The following model is considered. Let denote a graph, where is the set of edges, the set of vertices, and is partitioned as into a dot set and a circle set A dot denotes a discrete variable and a circle denotes a continuous variable. Thus the random variables are The absence of an edge between a pair of vertices means that the corresponding variable pair is independent conditionally on the other variables which is the pairwise Markov property with respect to The authors use a set of hyperedges to represent an observed data pattern. A normal graph represents a graphical model and a hypergraph represents an observed data pattern.
In terms of mixed graphs the decomposition of mixed graphical models with incomplete date is discussed. The authors present a partial imputation method which can be used in the EM algorithm and the Gibbs sampler to speed up their convergence. For a given mixed graphical model and an observed data pattern a large graph decomposes into several small ones so that the original likelihood can be factorized into a product of likelihoods with distinct parameters for small graphs. For the case where a graph cannot be decomposed due to its observed data pattern the authors impute missing data partially such that the graph can be decomposed.