observations of Y. The joint gpdf over U is the gpdf for U.

A Bayesian network for domain U represents a joint gpdf over U. This representation consists of a set of local conditional gpdfs combined with a set of conditional independence assertions that allow the construction of a global gpdf from the local gpdfs. As shown previously, the chain rule of probability can be used to ascertain these values:

(eq. 10)

One assumption imposed by Bayesian Network theory (and indirectly by the Product Rule of probability theory) is that each variable x_i, must be a set of variables that renders x_i and {x₁,...x_i-1} conditionally independent. In this way:

(eq. 11)[7]

A Bayesian Network Structure then encodes the assertions of conditional independence in equation 10 above. Essentially then, a Bayesian Network Structure B_s "is a directed acyclic graph such that (1) each variable in U corresponds to a node in B_s, and (2) the parents of the node corresponding to x_i are the nodes corresponding to the variables in [Pi]_i."[8]

"A Bayesian-network gpdf set Bp is the collection of local gpdfs for each node in the domain."[9]