Mixed graphical models with missing data and the partial imputation EM algorithm

In this paper,ve discuss graphical models for mixed types of continuous and discrete variables with incomplete data. We use a set of hyperedges to rep resent an observed data pattern. A hyperedge is a set of variables observed for a group of individuals. In a mixed graph with two types of vertices an d two types of edges, dots and circles represent discrete and continuous va riables respectively. A normal graph represents a graphical model and a hyp ergraph represents an observed data pattern. In terms of the mixed graph, w e discuss decomposition of mixed graphical models with incomplete data, and we present a partial imputation method which ran be used in the EM algorit hm and the Gibbs sampler to speed their convergence. For a given mixed grap hical model and an observed data pattern, we try to decompose a large graph into several small ones so that the original likelihood can be factored in to a product of likelihoods with distinct parameters for small graphs. For the case that a graph cannot be decomposed due to its observed data pattern , we can impute missing data partially so that the graph can be decomposed.