This paper outlines some of the issues in developing adaptive methods and presents some preliminary results. 1 Introduction

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The integrated likelihood (also called the marginal likelihood or the normalizing constant) is a central quantity in Bayesian model selection and model averaging. It is defined as the integral over the parameter space of the likelihood times the prior density. The Bayes factor for model comparison and Bayesian testing is a ratio of integrated likelihoods, and the model weights in Bayesian model averaging are proportional to the integrated likelihoods. We consider the estimation of the integrated likelihood from posterior simulation output, aiming at a generic method that uses only the likelihoods from the posterior simulation iterations. The key is the harmonic mean identity, which says that the reciprocal of the integrated likelihood is equal to the posterior harmonic mean of the likelihood. The simplest estimator based on the identity is thus the harmonic mean of the likelihoods. While this is an unbiased and simulation-consistent estimator, its reciprocal can have infinite variance and so it is unstable in general. We describe two methods for stabilizing the harmonic mean estimator. In the first one, the parameter space is reduced in such a way that the modified estimator involves a harmonic mean of heavier-tailed densities, thus resulting in a finite variance estimator. The resulting

Model search in regression with very large numbers of candidate predictors raises challenges for both model specification and computation, and standard approaches such as Markov chain Monte Carlo (MCMC) and step-wise methods are often infeasible or ineffective. We describe a novel shotgun stochastic search (SSS) approach that explores “interesting” regions of the resulting, very high-dimensional model spaces to quickly identify regions of high posterior probability over models. We describe algorithmic and modeling aspects, priors over the model space that induce sparsity and parsimony over and above the traditional dimension penalization implicit in Bayesian and likelihood analyses, and parallel computation using cluster computers. We discuss an example from gene expression cancer genomics, comparisons with MCMC and other methods, and theoretical and simulationbased aspects of performance characteristics in large-scale regression model search. We also provide software implementing the methods.

P against several alternative models such as a weighted averaging model (WTAV), which is an inefficient algorithm for combining the auditory and visual sources. For (2) The WTAV predicts that two sources can never be more informative than one. In direct contrasts, the FLMP has consistently and significantly outperformed the WTAV w a w v w w wa w v i j ( ) ( ) . / / da = + + = + - a v ( )( ) . / / da = + - - 1 1 1 Copyright 2001 Psychonomic Society, Inc. The research was supported by grants from the National Institute of Deafness and Other Communicative Disorders (PHS R01DC00236), the National Science Foundation (Challenge Grant CDA-9726363), Intel Corporation,and the University of California Digital Media Innovation Program. D.W.M. is highly appreciative of the encouraging support of Dan Friedman and Bill Rowe. We thank William Batchelder, James Cutting, In Jae Myung,Mark Pitt, and John Wixted for their constructive comments on an earlier version of the paper. Correspon

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We describe computationally efficient methods for Bayesian model selection. The methods select among mixtures in which each component is a directed acyclic graphical model (mixtures of DAGs or MDAGs), and can be applied to data sets in which some of the random variables are not always observed. The model-selection criterion that we consider is the posterior probability of the model (structure) given data. Our model-selection problem is difficult because (1) the number of possible model structures grows super-exponentially with the number of random variables and (2) missing data necessitates the use of computationally slow approximations of model posterior probability. We argue that simple search-and-score algorithms are infeasible for a variety of problems, and introduce a feasible approach in which parameter and structure search is interleaved and expected data is treated as real data. Our approach can be viewed as the combination of (1) a modified Cheeseman–Stutz asymptotic approximation for model posterior probability and (2) the Expectation–Maximization algorithm. We evaluate our procedure for selecting among MDAGs on synthetic and real examples.

Abstract: An important aspect of mixture modeling is the selection of the number of mixture components. In this paper, we discuss the Bayes factor as a selection tool. The discussion will focus on two aspects: computation of the Bayes factor and prior sensitivity. For the computation, we propose a variant of Chib’s estimator that accounts for the non-identifiability of the mixture components. To reduce the prior sensitivity of the Bayes factor, we propose to extend the model with a hyperprior. We further discuss the use of posterior predictive checks for examining the fit of the model. The ideas are illustrated by means of a psychiatric diagnosis example.

Single and multiple ratings of test items have become a stock component of standardized educational tests and surveys. For both formative and summative evaluation of raters, a number of multiple-read rating designs are now commonplace (Wilson and Hoskens, 1999), including designs with as many as six raters per item (e.g. Sykes, Heidorn and Lee, 1999). As digital image based distributed rating becomes commonplace, we expect the use of multiple raters as a routine part of test scoring to grow; increasing the number of raters also raises the possibility of improving the precision of examinee proficiency estimates. In this paper we develop Patz’s (1996) hierarchical rater model (HRM) for polytomously scored item response data, and show how it can be used, for example, to scale examinees and items, to model aspects of consensus among raters, and to model individual rater severity and consistency effects. The HRM treats examinee responses to open-ended items as unobserved discrete variables, and it explicitly models the “proficiency ” of raters in assigning accurate scores as well as the proficiency

In the present paper we discuss the problem of estimating model likelihoods from the MCMC output for a general mixture and switching model. Estimation is based on the method of bridge sampling (Meng and Wong, 1996), where the MCMC sample is combined with an iid sample from an importance density. The importance density is constructed in an unsupervised manner from the MCMC output using a mixture of complete data posteriors. Whereas the importance sampling estimator as well as the reciprocal importance sampling estimator are sensitive to the tail behaviour of the importance density, we demonstrate that the bridge sampling estimator is far more robust in this concern. Our case studies range from computing marginal likelihoods for a mixture of multivariate normal distributions, testing for the inhomogeneity of a discrete time Poisson process, to testing for the presence of Markov switching and order selection in the MSAR model.