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Sensory receptive fields are large enough that they can contain more than one perceptible stimulus. How, then, can the brain encode information abouteachof the stimuli that may be present at a given moment? We recently showed that when more than one stimulus is present, single neurons can fluctuate between coding one vs. the other(s) across some time period, suggesting a form of neural multiplexing of different stimuli (Caruso et al., 2018). Here we investigate (a) whether such coding fluctuations occur in early visual cortical areas; (b) how coding fluctuations are coordinated across the neural population; and (c) how coordinated coding fluctuations depend on the parsing of stimuli into separate vs. fused objects. We found coding fluctuations do occur in macaque V1 but only when the two stimuli form separate objects. Such separate objects evoked a novel pattern of V1 spike count (“noise”) correlations involving distinct distributions of positive and negative values. This bimodal correlation pattern was most pronounced among pairs of neurons showing the strongest evidence for coding fluctuations or multiplexing. Whether a given pair of neurons exhibited positive or negative correlations depended on whether the two neurons both responded better to the same object or had different object preferences. Distinct distributions of spike count correlations based on stimulus preferences were also seen in V4 for separate objects but not when two stimuli fused to form one object. These findings suggest multiple objects evoke different response dynamics than those evoked by single stimuli, lending support to the multiplexing hypothesis and suggesting a means by which information about multiple objects can be preserved despite the apparent coarseness of sensory coding.Significance StatementHow the brain separates information about multiple objects despite overlap in the neurons responsive to each item is not well understood. Here we show that some neurons in V1 exhibit coding fluctuations in response to two objects, and that these coding fluctuations are coordinated at the population level in ways that are not observed for single objects. Broadly similar results were obtained in V4. These response dynamics lend support to the hypothesis that information about individual objects may be multiplexed across the neural population, preserving information about each item despite the coarseness of sensory coding.

Linear quantile regression is a powerful tool to investigate how predictors may affect a response heterogeneously across different quantile levels. Unfortunately, existing approaches find it extremely difficult to adjust for any dependency between observation units, largely because such methods are not based upon a fully generative model of the data. For analyzing spatially indexed data, we address this difficulty by generalizing the joint quantile regression model of Yang and Tokdar (2017) and characterizing spatial dependence via a Gaussian or $t$ copula process on the underlying quantile levels of the observation units. A Bayesian semiparametric approach is introduced to perform inference of model parameters and carry out spatial quantile smoothing. An effective model comparison criteria is provided, particularly for selecting between different model specifications of tail heaviness and tail dependence. Extensive simulation studies and an application to particulate matter concentration in northeast US are presented to illustrate substantial gains in inference quality, accuracy and uncertainty quantification over existing alternatives., Comment: 30 pages, 10 figures

A novel statistical method is proposed and investigated for estimating a heavy tailed density under mild smoothness assumptions. Statistical analyses of heavy-tailed distributions are susceptible to the problem of sparse information in the tail of the distribution getting washed away by unrelated features of a hefty bulk. The proposed Bayesian method avoids this problem by incorporating smoothness and tail regularization through a carefully specified semiparametric prior distribution, and is able to consistently estimate both the density function and its tail index at near minimax optimal rates of contraction. A joint, likelihood driven estimation of the bulk and the tail is shown to help improve uncertainty assessment in estimating the tail index parameter and offer more accurate and reliable estimates of the high tail quantiles compared to thresholding methods., Comment: Combined article with all technical details uploaded here to complement JASA publication

We propose a Bayesian test of normality for univariate or multivariate data against alternative nonparametric models characterized by Dirichlet process mixture distributions. The alternative models are based on the principles of embedding and predictive matching. They can be interpreted to offer random granulation of a normal distribution into a mixture of normals with mixture components occupying a smaller volume the farther they are from the distribution center. A scalar parametrization based on latent clustering is used to cover an entire spectrum of separation between the normal distributions and the alternative models. An efficient sequential importance sampler is developed to calculate Bayes factors. Simulations indicate the proposed test can detect non-normality without favoring the nonparametric alternative when normality holds., 24 pages, 5 figures, 1 table

Conventional analysis of neuroscience data involves computing average neural activity over a group of trials and/or a period of time. This approach may be particularly problematic when assessing the response patterns of neurons to more than one simultaneously presented stimulus. In such cases, the brain must represent each individual component of the stimuli bundle, but trial-and-time-pooled averaging methods are fundamentally unequipped to address the means by which multi-item representation occurs. We introduce and investigate a novel statistical analysis framework that relates the firing pattern of a single cell, exposed to a stimuli bundle, to the ensemble of its firing patterns under each constituent stimulus. Existing statistical tools focus on what may be called "first order stochasticity" in trial-to-trial variation in the form of unstructured noise around a fixed firing rate curve associated with a given stimulus. Our analysis is based upon the theoretical premise that exposure to a stimuli bundle induces additional stochasticity in the cell's response pattern, in the form of a stochastically varying recombination of its single stimulus firing rate curves. We discuss challenges to statistical estimation of such "second order stochasticity" and address them with a novel dynamic admixture Poisson process (DAPP) model. DAPP is a hierarchical point process model that decomposes second order stochasticity into a Gaussian stochastic process and a random vector of interpretable features, and, facilitates borrowing of information on the latter across repeated trials through latent clustering. We present empirical evidence of the utility of the DAPP analysis with synthetic and real neural recordings., Comment: 26 pages, 7 figures

Baseball players must be able to see and react in an instant, yet it is hotly debated whether superior performance is associated with superior sensorimotor abilities. In this study, we compare sensorimotor abilities, measured through 8 psychomotor tasks comprising the Nike Sensory Station assessment battery, and game statistics in a sample of 252 professional baseball players to evaluate the links between sensorimotor skills and on-field performance. For this purpose, we develop a series of Bayesian hierarchical latent variable models enabling us to compare statistics across professional baseball leagues. Within this framework, we find that sensorimotor abilities are significant predictors of on-base percentage, walk rate and strikeout rate, accounting for age, position, and league. We find no such relationship for either slugging percentage or fielder-independent pitching. The pattern of results suggests performance contributions from both visual-sensory and visual-motor abilities and indicates that sensorimotor screenings may be useful for player scouting.

In spite of the recent surge of interest in quantile regression, joint estimation of linear quantile planes remains a great challenge in statistics and econometrics. We propose a novel parameterization that characterizes any collection of noncrossing quantile planes over arbitrarily shaped convex predictor domains in any dimension by means of unconstrained scalar, vector and function valued parameters. Statistical models based on this parameterization inherit a fast computation of the likelihood function, enabling penalized likelihood or Bayesian approaches to model fitting. We introduce a complete Bayesian methodology by using Gaussian process prior distributions on the function valued parameters and develop a robust and efficient Markov chain Monte Carlo parameter estimation. The resulting method is shown to offer posterior consistency under mild tail and regularity conditions. We present several illustrative examples where the new method is compared against existing approaches and is found to offer better accuracy, coverage and model fit. Supplementary materials for this article are available online.

Quantile regression is widely seen as an ideal tool to understand complex predictor-response relations. Its biggest promise rests in its ability to quantify whether and how predictor effects vary across response quantile levels. But this promise has not been fully met due to a lack of statistical estimation methods that perform a rigorous, joint analysis of all quantile levels. This gap has been recently bridged by Yang and Tokdar [18] . Here we demonstrate how their joint quantile regression method, as encoded in the R package qrjoint , offers a comprehensive and model-based regression analysis framework. This chapter is an R vignette where we illustrate how to fit models, interpret coefficients, improve and compare models and obtain predictions under this framework. Our case study is an application to ecology where we analyse how the abundance of red maple trees depends on topographical and geographical features of the location. A complete absence of the species contributes excess zeros in the response data. We treat such excess zeros as left censoring in the spirit of a Tobit regression analysis. By utilising the generative nature of the joint quantile regression model, we not only adjust for censoring but also treat it as an object of independent scientific interest.

Bayesian nonparametric regression under a rescaled Gaussian process prior offers smoothness-adaptive function estimation with near minimax-optimal error rates. Hierarchical extensions of this approach, equipped with stochastic variable selection, are known to also adapt to the unknown intrinsic dimension of a sparse true regression function. But it remains unclear if such extensions offer variable selection consistency, that is, if the true subset of important variables could be consistently learned from the data. It is shown here that variable consistency may indeed be achieved with such models at least when the true regression function has finite smoothness to induce a polynomially larger penalty on inclusion of false positive predictors. Our result covers the high-dimensional asymptotic setting where the predictor dimension is allowed to grow with the sample size. The proof utilizes Schwartz theory to establish that the posterior probability of wrong selection vanishes asymptotically. A necessary and challenging technical development involves providing sharp upper and lower bounds to small ball probabilities at all rescaling levels of the Gaussian process prior, a result that could be of independent interest.

We recently reported the existence of fluctuations in neural signals that may permit neurons to code multiple simultaneous stimuli sequentially across time. This required deploying a novel statistical approach to permit investigation of neural activity at the scale of individual trials. Here we present tests using synthetic data to assess the sensitivity and specificity of this analysis. We fabricated datasets to match each of several potential response patterns derived from single-stimulus response distributions. In particular, we simulated dual stimulus trial spike counts that reflected fluctuating mixtures of the single stimulus spike counts, stable intermediate averages, single stimulus winner-take-all, or response distributions that were outside the range defined by the single stimulus responses (such as summation or suppression). We then assessed how well the analysis recovered the correct response pattern as a function of the number of simulated trials and the difference between the simulated responses to each "stimulus" alone. We found excellent recovery of the mixture, intermediate, and outside categories (>97% percent correct), and good recovery of the single/winner-take-all category (>90% correct) when the number of trials was >20 and the single-stimulus response rates were 50Hz and 20Hz respectively. Both larger numbers of trials and greater separation between the single stimulus firing rates improved categorization accuracy. These results provide a benchmark, and guidelines for data collection, for use of this method to investigate coding of multiple items at the individual-trial time scale., Accepted for publication in Neurons, Behavior, Data analysis, and Theory

Discovering temporal evolution of themes from a time-stamped collection of text poses a challenging statistical learning problem. Dynamic topic models offer a probabilistic modeling framework to decompose a corpus of text documents into “topics”, i.e., probability distributions over vocabulary terms, while simultaneously learning the temporal dynamics of the relative prevalence of these topics. We extend the dynamic topic model of Blei and Lafferty (2006) by fusing its multinomial factor model on topics with dynamic linear models that account for time trends and seasonality in topic prevalence. A Markov chain Monte Carlo (MCMC) algorithm that utilizes Polya-Gamma data augmentation is developed for posterior sampling. Conditional independencies in the model and sampling are made explicit, and our MCMC algorithm is parallelized where possible to allow for inference in large corpora. Our model and inference algorithm are validated with multiple synthetic examples, and we consider the applied problem of modeling trends in real estate listings from the housing website Zillow. We demonstrate in synthetic examples that sharing information across documents is critical for accurately estimating document-specific topic proportions. Analysis of the Zillow corpus demonstrates that the method is able to learn seasonal patterns and locally linear trends in topic prevalence.

Gaussian process (GP) models are widely used to emulate propagation uncertainty in computer experiments. GP emulation sits comfortably within an analytically tractable Bayesian framework. Apart from propagating uncertainty of the input variables, a GP emulator trained on finitely many runs of the experiment also offers error bars for response surface estimates at unseen input values. This helps select future input values where the experiment should be run to minimize the uncertainty in the response surface estimation. However, traditional GP emulators use stationary covariance functions, which perform poorly and lead to suboptimal selection of future input points when the response surface has sharp local features, such as a jump discontinuity or an isolated tall peak. We propose an easily implemented nonstationary GP emulator, based on two stationary GPs, one nested into the other, and demonstrate its superior ability in handling local features and selecting future input points from the boundaries of such...

How the brain preserves information about multiple simultaneous items is poorly understood. We report that single neurons can represent multiple different stimuli by interleaving different signals across time. We record single units in an auditory region, the inferior colliculus, while monkeys localize 1 or 2 simultaneous sounds. During dual-sound trials, we find that some neurons fluctuate between firing rates observed for each single sound, either on a whole-trial or on a sub-trial timescale. These fluctuations are correlated in pairs of neurons, can be predicted by the state of local field potentials prior to sound onset, and, in one monkey, can predict which sound will be reported first. We find corroborating evidence of fluctuating activity patterns in a separate data set involving responses of inferotemporal cortex neurons to multiple visual stimuli. Alternation between activity patterns corresponding to each of multiple items may therefore be a general strategy to enhance the brain processing capacity, potentially linking such disparate phenomena as variable neural firing, neural oscillations, and limits in attentional/memory capacity.

We show that rate-adaptive multivariate density estimation can be performed using Bayesian methods based on Dirichlet mixtures of normal kernels with a prior distribution on the kernel's covariance matrix parameter. We derive sufficient conditions on the prior specification that guarantee convergence to a true density at a rate that is optimal minimax for the smoothness class to which the true density belongs. No prior knowledge of smoothness is assumed. The sufficient conditions are shown to hold for the Dirichlet location mixture of normals prior with a Gaussian base measure and an inverse-Wishart prior on the covariance matrix parameter. Locally H\"older smoothness classes and their anisotropic extensions are considered. Our study involves several technical novelties, including sharp approximation of finitely differentiable multivariate densities by normal mixtures and a new sieve on the space of such densities., Comment: 29 pages

Gaussian processes are widely used in nonparametric regression, classification and spatiotemporal modelling, facilitated in part by a rich literature on their theoretical properties. However, one of their practical limitations is expensive computation, typically on the order of n-super-3 where n is the number of data points, in performing the necessary matrix inversions. For large datasets, storage and processing also lead to computational bottlenecks, and numerical stability of the estimates and predicted values degrades with increasing n. Various methods have been proposed to address these problems, including predictive processes in spatial data analysis and the subset-of-regressors technique in machine learning. The idea underlying these approaches is to use a subset of the data, but this raises questions concerning sensitivity to the choice of subset and limitations in estimating fine-scale structure in regions that are not well covered by the subset. Motivated by the literature on compressive sensing, we propose an alternative approach that involves linear projection of all the data points onto a lower-dimensional subspace. We demonstrate the superiority of this approach from a theoretical perspective and through simulated and real data examples. Copyright 2013, Oxford University Press.

In studies involving functional data, it is commonly of interest to model the impact of predictors on the distribution of the curves, allowing flexible effects on not only the mean curve but also the distribution about the mean. Characterizing the curve for each subject as a linear combination of a high-dimensional set of potential basis functions, we place a sparse latent factor regression model on the basis coefficients. We induce basis selection by choosing a shrinkage prior that allows many of the loadings to be close to zero. The number of latent factors is treated as unknown through a highly-efficient, adaptive-blocked Gibbs sampler. Predictors are included on the latent variables level, while allowing different predictors to impact different latent factors. This model induces a framework for functional response regression in which the distribution of the curves is allowed to change flexibly with predictors. The performance is assessed through simulation studies and the methods are applied to data on blood pressure trajectories during pregnancy.

Predictive recursion is an accurate and computationally efficient algorithm for nonparametric estimation of mixing densities in mixture models. In semiparametric mixture models, however, the algorithm fails to account for any uncertainty in the additional unknown structural parameter. As an alternative to existing profile likelihood methods, we treat predictive recursion as a filter approximation by fitting a fully Bayes model, whereby an approximate marginal likelihood of the structural parameter emerges and can be used for inference. We call this the predictive recursion marginal likelihood. Convergence properties of predictive recursion under model misspecification also lead to an attractive construction of this new procedure. We show pointwise convergence of a normalized version of this marginal likelihood function. Simulations compare the performance of this new approach with that of existing profile likelihood methods and with Dirichlet process mixtures in density estimation. Mixed-effects models and an empirical Bayes multiple testing application in time series analysis are also considered. Copyright 2011, Oxford University Press.

We provide a short overview of importance sampling—a popular sampling tool used for Monte Carlo computing. We discuss its mathematical foundation and properties that determine its accuracy in Monte Carlo approximations. We review the fundamental developments in designing efficient importance sampling (IS) for practical use. This includes parametric approximation with optimization-based adaptation, sequential sampling with dynamic adaptation through resampling and population-based approaches that make use of Markov chain sampling. Copyright © 2009 John Wiley & Sons, Inc. For further resources related to this article, please visit the WIREs website.

Neurons in vitro and in vivo have epochs of bursting or "up state" activity during which firing rates are dramatically elevated. Various methods of detecting bursts in extracellular spike trains have appeared in the literature, the most widely used apparently being Poisson Surprise (PS). A natural description of the phenomenon assumes (1) there are two hidden states, which we label "burst" and "non-burst," (2) the neuron evolves stochastically, switching at random between these two states, and (3) within each state the spike train follows a time-homogeneous point process. If in (2) the transitions from non-burst to burst and burst to non-burst states are memoryless, this becomes a hidden Markov model (HMM). For HMMs, the state transitions follow exponential distributions, and are highly irregular. Because observed bursting may in some cases be fairly regular--exhibiting inter-burst intervals with small variation--we relaxed this assumption. When more general probability distributions are used to describe the state transitions the two-state point process model becomes a hidden semi-Markov model (HSMM). We developed an efficient Bayesian computational scheme to fit HSMMs to spike train data. Numerical simulations indicate the method can perform well, sometimes yielding very different results than those based on PS.

A novel method is proposed to compute the Bayes estimate for a logistic Gaussian process prior for density estimation. The method gains speed by drawing samples from the posterior of a finite-dimensional surrogate prior, which is obtained by imputation of the underlying Gaussian process. We establish that imputation results in quite accurate computation. Simulation studies show that accuracy and high speed can be combined. This fact, along with known flexibility of the logistic Gaussian priors for modeling smoothness and recent results on their large support, makes these priors and the resulting density estimate very attractive.

We discuss the Empirical Bayes approach to the problem of multiple testing and compare it with a very popular frequentist method of Benjamini and Hochberg aimed at controlling the false discovery rate. Our main focus is the ‘sparse mixture’ case, when only a small proportion of tested hypotheses is expected to be false. The specific parametric model we consider is motivated by the application to detecting genes responsible for quantitative traits, but it can be used in a variety of other applications. We define some Parametric Empirical Bayes procedures for multiple testing and compare them with the Benjamini and Hochberg method using computer simulations. We explain some similarities between these two approaches by placing them within the same framework of threshold tests with estimated critical values. Copyright © 2007 John Wiley & Sons, Ltd.

We establish weak and strong posterior consistency of Gaussian process priors studied by Lenk [1988. The logistic normal distribution for Bayesian, nonparametric, predictive densities. J. Amer. Statist. Assoc. 83 (402), 509–516] for density estimation. Weak consistency is related to the support of a Gaussian process in the sup-norm topology which is explicitly identified for many covariance kernels. In fact we show that this support is the space of all continuous functions when the usual covariance kernels are chosen and an appropriate prior is used on the smoothing parameters of the covariance kernel. We then show that a large class of Gaussian process priors achieve weak as well as strong posterior consistency (under some regularity conditions) at true densities that are either continuous or piecewise continuous. © 2005 Elsevier B.V. All rights reserved.

A wide variety of priors have been proposed for nonparametric Bayesian estimation of conditional distributions, and there is a clear need for theorems providing conditions on the prior for large support, as well as posterior consistency. Estimation of an uncountable collection of conditional distributions across different regions of the predictor space is a challenging problem, which differs in some important ways from density and mean regression estimation problems. Defining various topologies on the space of conditional distributions, we provide sufficient conditions for posterior consistency focusing on a broad class of priors formulated as predictor-dependent mixtures of Gaussian kernels. This theory is illustrated by showing that the conditions are satisfied for a class of generalized stick-breaking process mixtures in which the stick-breaking lengths are monotone, differentiable functions of a continuous stochastic process. We also provide a set of sufficient conditions for the case where stick-breaking lengths are predictor independent, such as those arising from a fixed Dirichlet process prior.

Minimax $L_2$ risks for high-dimensional nonparametric regression are derived under two sparsity assumptions: (1) the true regression surface is a sparse function that depends only on $d=O(\log n)$ important predictors among a list of $p$ predictors, with $\log p=o(n)$; (2) the true regression surface depends on $O(n)$ predictors but is an additive function where each additive component is sparse but may contain two or more interacting predictors and may have a smoothness level different from other components. For either modeling assumption, a practicable extension of the widely used Bayesian Gaussian process regression method is shown to adaptively attain the optimal minimax rate (up to $\log n$ terms) asymptotically as both $n,p\to\infty$ with $\log p=o(n)$., Comment: Published at http://dx.doi.org/10.1214/14-AOS1289 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We introduce a semi-parametric Bayesian framework for a simultaneous analysis of linear quantile regression models. A simultaneous analysis is essential to attain the true potential of the quantile regression framework, but is computa- tionally challenging due to the associated monotonicity constraint on the quantile curves. For a univariate covariate, we present a simpler equivalent characterization of the monotonicity constraint through an interpolation of two monotone curves. The resulting formulation leads to a tractable likelihood function and is embedded within a Bayesian framework where the two monotone curves are modeled via lo- gistic transformations of a smooth Gaussian process. A multivariate extension is suggested by combining the full support univariate model with a linear projection of the predictors. The resulting single-index model remains easy to flt and provides substantial and measurable improvement over the flrst order linear heteroscedastic model. Two illustrative applications of the proposed method are provided.

Whether the number of tropical cyclones (TCs) has increased in the last 150 years has become a matter of intense debate. We investigate the efiects of beliefs about TC detection capacities in the North Atlantic on trends in TC num- bers since the 1870s. While raw data show an increasing trend of TC counts, the capability to detect TCs and to determine intensities and changes in intensity has also increased dramatically over the same period. We present a model of TC activ- ity that allows investigating the relationship between what one believes about the increase in detection and what one believes about TC trends. Previous work has used assumptions on TC tracks, detection capacities or the relationship between TC activity and various climate parameters to provide estimates of year-by-year missed TCs. These estimates and the associated conclusions about trends cover a wide range of possibilities. We build on previous work to investigate the sensitivity of these conclusions to the assumed priors about detection. Our analysis shows that any inference on TC count trends is strongly sensitive to one's speciflcation of prior beliefs about TC detection. Overall, we regard the evidence on the trend in North Atlantic TC numbers to be ambiguous.

We propose a flexible and identifiable version of the two-groups model, motivated by hierarchical Bayes considerations, that features an empirical null and a semiparametric mixture model for the non-null cases. We use a computationally efficient predictive recursion marginal likelihood procedure to estimate the model parameters, even the nonparametric mixing distribution. This leads to a nonparametric empirical Bayes testing procedure, which we call PRtest, based on thresholding the estimated local false discovery rates. Simulations and real-data examples demonstrate that, compared to existing approaches, PRtest's careful handling of the non-null density can give a much better fit in the tails of the mixture distribution which, in turn, can lead to more realistic conclusions., 18 pages, 4 figures, 3 tables

We develop a novel Bayesian density regression model based on logistic Gaussian processes and subspace projection. Logistic Gaussian processes provide an attractive alternative to the popular stick-breaking processes for modeling a family of conditional densities that vary smoothly in the conditioning variable. Subspace projection offers dimension reduction of predictors through multiple linear combinations, offering an alternative to the zeroing out theme of variable selection. We illustrate that logistic Gaussian processes and subspace projection combine well to produce a computationally tractable and theoretically sound density regression procedure that offers good out of sample prediction, accurate estimation of subspace projection and satisfactory estimation of subspace dimensionality. We also demonstrate that subspace projection may lead to better prediction than variable selection when predictors are well chosen and possibly dependent on each other, each having a moderate influence on the response.

Mixture models have received considerable attention recently and Newton [Sankhy\={a} Ser. A 64 (2002) 306--322] proposed a fast recursive algorithm for estimating a mixing distribution. We prove almost sure consistency of this recursive estimate in the weak topology under mild conditions on the family of densities being mixed. This recursive estimate depends on the data ordering and a permutation-invariant modification is proposed, which is an average of the original over permutations of the data sequence. A Rao--Blackwell argument is used to prove consistency in probability of this alternative estimate. Several simulations are presented, comparing the finite-sample performance of the recursive estimate and a Monte Carlo approximation to the permutation-invariant alternative along with that of the nonparametric maximum likelihood estimate and a nonparametric Bayes estimate., Comment: Published in at http://dx.doi.org/10.1214/08-AOS639 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Here we explore general asymptotic properties of Predictive Recursion (PR) for nonparametric estimation of mixing distributions. We prove that, when the mixture model is mis-specified, the estimated mixture convergesalmost surely in total variation to the mixture that minimizes the Kullback-Leiblerdivergence,and a bound on the (Hellingercontrast)rate of convergence is obtained. Simulations suggest that this rate is nearly sharp in a minimax sense. Moreover, when the model is identifiable, almost sure weak convergence of the mixing distribution estimate follows. PR assumes that the support of the mixing distribution is known. To remove this requirement, we propose a generalization that incorporates a sequence of supports, increasing with the sample size, that combines the efficiency of PR with the flexibility of mixture sieves. Under mild conditions, we obtain a bound on the rate of convergence of these new estimates. AMS 2000 subject classifications: Primary 62G20; secondary 62G05, 62G07, 62G35. Keywords and phrases: Almost supermartingale, density estimation, empirical Bayes, Kullback-Leibler projection, mixture models.

In the spirit of modeling inference for microarrays as multiple testing for sparse mixtures, we present a similar approach to a simplified version of quantitative trait loci (QTL) mapping. Unlike in case of microarrays, where the number of tests usually reaches tens of thousands, the number of tests performed in scans for QTL usually does not exceed several hundreds. However, in typical cases, the sparsity $p$ of significant alternatives for QTL mapping is in the same range as for microarrays. For methodological interest, as well as some related applications, we also consider non-sparse mixtures. Using simulations as well as theoretical observations we study false discovery rate (FDR), power and misclassification probability for the Benjamini-Hochberg (BH) procedure and its modifications, as well as for various parametric and nonparametric Bayes and Parametric Empirical Bayes procedures. Our results confirm the observation of Genovese and Wasserman (2002) that for small p the misclassification error of BH is close to optimal in the sense of attaining the Bayes oracle. This property is shared by some of the considered Bayes testing rules, which in general perform better than BH for large or moderate $p$'s., Published in at http://dx.doi.org/10.1214/193940307000000158 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)