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.Comment: Published in at http://dx.doi.org/10.1214/08-STS257 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

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. 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.

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.Comment: Published in at http://dx.doi.org/10.1214/09-AOS735 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Phytonutrients reportedly extend the life span of Caenorhabditis elegans, Drosophila, and mice. We tested extracts of blueberry, pomegranate, green and black tea, cinnamon, sesame, and French maritime pine bark (Pycnogenol and taxifolin), as well as curcumin, morin, and quercetin for their effects on the life span of mice. While many of these phytonutrients reportedly extend the life span of model organisms, we found no significant effect on the life span of male F1 hybrid mice, even though the dosages used reportedly produce defined therapeutic end points in mice. The compounds were fed beginning at 12 months of age. The control and treatment groups were iso-caloric with respect to one another. A 40% calorically restricted and other groups not reported here did experience life span extension. Body weights were un-changed relative to controls for all but two supplemented groups, indicating most supplements did not change energy absorption or utilization. Tea extracts with morin decreased weight, whereas quercetin, taxifolin, and Pycnogenol together increased weight. These changes may be due to altered locomotion or fatty acid biosynthesis. Published reports of murine life span extension using curcumin or tea components may have resulted from induced caloric restriction. Together, our results do not support the idea that isolated phytonutrient anti-oxidants and anti-inflammatories are potential longevity therapeutics, even though consumption of whole fruits and vegetables is associated with enhanced health span and life span.