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.

/pdf/markov-chain-monte-carlo-can-we-trust-the-third-significant-4yjo9m0zbj.pdf

Markov chain Monte Carlo: Can we trust the third significant figure?

Markov chain Monte Carlo (MCMC) produces a correlated sample for estimating expectations with respect to a target distribution. A fundamental question is when should sampling stop so that we have good estimates of the desired quantities? The key to answering this question lies in assessing the Monte Carlo error through a multivariate Markov chain central limit theorem (CLT). The multivariate nature of this Monte Carlo error largely has been ignored in the MCMC literature. We present a multivariate framework for terminating simulation in MCMC. We define a multivariate effective sample size, estimating which requires strongly consistent estimators of the covariance matrix in the Markov chain CLT; a property we show for the multivariate batch means estimator. We then provide a lower bound on the number of minimum effective samples required for a desired level of precision. This lower bound depends on the problem only in the dimension of the expectation being estimated, and not on the underlying stochastic process. This result is obtained by drawing a connection between terminating simulation via effective sample size and terminating simulation using a relative standard deviation fixed-volume sequential stopping rule; which we demonstrate is an asymptotically valid procedure. The finite sample properties of the proposed method are demonstrated in a variety of examples.

Multivariate output analysis for 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.

Batch means and spectral variance estimators in markov chain monte carlo

/pdf/batch-means-and-spectral-variance-estimators-in-markov-chain-12ey25na7y.pdf

Batch means and spectral variance estimators in Markov chain Monte Carlo

The chimpanzees of Gombe: By Jane Goodall

In social primates, individuals use various tactics to compete for dominance rank. Grooming, displays and contact aggression are common components of a male chimpanzee's dominance repertoire. The optimal combination of these behaviors is likely to differ among males with individuals exhibiting a dominance "style" that reflects their tendency to use cooperative and/or agonistic dominance tactics. Here, we examine the grooming behavior of three alpha male chimpanzees at Gombe National Park, Tanzania. We found that (1) these males differed significantly in their tendency to groom with other males; (2) each male's grooming patterns remained consistent before, during and after his tenure as alpha, and (3) the three males tended to groom with high- middle- and low-ranking partners equally. We suggest that body mass may be one possible determinant of differences in grooming behavior. The largest male exhibited the lowest overall grooming rates, whereas the smallest male spent the most time grooming others. This is probably because large males are more effective at physically intimidating subordinates. To achieve alpha status, a small male may need to compensate for reduced size by investing more time and energy in grooming, thereby ensuring coalitionary support from others. Rates of contact aggression and charging displays conformed to this prediction, suggesting that each male exhibited a different dominance "style."

/pdf/alpha-male-chimpanzee-grooming-patterns-implications-for-4umcqu68vl.pdf

Alpha male chimpanzee grooming patterns: implications for dominance “style”

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.

Gibbs Sampling for a Bayesian Hierarchical Version of the General Linear Mixed Model

Auxiliary information is often used to improve the precision of estimators of the finite population cumulative distribution function through the use of superpopulation models. A variety of approaches are available to construct such estimators, including design-based, model-based and model-assisted methods. The superpopulation modeling framework can be either parametric or nonparametric, and the estimators can be constructed as either linear or nonlinear functions of the observations. In this article, we argue that model-assisted estimators based on a nonparametric model are a good overall choice for distribution function estimators, because they have good efficiency properties and are robust against model misspecification. When such estimators are constructed as linear functions of the data, they are also easily incorporated into the existing survey estimation paradigm through the use of survey weights. Theoretical properties of nonparametric distribution function estimators based on local linear regression are derived, and their practical behavior is evaluated in a simulation study.

Estimating Distribution Functions from Survey Data Using Nonparametric Regression

University of Minnesota Ph.D dissertation. July 2009. Major: Statistics. Advisor: Galin L. Jones. 1 computer file (PDF); viii-180, appendices A-B.

https://conservancy.umn.edu/bitstream/handle/11299/53661/Johnson_umn_0130E_10547.pdf;sequence=1

Geometric ergodicity of Gibbs samplers

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 algorithm. Strategies for combining component-wise updates include composition, random sequence and random scans. While these strategies can ease MCMC implementation and produce superior empirical performance compared 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.

Alicia A. Johnson

Papers

Alpha male chimpanzee grooming patterns: implications for dominance “style”

Gibbs Sampling for a Bayesian Hierarchical Version of the General Linear Mixed Model

Estimating Distribution Functions from Survey Data Using Nonparametric Regression

Geometric ergodicity of Gibbs samplers

Component-wise Markov chain Monte Carlo