Galin L. Jones

On the Markov chain central limit theorem

The goal of this expository paper is to describe conditions which guarantee a central limit theorem for functionals of general state space Markov chains. This is done with a view towards Markov chain Monte Carlo settings and hence the focus is on the connections between drift and mixing conditions and their implications. In particular, we consider three commonly cited central limit theorems and discuss their relationship to classical results for mixing processes. Several motivating examples are given which range from toy one-dimensional settings to complicated settings encountered in Markov chain Monte Carlo.

Fixed-Width Output Analysis for Markov Chain Monte Carlo

Markov chain Monte Carlo is a method of producing a correlated sample to estimate features of a target distribution through ergodic averages. A fundamental question is when sampling should stop; that is, at what point the ergodic averages are good estimates of the desired quantities. We consider a method that stops the simulation when the width of a confidence interval based on an ergodic average is less than a user-specified value. Hence calculating a Monte Carlo standard error is a critical step in assessing the simulation output. We consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution. We give sufficient conditions for the strong consistency of both methods and investigate their finite-sample properties in various examples.

Markov Chain Monte Carlo: Can We Trust the Third Significant Figure?

Current reporting of results based on Markov chain Monte Carlo computations could be improved. In particular, a measure of the accuracy of the resulting estimates is rarely reported. Thus we have little ability to objectively assess the quality of the reported estimates. We address this issue in that we discuss why Monte Carlo standard errors are important, how they can be easily calculated in Markov chain Monte Carlo and how they can be used to decide when to stop the simulation. We compare their use to a popular alternative in the context of two examples.

Batch means and spectral variance estimators in Markov chain Monte Carlo

Calculating a Monte Carlo standard error (MCSE) is an important step in the statistical analysis of the simulation output obtained from a Markov chain Monte Carlo experiment. An MCSE is usually based on an estimate of the variance of the asymptotic normal distribution. We consider spectral and batch means methods for estimating this variance. In particular, we establish conditions which guarantee that these estimators are strongly consistent as the simulation effort increases. In addition, for the batch means and overlapping batch means methods we establish conditions ensuring consistency in the mean-square sense which in turn allows us to calculate the optimal batch size up to a constant of proportionality. Finally, we examine the empirical finite-sample properties of spectral variance and batch means estimators and provide recommendations for practitioners.

Sufficient burn-in for Gibbs samplers for a hierarchical random effects model

We consider Gibbs and block Gibbs samplers for a Bayesian hierarchical version of the one-way random effects model. Drift and minorization conditions are established for the underlying Markov chains. The drift and minorization are used in conjunction with results from J. S. Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558– 566] and G. O. Roberts and R. L. Tweedie [Stochastic Process. Appl. 80 (1999) 211–229] to construct analytical upper bounds on the distance to stationarity. These lead to upper bounds on the amount of burn-in that is required to get the chain within a prespecified (total variation) distance of the stationary distribution. The results are illustrated with a numerical example. 1. Introduction. We consider a Bayesian hierarchical version of the standard normal theory one-way random effects model. The posterior density for this model is intractable in the sense that the integrals required for making inferences cannot be computed in closed form. Hobert and Geyer (1998) analyzed a Gibbs sampler and a block Gibbs sampler for this problem and showed that the Markov chains underlying these algorithms converge to the stationary (i.e., posterior) distribution at a geometric rate. However, Hobert and Geyer stopped short of constructing analytical upper bounds on the total variation distance to stationarity. In this article, we construct such upper bounds and this leads to a method for determining a sufficient burn-in. Our results are useful from a practical standpoint because they obviate troublesome, ad hoc convergence diagnostics [Cowles and Carlin (1996) and

Needle core length in sextant biopsy influences prostate cancer detection rate

OBJECTIVES Prostate cancer detection in biopsies increases with the number of sites and total tissue sampled. Its dependence on needle core fragment length is uncertain. METHODS We surveyed two consecutive series of sextant needle biopsies from two practices in 1998 to 2000: 251 patients from Pennsylvania (group P) and 1596 from Virginia (group V). We tabulated the gross needle core lengths per sextant site and classified the diagnoses as benign or into four nonbenign categories: high-grade prostatic intraepithelial neoplasia; atypical small acinar proliferation, suspicious; atypical small acinar proliferation, suspicious plus high-grade prostatic intraepithelial neoplasia; and cancer. Logistic regression analysis was used to correlate cancer or a nonbenign diagnosis with the total length (sum of six sites) and, after excluding the sites with more than one core, with the length per single core, and the anatomic site of origin (apex, mid-gland, base). RESULTS The mean total tissue length sampled was 108 +/- 27 mm (range 30 to 275) in group P and 81 +/- 22 mm (range 30 to 228) in group V. Sextant sites with a single core contained a mean of 12.8 +/- 3.5 mm tissue, with a 3.6-fold variation among the middle 95%. Group V core lengths at the apex averaged 11.8 mm, shorter (P = 0.0001) than mid (13.3 mm) or base (12.7 mm). A predictive value of longer length for a nonbenign diagnosis was noted in four of six sextants (P <0.04), with trend strongest at the apex, for which detection was influenced by abnormal digital rectal examination (P = 0.02) or ultrasound (P = 0.04) findings. CONCLUSIONS The length of single cores sampled by sextant biopsy can vary more than 3.6-fold and represents a quality assurance consideration. The effect of length on cancer or nonbenign detection was maximal at the prostatic apex where the cores were shortest.

Supplemental Vitamin C Appears to Slow Racing Greyhounds

During strenuous exercise, markers of oxidation increase and antioxidant capacity decreases. Antioxidants such as vitamin C may combat this oxidation stress. The benefits of vitamin C to greyhounds undertaking intense sprint exercise has not been investigated. The objective of this experiment was to determine whether a large dose (1 g or 57 mmol) of ascorbic acid influences performance and oxidative stress in greyhounds. Five adult female, trained racing greyhounds were assigned to receive each of three treatments for 4 wk per treatment: 1) no supplemental ascorbate; 2) 1 g oral ascorbate daily, administered after racing; 3) 1 g oral ascorbate daily, administered 1 h before racing. Dogs raced 500 m twice weekly. At the end of each treatment period, blood was collected before and 5 min, 60 min and 24 h after racing. Plasma ascorbate, alpha-tocopherol, thiobarbituric acid-reducing substances (TBARS) and Trolox equivalent antioxidant capacity (TEAC) concentrations were measured and adjusted to compensate for hemoconcentration after racing. TBARS, TEAC and alpha-tocopherol concentrations were unaffected by supplemental vitamin C. Plasma ascorbic acid concentrations 60 min after racing were higher in dogs that received vitamin C before racing than in dogs that either received no vitamin C or received vitamin C after racing. The dogs ran, on average, 0.2 s slower when supplemented with 1 g of vitamin C, equivalent to a lead of 3 m at the finish of a 500-m race. Supplementation with vitamin C, therefore, appeared to slow racing greyhounds.

Spatial Bayesian Variable Selection Models on Functional Magnetic Resonance Imaging Time-Series Data

A common objective of fMRI (functional magnetic resonance imaging) studies is to determine subject-specific areas of increased blood oxygenation level dependent (BOLD) signal contrast in response to a stimulus or task, and hence to infer regional neuronal activity. We posit and investigate a Bayesian approach that incorporates spatial and temporal dependence and allows for the task-related change in the BOLD signal to change dynamically over the scanning session. In this way, our model accounts for potential learning effects in addition to other mechanisms of temporal drift in task-related signals. We study the properties of the model through its performance on simulated and real data sets.

Component-Wise Markov Chain Monte Carlo: Uniform and Geometric Ergodicity under Mixing and Composition

It is common practice in Markov chain Monte Carlo to update the simulation one variable (or sub-block of variables) at a time, rather than conduct a single full-dimensional update. When it is possible to draw from each full-conditional distribution associated with the target this is just a Gibbs sampler. Often at least one of the Gibbs updates is replaced with a Metropolis-Hastings step, yielding a Metropolis-Hastings-within-Gibbs al- gorithm. Strategies for combining component-wise updates include compo- sition, random sequence and random scans. While these strategies can ease MCMC implementation and produce superior empirical performance com- pared to full-dimensional updates, the theoretical convergence properties of the associated Markov chains have received limited attention. We present conditions under which some component-wise Markov chains converge to the stationary distribution at a geometric rate. We pay particular attention to the connections between the convergence rates of the various component- wise strategies. This is important since it ensures the existence of tools that an MCMC practitioner can use to be as confident in the simulation results as if they were based on independent and identically distributed samples. We illustrate our results in two examples including a hierarchical linear mixed model and one involving maximum likelihood estimation for mixed models.

Gibbs Sampling for a Bayesian Hierarchical General Linear Model

We consider a Bayesian hierarchical version of the normal theory general linear model which is practically relevant in the sense that it is general enough to have many applications and it is not straightforward to sample directly from the corresponding posterior distribution. Thus we study a block Gibbs sampler that has the posterior as its invariant distribution. In particular, we establish that the Gibbs sampler converges at a geometric rate. This allows us to establish conditions for a central limit theorem for the ergodic averages used to estimate features of the posterior. Geometric ergodicity is also a key component for using batch means methods to consistently estimate the variance of the asymptotic normal distribution. Together, our results give practitioners the tools to be as confident in inferences based on the observations from the Gibbs sampler as they would be with inferences based on random samples from the posterior. Our theoretical results are illustrated with an application to data on the cost of health plans issued by health maintenance organizations.