Hierarchical Multinomial-Dirichlet model for the estimation of conditional probability tables

Abstract--We present a novel approach for estimating conditional probability tables, based on a joint, rather than independent, estimate of the conditional distributions belonging to the same table. We derive exact analytical expressions for the estimators and we analyse their properties both analytically and via simulation. We then apply this method to the estimation of parameters in a Bayesian network. Given the structure of the network, the proposed approach better estimates the joint distribution and significantly improves the classification performance with respect to traditional approaches. I. INTRODUCTION A Bayesian network is a probabilistic model constituted by a directed acyclic graph (DAG) and a set of conditional probability tables (CPTs), one for each node. The CPT of node X contains the conditional probability distributions of X given each possible configuration of its parents. Usually all variables are discrete and the conditional distributions are estimated adopting a Multinomial-Dirichlet model, where the Dirichlet prior is characterised by the vector of hyper-parameters α . Y et, Bayesian estimation of multinomials is sensitive to the choice of α and inappropriate values cause the estimator to perform poorly [1].