Search

Despite its extensive application in practice, the Metropolis-Hastings sampler can suffer from slow mixing and, in turn, statistical inefficiency. We introduce a modification to the Metropolis-Hast...

Customer reviews are an essential resource to reduce an online product’s uncertainty, which has been shown to be a critical factor for its purchase decision. Existing e-commerce platforms typically ask users to write free-form text reviews, which are sometimes augmented by a small set of predefined questions, e.g. “rate the product description’s accuracy from 1 to 5.” In this paper, we argue that this “passive” style of review solicitation is suboptimal in achieving low-uncertainty “review profiles” for products. Its key drawback is that some product aspects receive a very large number of reviews while other aspects do not have enough reviews to draw confident conclusions. Therefore, we hypothesize that we can achieve lower-uncertainty review profiles by carefully selecting which aspects users are asked to rate. To test this hypothesis, we propose various techniques to dynamically select which aspects to ask users to rate given the current review profile of a product. We use Bayesian inference principles to define reasonable review profile uncertainty measures; specifically, via an aspect’s rating variance. We compare our proposed aspect selection techniques to several baselines on several review profile uncertainty measures. Experimental results on two real-world datasets show that our methods lead to better review profile uncertainty compared to aspect selection baselines and traditional passive review solicitations. Moreover, we present and evaluate a hybrid solicitation method that combines the advantages of both active and passive review solicitations.

The fourth edition of Fundamentals of Probability with Stochastic Processes continues to be a great one- or two-semester elementary probability text. The text naturally requires an understanding of...

Monte Carlo experiments produce samples in order to estimate features of a given distribution. However, simultaneous estimation of means and quantiles has received little attention, despite being common practice. In this setting we establish a multivariate central limit theorem for any finite combination of sample means and quantiles under the assumption of a strongly mixing process, which includes the standard Monte Carlo and Markov chain Monte Carlo settings. We build on this to provide a fast algorithm for constructing hyperrectangular confidence regions having the desired simultaneous coverage probability and a convenient marginal interpretation. The methods are incorporated into standard ways of visualizing the results of Monte Carlo experiments enabling the practitioner to more easily assess the reliability of the results. We demonstrate the utility of this approach in various Monte Carlo settings including simulation studies based on independent and identically distributed samples and Bayesian analyses using Markov chain Monte Carlo sampling., Comment: 23 pages, 7 figures

Customer reviews have become an essential resource when people search for goods or services on the Internet. Previous work has shown that reducing a product's uncertainty is critical to its purchase decision. Thus, reviews are more effective when they reduce a product's uncertainty. Existing e-commerce platforms typically ask users to write free-form text reviews, which are sometimes augmented by a small set of predefined questions, e.g., “rate the product description's accuracy from 1 to 5.” In this paper, we argue that this “passive” style of review solicitation is suboptimal in achieving low-uncertainty “review profiles” for products. Its key drawback is that some product aspects receive a very large number of reviews while other aspects do not have enough reviews to draw confident conclusions. Therefore, we hypothesize that we can achieve lower-uncertainty review profiles by carefully selecting which aspects users are asked to rate. To test this hypothesis, we propose various techniques to dynamically select which aspects to ask users to rate given the current review profile of a product. We use Bayesian principles to define reasonable review profile uncertainty measures; specifically, we apply Bayesian inference to measure an aspect's rating variance. We compare our proposed aspect selection techniques to several baselines on several review profile uncertainty measures. Experimental results on two real-world datasets show that our methods lead to better review profile uncertainty compared to aspect selection baselines and traditional passive review solicitations.

Data Visualization: Charts, Maps, and Interactive Graphics, by Robert Grant, overviews a broad range of techniques and challenges encountered in visualizing quantitative information. This well-writ...

Several bivariate beta distributions have been proposed in the literature. In particular, Olkin and Liu (2003) proposed a 3 parameter bivariate beta model, which Arnold and Ng (2011) extend to 5 and 8 parameter models. The 3 parameter model allows for only positive correlation, while the latter models can accommodate both positive and negative correlation. However, these come at the expense of a density that is mathematically intractable. The focus of this research is on Bayesian estimation for the 5 and 8 parameter models. Since the likelihood does not exist in closed form, we apply approximate Bayesian computation, a likelihood free approach. Simulation studies have been carried out for the 5 and 8 parameter cases under various priors and tolerance levels. We apply the 5 parameter model to a real data set by allowing the model to serve as a prior to correlated proportions of a bivariate beta binomial model. Results and comparisons are then discussed., Comment: 22 pages, 3 figures

A current challenge for many Bayesian analyses is determining when to terminate high-dimensional Markov chain Monte Carlo simulations. To this end, we propose using an automated sequential stopping procedure that terminates the simulation when the computational uncertainty is small relative to the posterior uncertainty. Further, we show this stopping rule is equivalent to stopping when the effective sample size is sufficiently large. Such a stopping rule has previously been shown to work well in settings with posteriors of moderate dimension. In this article, we illustrate its utility in high-dimensional simulations while overcoming some current computational issues. As examples, we consider two complex Bayesian analyses on spatially and temporally correlated datasets. The first involves a dynamic space-time model on weather station data and the second a spatial variable selection model on fMRI brain imaging data. Our results show the sequential stopping rule is easy to implement, provides uncertainty est...

Summary Lag windows are commonly used in time series analysis, econometrics, steady-state simulation and Markov chain Monte Carlo to estimate time-average covariance matrices. In the presence of positive correlation in the underlying process, estimators of this matrix almost always exhibit significant negative bias, leading to undesirable finite-sample properties. We propose a new family of lag windows specifically designed to improve finite-sample performance by offsetting this negative bias. Any existing lag window can be adapted into a lugsail equivalent with no additional assumptions. We use these lag windows in spectral variance estimators and demonstrate their advantages in a linear regression model with autocorrelated and heteroskedastic residuals. We further employ the lugsail lag windows in weighted batch means estimators because of their computational efficiency on large simulation output. We obtain bias and variance results for these multivariate estimators and significantly weaken the mixing condition on the process. Superior finite-sample properties are demonstrated in a vector autoregressive process and a Bayesian logistic regression model.

Markov chain Monte Carlo (MCMC) algorithms are used to estimate features of interest of a distribution. The Monte Carlo error in estimation has an asymptotic normal distribution whose multivariate nature has so far been ignored in the MCMC community. We present a class of multivariate spectral variance estimators for the asymptotic covariance matrix in the Markov chain central limit theorem and provide conditions for strong consistency. We examine the finite sample properties of the multivariate spectral variance estimators and its eigenvalues in the context of a vector autoregressive process of order 1.

The naive importance sampling estimator, based on samples from a single importance density, can be numerically unstable. Instead, we consider generalized importance sampling estimators where samples from more than one probability distribution are combined. We study this problem in the Markov chain Monte Carlo context, where independent samples are replaced with Markov chain samples. If the chains converge to their respective target distributions at a polynomial rate, then under two finite moment conditions, we show a central limit theorem holds for the generalized estimators. Further, we develop an easy to implement method to calculate valid asymptotic standard errors based on batch means. We also provide a batch means estimator for calculating asymptotically valid standard errors of Geyer(1994) reverse logistic estimator. We illustrate the method using a Bayesian variable selection procedure in linear regression. In particular, the generalized importance sampling estimator is used to perform empirical Bayes variable selection and the batch means estimator is used to obtain standard errors in a high-dimensional setting where current methods are not applicable., Comment: 49 pages, 9 figures

The cosmic web plays a major role in the formation and evolution of galaxies and defines, to a large extent, their properties. However, the relation between galaxies and environment is still not well understood. Here we present a machine learning approach to study imprints of environmental effects on the mass assembly of haloes. We present a galaxy-LSS machine learning classifier based on galaxy properties sensitive to the environment. We then use the classifier to assess the relevance of each property. Correlations between galaxy properties and their cosmic environment can be used to predict galaxy membership to void/wall or filament/cluster with an accuracy of $93\%$. Our study unveils environmental information encoded in properties of haloes not normally considered directly dependent on the cosmic environment such as merger history and complexity. Understanding the physical mechanism by which the cosmic web is imprinted in a halo can lead to significant improvements in galaxy formation models. This is accomplished by extracting features from galaxy properties and merger trees, computing feature scores for each feature and then applying support vector machine to different feature sets. To this end, we have discovered that the shape and depth of the merger tree, formation time and density of the galaxy are strongly associated with the cosmic environment. We describe a significant improvement in the original classification algorithm by performing LU decomposition of the distance matrix computed by the feature vectors and then using the output of the decomposition as input vectors for support vector machine., Comment: Accepted for publication in MNRAS

We consider batch size selection for a general class of multivariate batch means variance estimators, which are computationally viable for high-dimensional Markov chain Monte Carlo simulations. We derive the asymptotic mean squared error for this class of estimators. Further, we propose a parametric technique for estimating optimal batch sizes and discuss practical issues regarding the estimating process. Vector auto-regressive, Bayesian logistic regression, and Bayesian dynamic space-time examples illustrate the quality of the estimation procedure where the proposed optimal batch sizes outperform current batch size selection methods., Comment: 38 pages, 5 figures

This paper proposes a family of weighted batch means variance estimators, which are computationally efficient and can be conveniently applied in practice. The focus is on Markov chain Monte Carlo simulations and estimation of the asymptotic covariance matrix in the Markov chain central limit theorem, where conditions ensuring strong consistency are provided. Finite sample performance is evaluated through auto-regressive, Bayesian spatial-temporal, and Bayesian logistic regression examples, where the new estimators show significant computational gains with a minor sacrifice in variance compared with existing methods., 52 pages, 6 figures, 3 tables

Nordihydroguaiaretic Acid (NDGA) is a lignin which constitutes about 12.5% of the dry weight of the leaves and twigs of the creosote bush, Larrea tridentate [Reviewed in (1–4)]. Aqueous extracts of creosote leaves and twigs have been used medicinally by indigenous North American tribes to treat over 50 health disorders, ranging from colds to cancer (2–4). NDGA was once classified as “generally recognized as safe” by the Federal Food and Drug Administration, and used as an antioxidant food additive. This classification was withdrawn after studies in rats showed that NDGA produced serious kidney toxicity and other pathologies, including stunted growth and internal hemorrhages [(5); Reviewed in (6)]. Human consumption of creosote leaf and stem extracts as dietary supplements led to cases of hepatitis, cirrhosis, and fulminant liver failure (4,7,8). Despite these findings, a recent Google internet search identified multiple vendors selling creosote leaf extracts as medicinal health aids. In vitro studies have shown that NDGA is an inhibitor of intercellular inflammatory signaling, tumor cell proliferation, insulin-like growth factor-1 (IGFIR) and HER2 receptor activation, and oxidative phosphorylation (4,9). Based on the therapeutic potential suggested by these results, the National Institute of Health Interventions Testing Program (NIH-ITP) undertook studies of the effects of NDGA on murine lifespan (10–12). They found that 2.5g of NDGA/kg diet produced a significant 12% increase in median lifespan for male mice, but not females (12). A second study found no effect on the lifespan of female mice (11). A third study censored after 70% mortality suggested that multiple NDGA doses may extend the lifespan of male mice (10). No necropsy, pathology, or toxicology results were reported, nor was food consumption reported. In Drosophila, a single dose study found a nonsignificant increase in lifespan (13). Thus, the effects of NDGA dosage on lifespan, and its effects on food consumption, end of life pathologies, energy disposition, and phylogenic conservation of the response are unclear. For these reasons, we conducted dose–response studies of the effects of NDGA on lifespan in mice and Drosophila melanogaster (Drosophila). We also investigated the effects of NDGA on food intake, body weight, and mortality-related pathologies in mice. In Drosophila, we characterized its effects on lifespan, food consumption, body weight, and locomotion Induced caloric restriction (CR) is a possible explanation for lifespan responses when food consumption is not monitored (14).

A modified conditional Metropolis–Hastings sampler for general state spaces is introduced. Under specified conditions, this modification can lead to substantial gains in statistical efficiency while maintaining the overall quality of convergence. Results are illustrated in two settings: a toy bivariate Normal model and a Bayesian version of the random effects model.

We develop a sample size methodology that achieves specified Type-1 and Type-2 error rates when comparing the survivor functions of multiple treatment groups versus a control group. The designs will control family-wise Type-1 error rate. We assume the family of Weibull distributions adequately describes the underlying survivor functions, and we separately consider three of the most common study scenarios: (a) complete samples; (b) Type-1 censoring with a common censoring time; and (c) Type-1 censoring with an accrual period. A mice longevity study comparing the effect on survival of multiple low-calorie diets is used to motivate our work on this problem.

Chronic treatment with β-adrenergic receptor (βAR) agonists increases mortality and morbidity while βAR antagonists (β-blockers) decrease all-cause mortality for those at risk of cardiac disease. Levels of sympathetic nervous system βAR agonists and βAR activity increase with age, and this increase may hasten the development of age-related mortality. Here, we show that β-blockers extend the life span of healthy metazoans. The β-blockers metoprolol and nebivolol, administered in food daily beginning at 12 months of age, significantly increase the mean and median life span of isocalorically fed, male C3B6F1 mice, by 10 and 6.4 %, respectively (P

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.

We consider quantile estimation using Markov chain Monte Carlo and establish conditions under which the sampling distribution of the Monte Carlo error is approximately Normal. Further, we investigate techniques to estimate the associated asymptotic variance, which enables construction of an asymptotically valid interval estimator. Finally, we explore the finite sample properties of these methods through examples and provide some recommendations to practitioners., 35 pages, 1 figure

Present data suggest that the consumption of individual dietary supplements does not enhance the health or longevity of healthy rodents or humans. It might be argued that more complex combinations of such agents might extend lifespan or health-span by more closely mimicking the complexity of micronutrients in fruits and vegetables, which appear to extend health-span and longevity. To test this hypothesis we treated long-lived, male, F1 mice with published and commercial combinations of dietary supplements and natural product extracts, and determined their effects on lifespan and health-span. Nutraceutical, vitamin or mineral combinations reported to extend the lifespan or health-span of healthy or enfeebled rodents were tested, as were combinations of botanicals and nutraceuticals implicated in enhanced longevity by a longitudinal study of human aging. A cross-section of commercial nutraceutical combinations sold as potential health enhancers also were tested, including Bone Restore®, Juvenon®, Life Extension Mix®, Ortho Core®, Ortho Mind®, Super K w k2®, and Ultra K2®. A more complex mixture of vitamins, minerals, botanical extracts and other nutraceuticals was compounded and tested. No significant increase in murine lifespan was found for any supplement mixture. Our diverse supplement mixture significantly decreased lifespan. Thus, our results do not support the hypothesis that simple or complex combinations of nutraceuticals, including antioxidants, are effective in delaying the onset or progress of the major causes of death in mice. The results are consistent with epidemiological studies suggesting that dietary supplements are not beneficial and even may be harmful for otherwise healthy individuals.

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.

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 pages

Markov chain Monte Carlo (MCMC) methods allow exploration of intractable probability distributions by constructing a Markov chain whose stationary distribution equals the desired distribution. The output from the Markov chain is typically used to estimate several features of the stationary distribution such as mean and variance parameters along with quantiles and so on. Unfortunately, most reported MCMC estimates do not include a clear notion of the associated uncertainty. For expectations one can assess the uncertainty by estimating the variance in an asymptotic normal distribution of the Monte Carlo error. For general functionals there is no such clear path. This article studies the applicability of subsampling bootstrap methods to assess the uncertainty in estimating general functionals from MCMC simulations.

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., 28 pages, 2 figures

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)

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., 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)

Statins, such as simvastatin, and ACE inhibitors (ACEis), such as ramipril, are standard therapies for the prevention and treatment of cardiovascular diseases. These types of drugs are commonly administered together. More recently, angiotensin II type 1 receptor (AT1R) antagonists, such as candesartan cilexetil (candesartan), have been used in the place of, or in combination with, ACEis. Here, we investigated the effects of simvastatin and ramipril single and combination therapy, and candesartan treatment on the lifespan of isocalorically fed, long-lived, B6C3F1 mice. Males were used for their relative endocrine simplicity and to minimize animal usage. The drugs were administered daily in food. The simvastatin and ramipril combination therapy significantly increased the mean and median lifespan by 9 %. In contrast, simvastatin, ramipril, or candesartan monotherapy was ineffective. All groups consumed the same number of calories. Simvastatin, alone or administered with ramipril, decreased body weight without changing caloric consumption, suggesting it may alter energy utilization in mice. Combination therapy elevated serum triglyceride and glucose levels, consistent with altered energy homeostasis. Few significant or consistent differences were found in mortality-associated pathologies among the groups. Simvastatin treatment did not reduce normal serum cholesterol or lipid levels in these mice, suggesting that the longevity effects may stem from the pleiotropic, non-cholesterol-related, effects of statins. Together, the results suggest that statins and ACEis together may enhance mouse longevity. Statins and ACE inhibitors are generally well-tolerated, and in combination, they have been shown to increase the lifespan of normotensive, normocholesterolemic humans.

Marine oils rich in ω-3 polyunsaturated fatty acids have been recommended as a preventive treatment for patients at risk for cardiovascular diseases. These oils also are the third most consumed dietary supplement in the USA. However, evidence for their health benefits is equivocal. We tested the daily, isocaloric administration of krill oil (1.17 g oil/kg diet) and Lovaza (Omacor; 4.40 g/kg diet), a pharmaceutical grade fish oil, beginning at 12 months of age, on the life span and mortality-related pathologies of long-lived, male, B6C3F1 mice. The oils were incorporated into the chemically defined American Institute of Nutrition (AIN)-93 M diet. An equivalent volume of soybean oil was removed. Krill oil was 3 % and Lovaza 11 % of the oil in the diets. When their effects were analyzed together, the marine oils significantly shortened life span by 6.6 % (P = 0.0321; log-rank test) relative to controls. Individually, Lovaza and krill oil non-significantly shortened median life span by 9.8 and 4.7 %, respectively. Lovaza increased the number of enlarged seminal vesicles (7.1-fold). Lovaza and krill oil significantly increased lung tumors (4.1- and 8.2-fold) and hemorrhagic diathesis (3.9- and 3.1-fold). Analysis of serum from treated mice found that Lovaza slightly increased blood urea nitrogen, while krill oil modestly increased bilirubin, triglycerides, and blood glucose levels. Taken together, the results do not support the idea that the consumption of isolated ω-3 fatty acid-rich oils will increase the life span or health of initially healthy individuals.