Often the goal of model selection is to choose a model for future prediction, and it is natural to measure the accuracy of a future prediction by squared error loss.

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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

Summary. Multicity time series studies of particulate matter and mortality and morbidity have provided evidence that daily variation in air pollution levels is associated with daily variation in mortality counts.These findings served as key epidemiological evidence for the recent review of the US national ambient air quality standards for particulate matter. As a result, methodological issues concerning time series analysis of the relationship between air pollution and health have attracted the attention of the scientific community and critics have raised concerns about the adequacy of current model formulations. Time series data on pollution and mortality are generally analysed by using log-linear, Poisson regression models for overdispersed counts with the daily number of deaths as outcome, the (possibly lagged) daily level of pollution as a linear predictor and smooth functions of weather variables and calendar time used to adjust for timevarying confounders. Investigators around the world have used different approaches to adjust for confounding, making it difficult to compare results across studies. To date, the statistical properties of these different approaches have not been comprehensively compared.To address these issues, we quantify and characterize model uncertainty and model choice in adjusting for seasonal and long-term trends in time series models of air pollution and mortality. First, we

Summary. Although factor analytic models have proven useful for covariance structure modeling and dimensionality reduction in a wide variety of applications, a challenging prob-lem is uncertainty in the number of latent factors. This article proposes an efficient Bayesian approach for model selection and averaging in hierarchical models having one or more factor analytic components. In particular, the approach relies on a method for embedding each of the smaller models within the largest possible model. Bayesian computation can proceed within the largest model, while moving between sub-models based on posterior model prob-abilities. The approach represents a type of parameter expansion, as one always samples within an encompassing model, incorporating extra parameters and latent variables when a smaller model is true. This results in a highly efficient stochastic search factor selection algo-rithm (SSFS) for identifying good factor models and performing model-averaged inferences. The approach is illustrated using simulated examples and a toxicology application.

Factor analytic models are widely used in social science applications to study latent traits, such as intelligence, creativity, stress and depression, that cannot be accurately measured with a single variable. In recent years, there has been a rise in the popularity of factor models due to their flexibility in characterizing multivariate data. For example, latent factor

J. K. Ghosh’s contribution to statistics:

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[A Bayesian approach based on incomplete measurements] Sampling, coding, and streaming even the most essential data, e.g., in medical imaging and weather-monitoring applications, produce a data deluge that severely stresses the available analog-to-digital converter, communication bandwidth, and digital-storage resources. Surprisingly, while the ambient data dimension is large in many problems, the relevant information in the data can reside in a much lower dimensional space. This observation has led to several important theoretical © DIGITAL STOCK & LUSPHIX and algorithmic developments under different low-dimensional modeling frameworks, such as compressive sensing (CS) [1], [2], matrix completion [3], [4], and general factor-model representations [5], [6]. These approaches have enabled new measurement systems, tools, and methods for information extraction from dimensionality-reduced or incomplete data. A key aspect of maximizing the potential of such techniques is to develop appropriate data models. In this article, we investigate this challenge from the perspective of nonparametric Bayesian analysis. Before detailing the Bayesian modeling techniques, we review the form of measurements. Specifically, we consider measurement systems based on dimensionality reduction, where we linearly project the signal of interest into a lower-dimensional space via y Ux þ d: (1) The signal is x 2 R d, the measurements are y 2 R d0, U is a d0 3 d matrix with d05 d,anddaccounts for noise. Such a projection process loses signal information in general, since U has a nontrivial null space. Hence, there has been significant interest over the last few decades in finding dimensionality reductions that preserve as much information as possible in the incomplete measurements y about certain signals x. One way to preserve information is for U to provide a stable embedding that approximately preserves pairwise distances between all signals in some set of interest. In some cases, this property allows the recovery of x from its measurement y.

Abstract: For the balanced ANOVA setup, we propose a new closed form Bayes factor without integral representation, which is however based on fully Bayes method, with reasonable model selection consistency for two asymptotic situations (either number of levels of the factor or number of replication in each level goes to infinity). Exact analytical calculation of the marginal density under a special choice of the priors enables such a Bayes factor.