Chapter 6 - Further Properties of Bayesian Networks

Abstract

This chapter deals with Bayesian networks. It begins by describing the conditional independencies entailed by the Markov condition. It also discusses the faithful condition, which entails the independency condition of the Markov condition. The probability distribution in the Bayesian network is said to be unfaithful to the direct acyclic graph (DAG) when it contains a conditional independency that is not entailed by the Markov condition. Certain DAGs are equivalent in the sense that they entail the same conditional independencies. This is explained in the chapter. A DAG entails a conditional independency if every probability distribution that satisfies the Markov condition with the DAG must have the conditional independency. In this light, the study illustrates few examples where one would expect these conditional independencies. It also considers two examples of probability distributions that satisfy the Markov condition with a DAG and contain a conditional independency that is not entailed by the DAG. Furthermore, this chapter indicates that a Bayesian network can have a large number of nodes and that the conditional probability of a given node can be affected by instantiating a distant node. This is explained by definitions and theorems. Finally, it explains Markov blankets and Markov boundaries, which are sets of variables that render a given variable conditionally independent of all other variables.