arXiv subjects

Malay Ghosh

Publications and source records attributed to Malay Ghosh.

At least 37 records · Page 2Linked to original sources

arXiv2019

On the Beta Prime Prior for Scale Parameters in High-Dimensional Bayesian Regression Models

We study high-dimensional Bayesian linear regression with a general beta prime distribution for the scale parameter. Under the assumption of sparsity, we show that appropriate selection of the hyperparameters in the beta prime prior leads to the (near) minimax posterior contraction rate when $p \gg n$. For finite samples, we propose a data-adaptive method for estimating the hyperparameters based on marginal maximum likelihood (MML). This enables our prior to adapt to both sparse and dense settings, and under our proposed empirical Bayes procedure, the MML estimates are never at risk of collapsing to zero. We derive efficient Monte Carlo EM and variational EM algorithms for implementing our model, which are available in the R package NormalBetaPrime. Simulations and analysis of a gene expression data set illustrate our model's self-adaptivity to varying levels of sparsity and signal strengths.

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arXiv2018

Bayesian Simultaneous Estimation for Means in $k$ Sample Problems

This paper is concerned with the simultaneous estimation of $k$ population means when one suspects that the $k$ means are nearly equal. As an alternative to the preliminary test estimator based on the test statistics for testing hypothesis of equal means, we derive Bayesian and minimax estimators which shrink individual sample means toward a pooled mean estimator given under the hypothesis. It is shown that both the preliminary test estimator and the Bayesian minimax shrinkage estimators are further improved by shrinking the pooled mean estimator. The performance of the proposed shrinkage estimators is investigated by simulation.

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arXiv2018

The Inverse Gamma-Gamma Prior for Optimal Posterior Contraction and Multiple Hypothesis Testing

We study the well-known problem of estimating a sparse $n$-dimensional unknown mean vector $θ= (θ_1, ..., θ_n)$ with entries corrupted by Gaussian white noise. In the Bayesian framework, continuous shrinkage priors which can be expressed as scale-mixture normal densities are popular for obtaining sparse estimates of $θ$. In this article, we introduce a new fully Bayesian scale-mixture prior known as the inverse gamma-gamma (IGG) prior. We prove that the posterior distribution contracts around the true $θ$ at (near) minimax rate under very mild conditions. In the process, we prove that the sufficient conditions for minimax posterior contraction given by Van der Pas et al. (2016) are not necessary for optimal posterior contraction. We further show that the IGG posterior density concentrates at a rate faster than those of the horseshoe or the horseshoe+ in the Kullback-Leibler (K-L) sense. To classify true signals ($θ_i \neq 0$), we also propose a hypothesis test based on thresholding the posterior mean. Taking the loss function to be the expected number of misclassified tests, we show that our test procedure asymptotically attains the optimal Bayes risk exactly. We illustrate through simulations and data analysis that the IGG has excellent finite sample performance for both estimation and classification.

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arXiv2018

High-Dimensional Multivariate Posterior Consistency Under Global-Local Shrinkage Priors

We consider sparse Bayesian estimation in the classical multivariate linear regression model with $p$ regressors and $q$ response variables. In univariate Bayesian linear regression with a single response $y$, shrinkage priors which can be expressed as scale mixtures of normal densities are popular for obtaining sparse estimates of the coefficients. In this paper, we extend the use of these priors to the multivariate case to estimate a $p \times q$ coefficients matrix $\mathbf{B}$. We derive sufficient conditions for posterior consistency under the Bayesian multivariate linear regression framework and prove that our method achieves posterior consistency even when $p>n$ and even when $p$ grows at nearly exponential rate with the sample size. We derive an efficient Gibbs sampling algorithm and provide the implementation in a comprehensive R package called MBSP. Finally, we demonstrate through simulations and data analysis that our model has excellent finite sample performance.

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arXiv2018

Bayes Minimax Competitors of Preliminary Test Estimators in k Sample Problems

In this paper, we consider the estimation of a mean vector of a multivariate normal population where the mean vector is suspected to be nearly equal to mean vectors of $k-1$ other populations. As an alternative to the preliminary test estimator based on the test statistic for testing hypothesis of equal means, we derive empirical and hierarchical Bayes estimators which shrink the sample mean vector toward a pooled mean estimator given under the hypothesis. The minimaxity of those Bayesian estimators are shown, and their performances are investigated by simulation.

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arXiv2017

Posterior Graph Selection and Estimation Consistency for High-dimensional Bayesian DAG Models

Covariance estimation and selection for high-dimensional multivariate datasets is a fundamental problem in modern statistics. Gaussian directed acyclic graph (DAG) models are a popular class of models used for this purpose. Gaussian DAG models introduce sparsity in the Cholesky factor of the inverse covariance matrix, and the sparsity pattern in turn corresponds to specific conditional independence assumptions on the underlying variables. A variety of priors have been developed in recent years for Bayesian inference in DAG models, yet crucial convergence and sparsity selection properties for these models have not been thoroughly investigated. Most of these priors are adaptations or generalizations of the Wishart distribution in the DAG context. In this paper, we consider a flexible and general class of these 'DAG-Wishart' priors with multiple shape parameters. Under mild regularity assumptions, we establish strong graph selection consistency and establish posterior convergence rates for estimation when the number of variables p is allowed to grow at an appropriate sub-exponential rate with the sample size n.

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arXiv2016

Approximate Bayesian Computation via Sufficient Dimension Reduction

Approximate Bayesian computation (ABC) has gained popularity in recent years owing to its easy implementation, nice interpretation and good performance. Its advantages are more visible when one encounters complex models where maximum likelihood estimation as well as Bayesian analysis via Markov chain Monte Carlo demand prohibitively large amount of time. This paper examines properties of ABC both from a theoretical as well as from a computational point of view.We consolidate the ABC theory by proving theorems related to its limiting behaviour. In particular, we consider partial posteriors, which serve as the first step towards approximating the full posteriors. Also, a new semi-automatic algorithm of ABC is proposed using sufficient dimension reduction (SDR) method. SDR has primarily surfaced in the frequentist literature. But we have demonstrated in this paper that it has connections with ABC as well.

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arXiv2016

Bayesian Multiple Testing Under Sparsity for Polynomial-Tailed Distributions

This paper considers Bayesian multiple testing under sparsity for polynomial-tailed distributions satisfying a monotone likelihood ratio property. Included in this class of distributions are the Student's t, the Pareto, and many other distributions. We prove some general asymptotic optimality results under fixed and random thresholding. As examples of these general results, we establish the Bayesian asymptotic optimality of several multiple testing procedures in the literature for appropriately chosen false discovery rate levels. We also show by simulation that the Benjamini-Hochberg procedure with a false discovery rate level different from the asymptotically optimal one can lead to high Bayes risk.

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arXiv2016

Bayesian Variable Selection and Estimation Based on Global-Local Shrinkage Priors

In this paper, we consider Bayesian variable selection problem of linear regression model with global-local shrinkage priors on the regression coefficients. We propose a variable selection procedure that select a variable if the ratio of the posterior mean to the ordinary least square estimate of the corresponding coefficient is greater than $1/2$. Under the assumption of orthogonal designs, we show that if the local parameters have polynomial-tailed priors, our proposed method enjoys the oracle property in the sense that it can achieve variable selection consistency and optimal estimation rate at the same time. However, if, instead, an exponential-tailed prior is used for the local parameters, the proposed method does not have the oracle property.

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arXiv2015

Bayesian Variable Selection and Estimation for Group Lasso

The paper revisits the Bayesian group lasso and uses spike and slab priors for group variable selection. In the process, the connection of our model with penalized regression is demonstrated, and the role of posterior median for thresholding is pointed out. We show that the posterior median estimator has the oracle property for group variable selection and estimation under orthogonal designs, while the group lasso has suboptimal asymptotic estimation rate when variable selection consistency is achieved. Next we consider bi-level selection problem and propose the Bayesian sparse group selection again with spike and slab priors to select variables both at the group level and also within a group. We demonstrate via simulation that the posterior median estimator of our spike and slab models has excellent performance for both variable selection and estimation.

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arXiv2015

Necessary and Sufficient Conditions for High-Dimensional Posterior Consistency under $g$-Priors

We examine necessary and sufficient conditions for posterior consistency under $g$-priors, including extensions to hierarchical and empirical Bayesian models. The key features of this article are that we allow the number of regressors to grow at the same rate as the sample size and define posterior consistency under the sup vector norm instead of the more conventional Euclidean norm. We consider in particular the empirical Bayesian model of George and Foster (2000), the hyper-$g$-prior of Liang et al. (2008), and the prior considered by Zellner and Siow (1980).

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arXiv2015

Asymptotic Properties of Bayes Risk of a General Class of Shrinkage Priors in Multiple Hypothesis Testing Under Sparsity

Consider the problem of simultaneous testing for the means of independent normal observations. In this paper, we study some asymptotic optimality properties of certain multiple testing rules induced by a general class of one-group shrinkage priors in a Bayesian decision theoretic framework, where the overall loss is taken as the number of misclassified hypotheses. We assume a two-groups normal mixture model for the data and consider the asymptotic framework adopted in Bogdan et al. (2011) who introduced the notion of asymptotic Bayes optimality under sparsity in the context of multiple testing. The general class of one-group priors under study is rich enough to include, among others, the families of three parameter beta, generalized double Pareto priors, and in particular the horseshoe, the normal-exponential-gamma and the Strawderman-Berger priors. We establish that within our chosen asymptotic framework, the multiple testing rules under study asymptotically attain the risk of the Bayes Oracle up to a multiplicative factor, with the constant in the risk close to the constant in the Oracle risk. This is similar to a result obtained in Datta and Ghosh (2013) for the multiple testing rule based on the horseshoe estimator introduced in Carvalho et al. (2009, 2010). We further show that under very mild assumption on the underlying sparsity parameter, the induced decision rules based on an empirical Bayes estimate of the corresponding global shrinkage parameter proposed by van der Pas et al. (2014), attain the optimal Bayes risk up to the same multiplicative factor asymptotically. We provide a unifying argument applicable for the general class of priors under study. In the process, we settle a conjecture regarding optimality property of the generalized double Pareto priors made in Datta and Ghosh (2013). Our work also shows that the result in Datta and Ghosh (2013) can be improved further.

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arXiv2013

Two-stage Benchmarking as Applied to Small Area Estimation

There has been recent growth in small area estimation due to the need for more precise estimation of small geographic areas, which has led to groups such as the U.S. Census Bureau, Google, and the RAND corporation utilizing small area estimation procedures. We develop novel two-stage benchmarking methodology using a single weighted squared error loss function that combines the loss at the unit level and the area level without any specific distributional assumptions. We consider this loss while benchmarking the weighted means at each level or both the weighted means and weighted variability at the unit level. Multivariate extensions are immediate. We analyze the behavior of our methods using a complex study from the National Health Interview Survey (NHIS) from 2000, which estimates the proportion of people that do not have health insurance for many domains of an Asian subpopulation. Finally, the methodology is explored via simulated data under the proposed model. We ultimately conclude that three proposed benchmarked Bayes estimators do not dominate each other, leaving much exploration for future research.

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arXiv2013

Bayesian semiparametric analysis for two-phase studies of gene-environment interaction

The two-phase sampling design is a cost-efficient way of collecting expensive covariate information on a judiciously selected subsample. It is natural to apply such a strategy for collecting genetic data in a subsample enriched for exposure to environmental factors for gene-environment interaction (G x E) analysis. In this paper, we consider two-phase studies of G x E interaction where phase I data are available on exposure, covariates and disease status. Stratified sampling is done to prioritize individuals for genotyping at phase II conditional on disease and exposure. We consider a Bayesian analysis based on the joint retrospective likelihood of phases I and II data. We address several important statistical issues: (i) we consider a model with multiple genes, environmental factors and their pairwise interactions. We employ a Bayesian variable selection algorithm to reduce the dimensionality of this potentially high-dimensional model; (ii) we use the assumption of gene-gene and gene-environment independence to trade off between bias and efficiency for estimating the interaction parameters through use of hierarchical priors reflecting this assumption; (iii) we posit a flexible model for the joint distribution of the phase I categorical variables using the nonparametric Bayes construction of Dunson and Xing [J. Amer. Statist. Assoc. 104 (2009) 1042-1051].

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arXiv2013

On estimation of mean squared errors of benchmarked empirical Bayes estimators

We consider benchmarked empirical Bayes (EB) estimators under the basic area-level model of Fay and Herriot while requiring the standard benchmarking constraint. In this paper we determine the excess mean squared error (MSE) from constraining the estimates through benchmarking. We show that the increase due to benchmarking is O(m^{-1}), where m is the number of small areas. Furthermore, we find an asymptotically unbiased estimator of this MSE and compare it to the second-order approximation of the MSE of the EB estimator or, equivalently, of the MSE of the empirical best linear unbiased predictor (EBLUP), that was derived by Prasad and Rao (1990). Morever, using methods similar to those of Butar and Lahiri (2003), we compute a parametric bootstrap estimator of the MSE of the benchmarked EB estimator under the Fay-Herriot model and compare it to the MSE of the benchmarked EB estimator found by a second-order approximation. Finally, we illustrate our methods using SAIPE data from the U.S. Census Bureau, and in a simulation study.

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arXiv2011

Rejoinder

Rejoinder of ``Objective Priors: An Introduction for Frequentists'' by M. Ghosh [arXiv:1108.2120]

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arXiv2011

Objective Priors: An Introduction for Frequentists

Bayesian methods are increasingly applied in these days in the theory and practice of statistics. Any Bayesian inference depends on a likelihood and a prior. Ideally one would like to elicit a prior from related sources of information or past data. However, in its absence, Bayesian methods need to rely on some "objective" or "default" priors, and the resulting posterior inference can still be quite valuable. Not surprisingly, over the years, the catalog of objective priors also has become prohibitively large, and one has to set some specific criteria for the selection of such priors. Our aim is to review some of these criteria, compare their performance, and illustrate them with some simple examples. While for very large sample sizes, it does not possibly matter what objective prior one uses, the selection of such a prior does influence inference for small or moderate samples. For regular models where asymptotic normality holds, Jeffreys' general rule prior, the positive square root of the determinant of the Fisher information matrix, enjoys many optimality properties in the absence of nuisance parameters. In the presence of nuisance parameters, however, there are many other priors which emerge as optimal depending on the criterion selected. One new feature in this article is that a prior different from Jeffreys' is shown to be optimal under the chi-square divergence criterion even in the absence of nuisance parameters. The latter is also invariant under one-to-one reparameterization.

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arXiv2009

A Conversation with Pranab Kumar Sen

Pranab Kumar Sen was born on November 7, 1937 in Calcutta, India. His father died when Pranab was 10 years old, so his mother raised the family of seven children. Given his superior performance on an exam, Pranab nearly went into medical school, but did not because he was underage. He received a B.Sc. degree in 1955 and an M.Sc. degree in 1957 in statistics from Calcutta University, topping the class both times. Dr. Sen's dissertation on order statistics and nonparametrics, under the direction of Professor Hari Kinkar Nandi, was completed in 1961. After teaching for three years at Calcutta University, 1961--1964, Professor Sen came to Berkeley as a Visiting Assistant Professor in 1964. In 1965, he joined the Departments of Statistics and Biostatistics at the University of North Carolina at Chapel Hill, where he has remained. Professor Sen's pioneering contributions have touched nearly every area of statistics. He is the first person who, in joint collaboration with Professor S. K. Chatterjee, developed multivariate rank tests as well as time-sequential nonparametric methods. He is also the first person who carried out in-depth research in sequential nonparametrics culminating in his now famous Wiley book Sequential Nonparametrics: Invariance Principles and Statistical Inference and SIAM monograph. Professor Sen has over 600 research publications. In addition, he has authored or co-authored 11 books and monographs, and has edited or co-edited 11 more volumes. He has supervised over 80 Ph.D. students, many of whom have achieved distinction both nationally and internationally. Professor Sen is the founding co-editor of two international journals: Sequential Analysis and Statistics and Decisions.

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