# 3. Bayesian Networks

# 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 (2^{5}-1=31 discrete parameters) might be manageable. An expert system for monitoring patients with 37 variables resulting in a joint distribution of over 2^{37} parameters would not be manageable[6].

# 3.2 Definition

Consider a domain U of *n* variables, *x*_{1},...*x _{n. }*Each variable may be discrete having a finite or countable number of states, or continuous. Given a subset X of variables

*x*where

_{i }*x*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

_{i}**"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:

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*and {x

_{i}_{1},...x

_{i-1}} conditionally independent. In this way:

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 B*p* is the collection of local gpdfs for each node in the domain."[9]