Mary Kathryn Cowles

Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review

Abstract A critical issue for users of Markov chain Monte Carlo (MCMC) methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest. Research into methods of computing theoretical convergence bounds holds promise for the future but to date has yielded relatively little of practical use in applied work. Consequently, most MCMC users address the convergence problem by applying diagnostic tools to the output produced by running their samplers. After giving a brief overview of the area, we provide an expository review of 13 convergence diagnostics, describing the theoretical basis and practical implementation of each. We then compare their performance in two simple models and conclude that all of the methods can fail to detect the sorts of convergence failure that they were designed to identify. We thus recommend a combination of strategies aimed at evaluating and accelerating MCMC sampler convergence, including ap...

Accelerating Monte Carlo Markov chain convergence for cumulative-link generalized linear models

The ordinal probit, univariate or multivariate, is a generalized linear model (GLM) structure that arises frequently in such disparate areas of statistical applications as medicine and econometrics. Despite the straightforwardness of its implementation using the Gibbs sampler, the ordinal probit may present challenges in obtaining satisfactory convergence.We present a multivariate Hastings-within-Gibbs update step for generating latent data and bin boundary parameters jointly, instead of individually from their respective full conditionals. When the latent data are parameters of interest, this algorithm substantially improves Gibbs sampler convergence for large datasets. We also discuss Monte Carlo Markov chain (MCMC) implementation of cumulative logit (proportional odds) and cumulative complementary log-log (proportional hazards) models with latent data.

Bayesian Tobit Modeling of Longitudinal Ordinal Clinical Trial Compliance Data with Nonignorable Missingness

Abstract In the Lung Health Study (LHS), compliance with the use of inhaled medication was assessed at each follow-up visit both by self-report and by weighing the used medication canisters. One or both of these assessments were missing if the participant failed to attend the visit or to return all canisters. Approximately 30% of canister-weight data and 5% to 15% of self-report data were missing at different visits. We use Gibbs sampling with data augmentation and a multivariate Hastings update step to implement a Bayesian hierarchical model for LHS inhaler compliance. Incorporating individual-level random effects to account for correlations among repeated measures on the same participant, our model is a longitudinal extension of the Tobit models used in econometrics to deal with partially unobservable data. It enables (a) assessment of the relationships among visit attendance, canister return, self-reported compliance level, and canister weight compliance, and (b) determination of demographic, physiolog...

A simulation approach to convergence rates for Markov chain Monte Carlo algorithms

Markov chain Monte Carlo (MCMC) methods, including the Gibbs sampler and the Metropolis–Hastings algorithm, are very commonly used in Bayesian statistics for sampling from complicated, high-dimensional posterior distributions. A continuing source of uncertainty is how long such a sampler must be run in order to converge approximately to its target stationary distribution. A method has previously been developed to compute rigorous theoretical upper bounds on the number of iterations required to achieve a specified degree of convergence in total variation distance by verifying drift and minorization conditions. We propose the use of auxiliary simulations to estimate the numerical values needed in this theorem. Our simulation method makes it possible to compute quantitative convergence bounds for models for which the requisite analytical computations would be prohibitively difficult or impossible. On the other hand, although our method appears to perform well in our example problems, it cannot provide the guarantees offered by analytical proof.

Bayesian Statistical Modelling

Preface. Chapter 1 Introduction: The Bayesian Method, its Benefits and Implementation. Chapter 2 Bayesian Model Choice, Comparison and Checking. Chapter 3 The Major Densities and their Application. Chapter 4 Normal Linear Regression, General Linear Models and Log-Linear Models. Chapter 5 Hierarchical Priors for Pooling Strength and Overdispersed Regression Modelling. Chapter 6 Discrete Mixture Priors. Chapter 7 Multinomial and Ordinal Regression Models. Chapter 8 Time Series Models. Chapter 9 Modelling Spatial Dependencies. Chapter 10 Nonlinear and Nonparametric Regression. Chapter 11 Multilevel and Panel Data Models. Chapter 12 Latent Variable and Structural Equation Models for Multivariate Data. Chapter 13 Survival and Event History Analysis. Chapter 14 Missing Data Models. Chapter 15 Measurement Error, Seemingly Unrelated Regressions, and Simultaneous Equations. Appendix 1 A Brief Guide to Using WINBUGS. Index.

Automatic Nonuniform Random Variate Generation

considers the four symmetries now adapted to a fixed filtration, using martingale techniques to gain new characterizations. It closes with an authoritative treatment of Palm measures. Chapter 3 is on weak convergence, for instance characterizing convergence in law of exchangeable processes by convergence of their directing mixing measures. For me, the highlight of the chapter is the discussion of subsequence principles that concludes it. The results in Chapters 4 and 5 are closely related but use rather different methods. The former focuses on invariance under predictable transformations, and the latter focuses on decoupling, where many remarkable identities appear, generalizing classical equations of Wald for optionally stopped random walk. Chapter 6 is concerned with random sets and point processes of excursions, exploiting the implications of local or global homogeneity and of invariance under reflections. The final chapters cover multivariate versions of the latter three symmetries. Chapter 7 discusses exchangeable and contractible arrays, and Chapter 8 covers rotatable arrays, both in the multivariate case. Chapter 9 deals with exchangeable random measures in the plane. In a book of this nature, there is much new mathematics, both explicit and implicit, as for instance in Section 7.8, where Kingman’s paintbox representation for exchangeable partitions is significantly extended. The treatment here is understandably rather limited in scope; past work on random partitions was largely motivated by problems arising in population genetics. Although it would take the author far off topic to go into that area, the reader should consult Pitman (2006) to see where it all leads to. Chapter 9 is followed by appendixes on particular points of measure theory and real analysis, and a thorough set of historical and bibliographical notes, giving a comprehensive set of attributions and references and of course further readings. The author’s reference for the probability theory that his book starts from is naturally the text by Kallenberg (2002). The standard of exposition is uncompromising—the reader has to be a poet in posse, as Loève put it (in the preface to the third edition of his treatise, reprinted in Loève 1977)—but uniformly clear. I suppose the one thing that might help the reader more, particularly toward the end of the book, would be more explicit description of the logical dependence of the various sections and results on the earlier material. That would further empower the randomaccess (as opposed to the sequential-access) reader to whom I referred in the first paragraph of this review. Perhaps the author could provide something on his website. But it is inappropriate to carp. As a whole, this is a grand conception, superbly realized.

Generalized, Linear, and Mixed Models

This book bridges a gap in the smoothing and image processing literature. Books on smoothing usually concentrate on estimating smooth functions and may devote one section to estimating functions with jumps. The book provides comprehensive coverage of statistical methodologies for jump detection and estimation in the nonparametric regression setup as well as the estimation of regression functions with such jumps. It surveys common edge detection methods in image processing and relates edge detection to jump analysis. The book comprises seven chapters, each of which concludes with exercises. Chapter 1 introduces image processing and nonparametric regression and provides an overview of the book. Chapter 2 reviews simple inference concepts and common smoothing techniques for estimating smooth regression functions, including various kernel, spline, and orthogonal basis methods. Chapter 3 presents the estimation of a one-dimensional regression function with jumps. To estimate such a function by smoothing methods, one needs to know the number of jumps and their corresponding jump locations. The author considers cases when the number of jumps is known and unknown. Jump detection approaches presented are kernel and local polynomial methods with one-sided kernels, spline methods with special truncated power functions, and wavelet methods. Once the numbers of jumps and jump locations are determined, smoothing methods are used to estimate smooth curves between successive jump points. Chapters 4 and 5 investigate the two-dimensional case. In two dimensions, the regression surface often jumps along some curve. Chapter 4 deals with locating such a curve, and Chapter 5 is devoted to recovering the jump regression surface. Intrinsic difficulties present when moving from one dimension to two dimensions. Jump curves may not have explicit mathematical expressions, and the jump size at a fixed point on the curve may vary as the direction changes. The jump detection and estimation methods presented in Chapter 4 include local linear and kernel based methods, contrast statistic-based methods, tracking algorithms based on kernel-weighted local log-likelihood ratios, and wavelet methods. The theory for jump detection and estimation methods is discussed. The well-known Sobel edge detector in image processing is linked to the smoothed differences based on two one-sided kernels with two directions. Estimation of jump regression surfaces is studied in Chapter 5. The methods described in this chapter are mostly developed by the author and are based on local smoothing with various schemes like thresholding and gradients. Chapter 6 features edge detection methods in image processing, which is analogous to jump-curve estimation in Chapter 4. Various methods and their mathematical approaches are provided. These methods are described mostly in deterministic fashion. Chapter 7 handles image restoration with edges preserved. It parallels Chapter 5 for recovering a jump surface. Methods include Fourier, Markov random field, regularization, local smoothing, and diffusion filtering approaches. The chapter should offer a section to describe the wavelet based methodology for image restoration. Wavelets are very popular and effective in image processing, and in fact are much better than some of the methods listed in the chapter. The book is written for both statisticians and engineers. It presents significant material in a terse but careful way and covers both statistics and image processing. This book can serve as a course text for a doctoral-level special topic course on change points in nonparametric regression and image processing or a supplemental course text for graduate-level elective courses on smoothing or image processing. I would also recommend this book as a useful reference for researchers with interests in image processing and nonparametric regression. It has much to offer that is hard to find elsewhere.