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15
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 (
Computing Maximum Likelihood Estimates in loglinear models
, 2006
"... We develop computational strategies for extended maximum likelihood estimation, as defined in Rinaldo (2006), for general classes of loglinear models of widespred use, under Poisson and productmultinomial sampling schemes. We derive numerically efficient procedures for generating and manipulating ..."
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Cited by 13 (3 self)
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We develop computational strategies for extended maximum likelihood estimation, as defined in Rinaldo (2006), for general classes of loglinear models of widespred use, under Poisson and productmultinomial sampling schemes. We derive numerically efficient procedures for generating and manipulating design matrices and we propose various algorithms for computing the extended maximum likelihood estimates of the expectations of the cell counts. These algorithms allow to identify the set of estimable cell means for any given observable table and can be used for modifying traditional goodnessoffit tests to accommodate for a nonexistent MLE. We describe and take advantage of the connections between extended maximum likelihood
Gibbs Variable Selection using BUGS
 Artificial Intelligence
, 1999
"... In this paper we discuss and present in detail the implementation of Gibbs variable selection as defined by Dellaportas et al. (2000, 2002) using the BUGS software (Spiegelhalter et al., 1996a,b,c). The specification of the likelihood, prior and pseudoprior distributions of the parameters as well a ..."
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Cited by 9 (0 self)
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In this paper we discuss and present in detail the implementation of Gibbs variable selection as defined by Dellaportas et al. (2000, 2002) using the BUGS software (Spiegelhalter et al., 1996a,b,c). The specification of the likelihood, prior and pseudoprior distributions of the parameters as well as the prior term and model probabilities are described in detail. Guidance is also provided for the calculation of the posterior probabilities within BUGS environment when the number of models is limited. We illustrate the application of this methodology in a variety of problems including linear regression, loglinear and binomial response models.
Data augmentation in multiway contingency tables with fixed marginal totals
 JOURNAL OF STATISTICAL PLANNING AND INFERENCE
, 2006
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The mode oriented stochastic search (MOSS) algorithm for loglinear models with conjugate priors
, 2008
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Rate of Convergence of the Gibbs Sampler by Gaussian Approximation
, 1997
"... this article we approximate the rate of convergence of the Gibbs sampler by a normal approximation of the target distribution. Based on this approximation, we consider many implementational issues for the Gibbs sampler, e.g., updating strategy, parameterization and blocking. We give theoretical resu ..."
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Cited by 4 (3 self)
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this article we approximate the rate of convergence of the Gibbs sampler by a normal approximation of the target distribution. Based on this approximation, we consider many implementational issues for the Gibbs sampler, e.g., updating strategy, parameterization and blocking. We give theoretical results to justify our approximation and illustrate our methods in a number of realistic examples. Key words: Correlation Structure; Gaussian distribution; Generalized linear models; Gibbs sampler; Markov chain Monte Carlo; Parameterization; Random scan; Rates of convergence.
A conjugate prior for discrete hierarchical loglinear models
, 2009
"... In Bayesian analysis of multiway contingency tables, the selection of a prior distribution for either the loglinear parameters or the cell probabilities parameters is a major challenge. In this paper, we define a flexible family of conjugate priors for the wide class of discrete hierarchical logl ..."
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Cited by 2 (1 self)
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In Bayesian analysis of multiway contingency tables, the selection of a prior distribution for either the loglinear parameters or the cell probabilities parameters is a major challenge. In this paper, we define a flexible family of conjugate priors for the wide class of discrete hierarchical loglinear models, which includes the class of graphical models. These priors are defined as the Diaconis–Ylvisaker conjugate priors on the loglinear parameters subject to “baseline constraints” under multinomial sampling. We also derive the induced prior on the cell probabilities and show that the induced prior is a generalization of the hyper Dirichlet prior. We show that this prior has several desirable properties and illustrate its usefulness by identifying the most probable decomposable, graphical and hierarchical loglinear models for a sixway contingency table.
Specification of prior distributions under model uncertainty
, 2008
"... We consider the specification of prior distributions for Bayesian model comparison, focusing on regressiontype models. We propose a particular joint specification of the prior distribution across models so that sensitivity of posterior model probabilities to the dispersion of prior distributions fo ..."
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We consider the specification of prior distributions for Bayesian model comparison, focusing on regressiontype models. We propose a particular joint specification of the prior distribution across models so that sensitivity of posterior model probabilities to the dispersion of prior distributions for the parameters of individual models (Lindley’s paradox) is diminished. We illustrate the behavior of inferential and predictive posterior quantities in linear and loglinear regressions under our proposed prior densities with a series of simulated and real data examples.
Summary Bayesian Inference for Categorical Data Analysis
"... This article surveys Bayesian methods for categorical data analysis, with primary emphasis on contingency table analysis. Early innovations were proposed by Good (1953, 1956, 1965) for smoothing proportions in contingency tables and by Lindley (1964) for inference about odds ratios. These approache ..."
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This article surveys Bayesian methods for categorical data analysis, with primary emphasis on contingency table analysis. Early innovations were proposed by Good (1953, 1956, 1965) for smoothing proportions in contingency tables and by Lindley (1964) for inference about odds ratios. These approaches primarily used conjugate beta and Dirichlet priors. Altham (1969, 1971) presented Bayesian analogs of smallsample frequentist tests for 2×2 tables using such priors. An alternative approach using normal priors for logits received considerable attention in the 1970s by Leonard and others (e.g., Leonard 1972). Adopted usually in a hierarchical form, the logitnormal approach allows greater flexibility and scope for generalization. The 1970s also saw considerable interest in loglinear modeling. The advent of modern computational methods since the mid1980s has led to a growing literature on fully Bayesian analyses with models for categorical data, with main emphasis on general
Bayesian Methods for Twoway Contingency Tables
"... this paper, we consider Bayesian estimation and model determination for twoway contingency tables. The analysis is based on partitioning the parameter space of a saturated loglinear model into parameter subspaces identified by considering the natural symmetries of the data. Prior distributions may ..."
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this paper, we consider Bayesian estimation and model determination for twoway contingency tables. The analysis is based on partitioning the parameter space of a saturated loglinear model into parameter subspaces identified by considering the natural symmetries of the data. Prior distributions may then be chosen to respect invariance restrictions required by the available prior information. We focus particular attention on the case of a square contingency table, where prior information is invariant to permutation of the row (column) category levels, and to interchanging the row and column variable. The resulting class of invariant loglinear models includes many familiar models for this kind of data (Symmetry, QuasiSymmetry, QuasiIndependence etc.) Posterior inference is reasonably straightforward, and inference related to model determination may be presented either through posterior probabilities of individual models, or by considering marginal posterior distributions within the invariant parameter spaces. The analysis of an intermarriage table is presented as an example.