Model Identifiability in Naive Bayesian Networks

Bayesian networks are a useful tool for representing the probabilistic relationships between a set of random variables. When using such models it is important to consider issues of identifiability, in effect to consider whether or not it is possible to estimate the parameters in our model uniquely . Identifiability is an important consideration as lack of identifiability can lead to problems of model fitting and interpretation. Here we consider the identifiability of a particular type of Bayesian network, that of the naive Bayesian network with a binary, unobservable, root node and binary observable nodes. I. BAYESIAN NETWORKS A Bayesian Network is a means of representing relationships between a number of variables in some domain, 3971-40258 . A directed acyclic graph (DAG) is used to represent the relationships between variables; nodes in the graph represent the variables themselves and arcs between them are used to indicate cause and effect, arcs being drawn from the causative variable to its immediate effect. Associated with each variable, , are two subsets of known as the parents and children of . These are, respectively, the set of all variables having a direct effect upon and the set of all variables which are directly affected by . Figure 1 shows a simple DAG representing the relationship between cause, disease and symptom for two different strains of the hepatitis virus. Here the parents of RECENT VISIT TO ASIA USER DRUG TEST POSITIVE FOR HEPATITIS A JAUNDICE HEPATITIS B Fig. 1. A simple Bayesian network representing the relationship between cause, disease and symptoms for two different strains of the hepatitis virus