Abstract. Various noninformative prior distributions have been suggested for scale parameters in hierarchical models. We construct a new folded-noncentral-t family of conditionally conjugate priors for hierarchical standard deviation parameters, and then consider noninformative and weakly informative priors in this family. We use an example to illustrate serious problems with the inverse-gamma family of “noninformative ” prior distributions. We suggest instead to use a uniform prior on the hierarchical standard deviation, using the half-t family when the number of groups is small and in other settings where a weakly informative prior is desired. We also illustrate the use of the half-t family for hierarchical modeling of multiple variance parameters such as arise in the analysis of variance.

This article presents a mathematical denition of texture { the Julesz ensemble h), which is the set of all images (defined on Z²) that share identical statistics h. Then texture modeling is posed as an inverse problem: given a set of images sampled from an unknown Julesz ensemble h ), we search for the statistics h which define the ensemble. A Julesz ensemble h) has an associated probability distribution q(I; h), which is uniform over the images in the ensemble and has zero probability outside. In a companion paper [32], q(I; h) is shown to be the limit distribution of the FRAME (Filter, Random Field, And Minimax Entropy) model[35] as the image lattice ! Z². This conclusion establishes the intrinsic link between the scientific definition of texture on Z² and the mathematical models of texture on finite lattices. It brings two advantages to computer vision. 1). The engineering practice of synthesizing texture images by matching statistics has been put on a mathematical fou...

Hierarchical linear and generalized linear models can be fit using Gibbs samplers and Metropolis algorithms; these models, however, often have many parameters, and convergence of the seemingly most natural Gibbs and Metropolis algorithms can sometimes be slow. We examine solutions that involve reparameterization and over-parameterization. We begin with parameter expansion using working parameters, a strategy developed for the EM algorithm by Meng and van Dyk (1997) and Liu, Rubin, and Wu (1998). This strategy can lead to algorithms that are much less susceptible to becoming stuck near zero values of the variance parameters than are more standard algorithms. Second, we consider a simple rotation of the regression coefficients based on an estimate of their posterior covariance matrix. This leads to a Gibbs algorithm based on updating the transformed parameters one at a time or a Metropolis algorithm with vector jumps; either of these algorithms can perform much better (in terms of total CPU time) than the two standard algorithms: one-at-a-time updating of untransformed parameters or vector updating using a linear regression at each step. We present an innovative evaluation of the algorithms in terms of how quickly they can get away from remote areas of parameter space, along with some more standard evaluation of computation and convergence speeds. We illustrate our methods with examples from our applied work. Our ultimate goal is to develop a fast and reliable method for fitting a hierarchical linear model as easily as one can now fit a non-hierarchical model, and to increase understanding of Gibbs samplers for hierarchical models in general. Keywords: Bayesian computation, blessing of dimensionality, Markov chain Monte Carlo, multilevel modeling, mixed effects models, PX-EM algorithm, random effects regression, redundant

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We focus on sparse modeling of high-dimensional covariance matrices using Bayesian latent factor models. We propose a multiplicative gamma process shrinkage prior on the factor loadings which allows introduction of infinitely many factors, with the loadings increasingly shrunk toward zero as the column index increases. We use our prior on a parameter expanded loadings matrix to avoid the order dependence typical in factor analysis models and develop a highly efficient Gibbs sampler that scales well as data dimensionality increases. The gain in efficiency is achieved by the joint conjugacy property of the proposed prior, which allows block updating of the loadings matrix. We propose an adaptive Gibbs sampler for automatically truncating the infinite loadings matrix through selection of the number of important factors. Theoretical results are provided on the support of the prior and truncation approximation bounds. A fast algorithm is proposed to produce approximate Bayes estimates. Latent factor regression methods are developed for prediction and variable selection in applications with high-dimensional correlated predictors. Operating characteristics are assessed through simulation studies and the approach is applied to predict survival after chemotherapy from gene expression data.

This paper shows (i) improvements over state-of-the-art local feature recognition systems, (ii) how to formulate principled models for automatic local feature selection in object class recognition when there is little supervised data, and (iii) how to formulate sensible spatial image context models using a conditional random field for integrating local features and segmentation cues (superpixels). By adopting sparse kernel methods, Bayesian learning techniques and data association with constraints, the proposed model identifies the most relevant sets of local features for recognizing object classes, achieves performance comparable to the fully supervised setting, and obtains excellent results for image classification.

Among the computationally intensive methods for fitting complex multilevel models, the Gibbs sampler is especially popular owing to its simplicity and power to effectively generate samples from a high-dimensional probability distribution. The Gibbs sampler, however, is often justifiably criticized for its sometimes slow convergence, especially when it is used to fit highly structured complex models. The recently proposed Partially Collapsed Gibbs (PCG) sampler offers a new strategy for improving the convergence characteristics of a Gibbs sampler. A PCG sampler achieves faster convergence by reducing the conditioning in some or all of the component draws of its parent Gibbs sampler. Although this strategy can significantly improve convergence, it must be implemented with care to be sure that the desired stationary distribution is preserved. In some cases the set of conditional distributions sampled in a PCG sampler may be functionally incompatible and permuting the order of draws can change the stationary distribution of the chain. In this article, we draw an analogy between the PCG sampler and certain efficient EM-type algorithms that helps to explain the computational advantage of PCG samplers and to suggest when they might be used in practice.

In recent years, a variety of extensions and refinements have been developed for data augmentation based model fitting routines. These developments aim to extend the application, improve the speed, and/or simplify the implementation of data augmentation methods, such as the deterministic EM algorithm for mode finding and stochastic Gibbs sampler and other auxiliary-variable based methods for posterior sampling. In this overview article we graphically illustrate and compare a number of these extensions all of which aim to maintain the simplicity and computation stability of their predecessors. We particularly emphasize the usefulness of identifying similarities between the deterministic and stochastic counterparts as we seek more efficient computational strategies. We also demonstrate the applicability of data augmentation methods for handling complex models

Factor analytic models are widely used in social sciences. These models have also proven useful for sparse modeling of the covariance structure in multidimensional data. Normal prior distributions for factor loadings and inverse gamma prior distributions for residual variances are a popular choice because of their conditionally conjugate form. However, such prior distributions require elicitation of many hyperparameters and tend to result in poorly behaved Gibbs samplers. In addition, one must choose an informative specification, as high variance prior distributions face problems due to impropriety of the posterior distribution. This article proposes a default, heavy tailed prior distribution specification, which is induced through parameter expansion while facilitating efficient posterior computation. We also develop an approach to allow uncertainty in the number of factors. The methods are illustrated through simulated examples and epidemiology and toxicology applications.

The use of Gibbs samplers driven by improper posteriors has been a controversial issue in the statistics literature over the last few years. Recently, Gelfand and Sahu (1999), Liu and Wu (1999), Meng and van Dyk (1999), and van Dyk and Meng (2001) have given examples demonstrating that it is possible to make valid statistical inferences through such Gibbs samplers. Furthermore, these authors provide theoretical and empirical evidence that there are actually computational advantages to using these non-positive recurrent Markov chains rather than more standard positive recurrent chains. These results provide motivation for a general study of the behavior of the Gibbs Markov chain when it is not positive recurrent. This paper concerns stability relationships among the two-variable Gibbs sampler and its subchains. We show that these three Markov chains always share the same stability; that is, they are either all positive recurrent, all null recurrent, or all transient. In additi...