Relative fixed-width stopping rules for Markov chain Monte Carlo simulations

Markov chain Monte Carlo (MCMC) simulations are commonly employed for estimating features of a target distribution, particularly for Bayesian inference. A fundamental challenge is determining when these simulations should stop. We consider a sequential stopping rule that terminates the simulation when the width of a confidence interval is sufficiently small relative to the size of the target parameter. Specifically, we propose relative magnitude and relative standard deviation stopping rules in the context of MCMC. In each setting, we develop sufficient conditions for asymptotic validity, that is conditions to ensure the simulation will terminate with probability one and the resulting confidence intervals will have the proper coverage probability. Our results are applicable in a wide variety of MCMC estimation settings, such as expectation, quantile, or simultaneous multivariate estimation. Finally, we investigate the finite sample properties through a variety of examples and provide some recommendations to practitioners.Comment: 24 page

You can go your own way: effectiveness of participant-driven versus experimenter-driven processing strategies in memory training and transfer

Cognitive training programs that instruct specific strategies frequently show limited transfer. Open-ended approaches can achieve greater transfer, but may fail to benefit many older adults due to age deficits in self-initiated processing. We examined whether a compromise that encourages effort at encoding without an experimenter-prescribed strategy might yield better results. Older adults completed memory training under conditions that either (1) mandated a specific strategy to increase deep, associative encoding, (2) attempted to suppress such encoding by mandating rote rehearsal, or (3) encouraged time and effort toward encoding but allowed for strategy choice. The experimenter-enforced associative encoding strategy succeeded in creating integrated representations of studied items, but training-task progress was related to pre-existing ability. Independent of condition assignment, self-reported deep encoding was associated with positive training and transfer effects, suggesting that the most beneficial outcomes occur when environmental support guiding effort is provided but participants generate their own strategies

Exact sampling for intractable probability distributions via a Bernoulli factory

Many applications in the field of statistics require Markov chain Monte Carlo methods. Determining appropriate starting values and run lengths can be both analytically and empirically challenging. A desire to overcome these problems has led to the development of exact, or perfect, sampling algorithms which convert a Markov chain into an algorithm that produces i.i.d. samples from the stationary distribution. Unfortunately, very few of these algorithms have been developed for the distributions that arise in statistical applications, which typically have uncountable support. Here we study an exact sampling algorithm using a geometrically ergodic Markov chain on a general state space. Our work provides a significant reduction to the number of input draws necessary for the Bernoulli factory, which enables exact sampling via a rejection sampling approach. We illustrate the algorithm on a univariate Metropolis-Hastings sampler and a bivariate Gibbs sampler, which provide a proof of concept and insight into hyper-parameter selection. Finally, we illustrate the algorithm on a Bayesian version of the one-way random effects model with data from a styrene exposure study.Comment: 28 pages, 2 figure

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.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

Get the gist? The effects of processing depth on false recognition in short-term and long-term memory

Gist-based processing has been proposed to account for robust false memories in the converging-associates task. The deep-encoding processes known to enhance verbatim memory also strengthen gist memory and increase distortions of long-term memory (LTM). Recent research has demonstrated that compelling false memory illusions are relatively delay-invariant, also occurring under canonical short-term memory (STM) conditions. To investigate the contributions of gist to false memory at short and long delays, processing depth was manipulated as participants encoded lists of four semantically related words and were probed immediately, following a filled 3- to 4-s retention interval, or approximately 20 min later, in a surprise recognition test. In two experiments, the encoding manipulation dissociated STM and LTM on the frequency, but not the phenomenology, of false memory. Deep encoding at STM increases false recognition rates at LTM, but confidence ratings and remember/know judgments are similar across delays and do not differ as a function of processing depth. These results suggest that some shared and some unique processes underlie false memory illusions at short and long delays