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Markov Chain Monte Carlo Model Determination for Hierarchical and Graphical Loglinear Models
 Biometrika
, 1996
"... this paper, we will only consider undirected graphical models. For details of Bayesian model selection for directed graphical models see Madigan et al (1995). An (undirected) graphical model is determined by a set of conditional independence constraints of the form `fl 1 is independent of fl 2 condi ..."
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Cited by 55 (8 self)
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this paper, we will only consider undirected graphical models. For details of Bayesian model selection for directed graphical models see Madigan et al (1995). An (undirected) graphical model is determined by a set of conditional independence constraints of the form `fl 1 is independent of fl 2 conditional on all other fl i 2 C'. Graphical models are so called because they can each be represented as a graph with vertex set C and an edge between each pair fl 1 and fl 2 unless fl 1 and fl 2 are conditionally independent as described above. Darroch, Lauritzen and Speed (1980) show that each graphical loglinear model is hierarchical, with generators given by the cliques (complete subgraphs) of the graph. The total number of possible graphical models is clearly given by 2 (
Symmetry analysis of reversible markov chains
 Internet Mathematics
, 2005
"... We show how to use subgroups of the symmetry group of a reversible Markov chain to give useful bounds on eigenvalues and their multiplicity. We supplement classical representation theoretic tools involving a group commuting with a selfadjoint operator with criteria for an eigenvector to descend to ..."
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Cited by 33 (11 self)
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We show how to use subgroups of the symmetry group of a reversible Markov chain to give useful bounds on eigenvalues and their multiplicity. We supplement classical representation theoretic tools involving a group commuting with a selfadjoint operator with criteria for an eigenvector to descend to an orbit graph. As examples, we show that the Metropolis construction can dominate a maxdegree construction by an arbitrary amount and that, in turn, the fastest mixing Markov chain can dominate the Metropolis construction by an arbitrary amount. 1
Bayesian Selection of LogLinear Models
 Canadian Journal of Statistics
, 1995
"... A general methodology is presented for finding suitable Poisson loglinear models with applications to multiway contingency tables. Mixtures of multivariate normal distributions are used to model prior opinion when a subset of the regression vector is believed to be nonzero. This prior distribution ..."
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Cited by 7 (2 self)
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A general methodology is presented for finding suitable Poisson loglinear models with applications to multiway contingency tables. Mixtures of multivariate normal distributions are used to model prior opinion when a subset of the regression vector is believed to be nonzero. This prior distribution is studied for two and threeway contingency tables, in which the regression coefficients are interpretable in terms of oddsratios in the table. Efficient and accurate schemes are proposed for calculating the posterior model probabilities. The methods are illustrated for a large number of twoway simulated tables and for two threeway tables. These methods appear to be useful in selecting the best loglinear model and in estimating parameters of interest that reflect uncertainty in the true model. Key words and phrases: Bayes factors, Laplace method, Gibbs sampling, Model selection, Odds ratios. AMS subject classifications: Primary 62H17, 62F15, 62J12. 1 Introduction 1.1 Bayesian testing...
Testing HardyWeinberg Equilibrium: an Objective Bayesian Analysis
"... Summary: We analyze the general (multiallelic) HardyWeinberg equilibrium problem from an objective Bayesian testing standpoint. We argue that for small or moderate sample sizes the answer is rather sensitive to the prior chosen, and this suggests to carry out a Bayesian sensitivity analysis to the ..."
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Cited by 3 (2 self)
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Summary: We analyze the general (multiallelic) HardyWeinberg equilibrium problem from an objective Bayesian testing standpoint. We argue that for small or moderate sample sizes the answer is rather sensitive to the prior chosen, and this suggests to carry out a Bayesian sensitivity analysis to the prior. This objective is achieved through the identification of a class of priors specifically designed for this testing problem. In this paper we consider the class of intrinsic priors under the full model, indexed by a tuning quantity, the training sample size. These priors are objective, satisfy Savage’s continuity condition and have proved to behave extremely well for many statistical testing problems. We compute the posterior probability of the HardyWeinberg equilibrium model for the class of intrinsic priors, and thus provide a range of plausible answers. If our decision of rejecting the null hypothesis does not change as the intrinsic prior varies over this class, we conclude that our analysis is robust. On the other hand, when the sample size grows to infinity any sensitivity to the prior disappears because under rather general conditions the Bayes factor for intrinsic priors is consistent.