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algorithm assume asymptotic block-recursive calculated canonical parameter canonical statistic CG density CG distribution chain components chain graph model cliques of Q complete concentration matrix conditional distribution conditional independence consider continuous variables corresponding counts covariance matrix covariance selection model decomposable graph decomposable models defined denote deviance test direct join directed acyclic graph discrete variables edge equivalent example factorization follows given graph Q graphical models Hence hierarchical model holds homogeneous hyperedges hypergraph implies inverse iterative Lauritzen Lemma likelihood equations likelihood function likelihood ratio linear log-affine model log-linear models marginal table marked graph Markov property maximum likelihood estimate model with graph moral graph multinomial sampling multivariate normal nd(a normal distribution notation obtained pairwise partitioned perfect sequence positive definite probability Proof Proposition quadratic random variables regular exponential model restrictions result satisfies saturated model Section simplicial space subgraph subsets sufficient statistic Theorem triangulated undirected graph vertex vertices zero
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Kevin Patrick Murphy - 2002
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Stephen P Brooks - 1998 - The Statistician
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