*(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 $F\xb7$ A dot denotes a discrete variable and a circle denotes a continuous variable. Thus the random variables are ${X}_{V}={\left({X}_{v}\right)}_{v\in V}\xb7$ 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 $G\xb7$ 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) |

05C90 | Applications of graph theory |

62-09 | Graphical methods in statistics |

60E99 | Distribution theory in probability theory |