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

*(English)*Zbl 0972.62001The 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.

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.

Reviewer: Yu.V.Kozachenko (Kyïv)

##### MSC:

62-07 | Data analysis (statistics) (MSC2010) |

05C90 | Applications of graph theory |

62-09 | Graphical methods in statistics (MSC2010) |

60E99 | Distribution theory |