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Mixed graphical models with missing data and the partial imputation EM algorithm. (English) Zbl 0972.62001
The following model is considered. Let $$G =(V, E)$$ denote a graph, where $$E$$ is the set of edges, $$V$$ the set of vertices, and $$V$$ is partitioned as $$V = \Delta \cup \Gamma$$ into a dot set $$\Delta$$ and a circle set $$F.$$ A dot denotes a discrete variable and a circle denotes a continuous variable. Thus the random variables are $$X_V = (X_v)_{v\in V}.$$ 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 $$G.$$ 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.

##### MSC:
 62-07 Data analysis (statistics) (MSC2010) 05C90 Applications of graph theory 62-09 Graphical methods in statistics (MSC2010) 60E99 Distribution theory
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