Factorization of posteriors and partial imputation algorithm for graphical models with missing data

Abstract

In this paper, we discuss factorization of a posterior distribution and present a partial imputation algorithm for a graphical model with incomplete data. We use an ordinary graph to represent a graphical model and introduce a hypergraph to represent an observed data pattern where each hyperedge is a set of variables observed for a group of individuals. First, in terms of a decomposition of such a mixed graph, we discuss factorization of a joint posterior distribution into several marginal posterior distributions so that calculation of posterior distribution can be localized. Then, for a mixed graph which cannot be decomposed without loss of information, we present a partial imputation algorithm which imputes only a part of missing data and reduces unnecessary imputation of an ordinary Gibbs sampler. Finally, we discuss the efficiency improved by a decomposition and the partial imputation algorithm.