arXiv subjects

Arijit Chakrabarti

Publications and source records attributed to Arijit Chakrabarti.

16 recordsLinked to original sources

arXiv2026

Asymptotic Bayes Optimality Under Sparsity of the Gavrilov-Benjamini-Sarkar Step-Down Testing Procedure

In this article, we investigate the asymptotic Bayes optimality under sparsity (ABOS) of the Gavrilov-Benjamini-Sarkar (GBS) step-down multiple testing procedure of Gavrilov et al. (2009) in the sparse Gaussian sequence model. While the asymptotic optimality properties of the Benjamini-Hochberg procedure have been extensively studied, corresponding results for the GBS procedure remain unavailable despite its favorable finite-sample performance and widespread applicability. Within the spike-and-slab Bayesian formulation and the asymptotic decision-theoretic framework of Bogdan et al. (2011), we establish that the GBS procedure is ABOS over a broad class of sparse asymptotic regimes. Existing ABOS analyses of the Benjamini-Hochberg procedure rely on approximating the random rejection threshold by a suitable deterministic surrogate. In contrast, our approach analyzes the Bayes risk directly, separately controlling the false discovery and false nondiscovery components. The analysis combines two key ingredients: a new finite-sample inequality for shifted order statistics associated with the GBS critical constants and a signal-crossing argument for ordered alternative p-values that exploits the procedure's sequential step-down structure. Together, these tools yield the desired ABOS property without resorting to threshold-localization arguments. To the best of our knowledge, this is the first asymptotic decision-theoretic analysis of the GBS procedure and the first proof of its asymptotic Bayes optimality under sparsity.

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arXiv2026

Bayesian Model Pursuit and Near-Oracle Sparse Signal Discovery Under Dependence

Sparse signal discovery is a fundamental problem in large-scale inference, where the goal is to identify a small number of active signals hidden among a large collection of null effects. Despite the prevalence of dependence in modern applications, relatively little is known about how much dependence can be exploited for efficient sparse signal recovery from a Bayes-risk perspective. In this paper, we develop a Bayesian Step-Down (BSD) procedure for sparse signal discovery under arbitrary known covariance dependence. BSD adopts a posterior-guided model-pursuit strategy that sequentially accumulates evidence for competing sparse signal configurations while explicitly incorporating the data's covariance structure. To assess its effectiveness, we introduce a Bayes Oracle for a class of sparse one-factor dependence models and compare BSD with the Oracle, the recently proposed MRD-GBS procedure of Ghosh and Chakrabarti (2026), the original MRD procedure of Cohen et al. (2009), and the Benjamini-Hochberg method. Our simulation studies reveal a striking phenomenon: across a broad range of dimensions, sparsity levels, and dependence structures, BSD exhibits near-oracle behavior and is often virtually indistinguishable from the Bayes Oracle in terms of Bayes risk and support recovery performance. Remarkably, a similarly close agreement is observed between BSD and MRD-GBS despite their fundamentally different Bayesian and frequentist foundations. These findings provide new insight into the attainable Bayes-risk frontier for sparse signal discovery under dependence and suggest that BSD may serve as a useful benchmark when exact Oracle calculations are unavailable. Finally, we show that BSD admits a residual representation, thereby yielding admissibility under arbitrary covariance dependence and substantial computational simplifications.

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arXiv2026

Covariance-Adaptive Residualization and Stagewise Calibration for Dependent Multiple Testing

In this paper, we study simultaneous hypothesis testing for multivariate Gaussian means under arbitrary covariance dependence. Building upon the Maximum Residual Down (MRD) procedure of Cohen et al. (2009), we investigate a systematic stagewise calibration strategy based on the generalized step-down critical constants of Gavrilov et al. (2009). The proposed methodology retains the covariance-adaptive residualization mechanism of MRD while replacing the original model-dependent threshold specification with a simple and principled calibration rule. Since the resulting procedure belongs to the class of monotone residual-based step-down procedures studied by Ghosh and Chakrabarti (2026), its admissibility follows directly from their general theory. We also derive alternative representations of the MRD residual statistics that express all active residuals through a single active precision matrix, substantially reducing computational complexity while revealing a direct connection between covariance-adaptive residualization and active precision-matrix geometry. Extensive simulation studies under a broad range of dependence structures demonstrate that the proposed methodology frequently achieves substantially lower normalized misclassification risk than several widely used marginal testing procedures. Under several structured dependence models, it also exhibits remarkably strong signal-recovery behavior, simultaneously attaining false discovery rates close to the nominal level, extremely small false non-discovery rates, powers approaching one, and average numbers of rejections close to the expected number of true signals. These findings suggest that covariance-adaptive residualization and stagewise calibration play complementary roles in exploiting dependence information for large-scale multiple testing under arbitrary covariance structures.

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arXiv2026

Admissibility of Adaptive Monotone Step-Down Multiple Testing Procedures Under Arbitrary Covariance Dependence

In this paper, we consider the problem of simultaneous testing of multivariate normal means under arbitrary covariance dependence. Specifically, let $\boldsymbol{X}\sim N_n(\boldsymbol{\theta},\boldsymbol{\Sigma})$, where $\boldsymbol{\theta}\in\mathbb{R}^n$ is unknown and $\boldsymbol{\Sigma}$ is a known positive definite covariance matrix. The objective is to test $H_{0i}:\theta_i=0$ against $H_{Ai}:\theta_i\neq 0$, simultaneously for $i=1,\ldots,n$. We establish a general admissibility theorem for a broad class of monotone residual-based step-down multiple testing procedures which iteratively rank the active hypotheses using statistics obtained through locally adaptive strictly increasing transformations of suitably standardized residual statistics arising from conditional normal distributions. Our main result shows that every such procedure is admissible with respect to a vector-valued loss function whose components are the usual individual $0$--$1$ testing losses. The proof relies on a delicate geometric analysis of the induced acceptance regions together with structural invariance properties of the adaptive stagewise rejection indices. The theorem substantially extends the admissibility theory developed for the maximum residual down procedure of Cohen et al. (2009) and reveals that admissibility under dependence is fundamentally driven by the monotone ordering structure induced by the residual statistics rather than by the precise functional form of the testing rule itself.

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arXiv2025

Asymptotic Bayes Optimality for Sparse Count Data

Consider a situation of analyzing high-dimensional count data containing an excess of near-zero counts with a small number of moderate or large counts. Assuming that the observations are modeled by a Poisson distribution, we are interested in simultaneous testing of whether the mean of the $i^{\text{th}}$ observation is small or large. In this work, we study some optimal properties (in terms of Bayes risk) of multiple-testing rules when the mean parameter is modeled by both two-group and a general class of one-group shrinkage priors, proposed by Polson and Scott (2010). Here, first, we model each mean by a two-group prior, and under additive $0-1$ loss function, obtain an expression for the optimal Bayes risk under some assumption similar in the spirit of Bogdan et al. (2011). Next, assuming that the observations are truly generated from a two-group mixture model and modelling each mean parameter by the broad class of one-group priors, we study the Bayes risk induced by our chosen class of priors. We have been able to show that, when the underlying level of sparsity is known, under some proposed assumptions, the Bayes risk corresponding to our broad class of priors attains the optimal Bayes risk, upto a multiplicative constant. When this sparsity pattern is unknown, motivated by Yano et al. (2021), we use an empirical Bayes estimate of the global shrinkage parameter. In this case, also, we show that the modified decision rule attains the optimal Bayes risk, upto a multiplicative constant. In this way, as an alternative solution for two-group prior, we propose a broad class of global-local priors having similar optimal properties in terms of Bayes risk for quasi-sparse count data. Finally, the theoretical results are verified using simulation studies followed by a real data analysis.

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arXiv2025

Consistent Group selection using Global-local prior in High dimensional setup

We consider the problem of model selection when grouping structure is inherent within the regressors. Using a Bayesian approach, we model the mean vector by a one-group global-local shrinkage prior belonging to a broad class of such priors that includes the horseshoe prior. In the context of variable selection, this class of priors was studied by Tang et al. (2018). A modified form of the usual class of global-local shrinkage priors with polynomial tail on the group regression coefficients is proposed. The resulting threshold rule selects the active group if within a group, the ratio of the $L_2$ norm of the posterior mean of its group coefficient to that of the corresponding ordinary least square group estimate is greater than a half. In the theoretical part of this article, we have used the global shrinkage parameter either as a tuning one or an empirical Bayes estimate of it depending on the knowledge regarding the underlying sparsity of the model. When the proportion of active groups is known, using $τ$ as a tuning parameter, we have proved that our method is oracle. In case this proportion is unknown, we propose an empirical Bayes estimate of $τ$. Even if this empirical Bayes estimate is used, then also our half-thresholding rule captures the truly important groups and obtains optimal estimation rate of the group coefficients simultaneously. Though our theoretical works rely on a special form of the design matrix, for general design matrices also, our simulation results show that the half-thresholding rule yields results similar to that of Yang and Narisetty (2020). As a consequence of this, in a high dimensional sparse group selection problem, instead of using the so-called `gold standard' spike and slab prior, one can use the one-group global-local shrinkage priors with polynomial tail to obtain similar results.

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arXiv2025

Sharp Asymptotic Minimaxity for Multiple Testing Using One-Group Shrinkage Priors

This paper investigates asymptotic minimaxity properties of Bayesian multiple testing rules in the sparse Gaussian sequence model using a broad class of global-local scale mixtures of normals as priors for the means. Minimaxity is studied under standard misclassification loss and the composite loss given by the sum of the false discovery proportion (FDP) and false non-discovery proportion (FNP). When the sparsity level is known, we show that by suitably choosing the global shrinkage parameter based on the sparsity level, our proposed testing rule achieves the exact minimax risk asymptotically for both losses under the ''beta-min'' separation condition. When the sparsity level is unknown, both empirical Bayes and fully Bayesian adaptations of the same rule are shown to achieve exact minimax risk asymptotically under suitable assumptions on sparsity. Our results reveal that minimaxity is attained for ''horseshoe-type'' priors that are broad enough to include the horseshoe, Strawderman-Berger, standard double Pareto, and certain inverse-gamma priors, among others. For non-''horseshoe-type'' priors, minimaxity fails to hold for either loss function. To the best of our knowledge, these are the first results of their kind for multiple hypothesis testing based on global-local shrinkage priors.

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arXiv2024

Posterior Contraction rate and Asymptotic Bayes Optimality for one-group shrinkage priors in sparse normal means problem

We consider a high-dimensional sparse normal means model where the goal is to estimate the mean vector assuming the proportion of non-zero means is unknown. We model the mean vector by a one-group global-local shrinkage prior belonging to a broad class of such priors that includes the horseshoe prior. We address some questions related to asymptotic properties of the resulting posterior distribution of the mean vector for the said class priors. We consider two ways to model the global parameter in this paper. Firstly by considering this as an unknown fixed parameter and then by an empirical Bayes estimate of it. In the second approach, we do a hierarchical Bayes treatment by assigning a suitable non-degenerate prior distribution to it. We first show that for the class of priors under study, the posterior distribution of the mean vector contracts around the true parameter at a near minimax rate when the empirical Bayes approach is used. Next, we prove that in the hierarchical Bayes approach, the corresponding Bayes estimate attains the minimax risk asymptotically under the squared error loss function. We also show that the posterior contracts around the true parameter at a near minimax rate. These results generalize those of van der Pas et al. (2014) \cite{van2014horseshoe}, (2017) \cite{van2017adaptive}, proved for the horseshoe prior. We have also studied in this work the asymptotic Bayes optimality of global-local shrinkage priors where the number of non-null hypotheses is unknown. Here our target is to propose some conditions on the prior density of the global parameter such that the Bayes risk induced by the decision rule attains Optimal Bayes risk, up to some multiplicative constant. Using our proposed condition, under the asymptotic framework of Bogdan et al. (2011) \cite{bogdan2011asymptotic}, we are able to provide an affirmative answer to satisfy our hunch.

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arXiv2020

High dimensional PCA: a new model selection criterion

Given a random sample from a multivariate population, estimating the number of large eigenvalues of the population covariance matrix is an important problem in Statistics with wide applications in many areas. In the context of Principal Component Analysis (PCA), the linear combinations of the original variables having the largest amounts of variation are determined by this number. In this paper, we study the high dimensional asymptotic regime where the number of variables grows at the same rate as the number of observations, and use the spiked covariance model proposed in Johnstone (2001), under which the problem reduces to model selection. Our focus is on the Akaike Information Criterion (AIC) which is known to be strongly consistent from the work of Bai et al. (2018). However, Bai et al. (2018) requires a certain "gap condition" ensuring the dominant eigenvalues to be above a threshold strictly larger than the BBP threshold (Baik et al. (2005), both quantities depending on the limiting ratio of the number of variables and observations. It is well-known that, below the BBP threshold, a spiked covariance structure becomes indistinguishable from one with no spikes. Thus the strong consistency of AIC requires some extra signal strength. In this paper, we investigate whether consistency continues to hold even if the "gap" is made smaller. We show that strong consistency under arbitrarily small gap is achievable if we alter the penalty term of AIC suitably depending on the target gap. Furthermore, another intuitive alteration of the penalty can indeed make the gap exactly zero, although we can only achieve weak consistency in this case. We compare the two newly-proposed estimators with other existing estimators in the literature via extensive simulation studies, and show, by suitably calibrating our proposals, that a significant improvement in terms of mean-squared error is achievable.

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arXiv2018

A New Step-down Procedure for Simultaneous Hypothesis Testing Under Dependence

In this article, we consider the problem of simultaneous testing of hypotheses when the individual test statistics are not necessarily independent. Specifically, we consider the problem of simultaneous testing of point null hypotheses against two-sided alternatives for the mean parameters of normally distributed random variables. We assume that conditionally given the vector of means, these random variables jointly follow a multivariate normal distribution with a known but arbitrary covariance matrix. We consider a Bayesian framework where each unknown mean parameter is modeled through a two-component "spike and slab" mixture prior. This way, unconditionally the test statistics jointly have a mixture of multivariate normal distributions. A new testing procedure is developed that uses the dependence among the test statistics and works in a "step-down" manner. This procedure is general enough to be applied for non-normal data. A decision theoretic justification in favor of the proposed testing procedure has been provided by showing that unlike many traditional p-value based stepwise procedures, this new method possesses a certain "convexity property" which makes it admissible with respect to a vector risk function that captures the risks for the individual testing problems. An alternative representation of the proposed test statistics has also been established resulting in great simplification in the computational complexity. It is demonstrated through extensive simulations that for various forms of dependence and a wide range of sparsity levels, the proposed testing procedure compares quite favorably with several existing multiple testing procedures available in the literature in terms of overall misclassification probability.

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arXiv2015

Asymptotic Minimaxity, Optimal Posterior Concentration and Asymptotic Bayes Optimality of Horseshoe-type Priors Under Sparsity

In this article, we investigate certain asymptotic optimality properties of a very broad class of one-group continuous shrinkage priors for simultaneous estimation and testing of a sparse normal mean vector. Asymptotic optimality of Bayes estimates and posterior concentration properties corresponding to the general class of one-group priors under consideration are studied where the data is assumed to be generated according to a multivariate normal distribution with a fixed unknown mean vector. Under the assumption that the number of non-zero means is known, we show that Bayes estimators arising out of this general class of shrinkage priors under study, attain the minimax risk, up to some multiplicative constant, under the $l_2$ norm. In particular, it is shown that for the horseshoe-type priors such as the three parameter beta normal mixtures with parameters $a=0.5, b>0$ and the generalized double Pareto prior with shape parameter $α=1$, the corresponding Bayes estimates become asymptotically minimax. Moreover, posterior distributions arising out of this general class of one-group priors are shown to contract around the true mean vector at the minimax $l_2$ rate for a wide range of values of the global shrinkage parameter depending on the proportion of non-zero components of the underlying mean vector. An important and remarkable fact that emerges as a consequence of one key result essential for proving the aforesaid minimaxity result is that, within the asymptotic framework of Bogdan et al. (2011), the natural thresholding rules due to Carvalho et al. (2010) based on the horseshoe-type priors, asymptotically attain the optimal Bayes risk w.r.t. a $0-1$ loss, up to the correct multiplicative constant and are thus, asymptotically Bayes optimal under sparsity (ABOS).

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arXiv2015

Posterior Concentration Properties of a General Class of Shrinkage Priors around Nearly Black Vectors

Suppose we have data generated according to a multivariate normal distribution with a fixed unknown mean vector that is sparse in the sense of being nearly black. Optimality of Bayes estimates and posterior concentration properties in terms of the minimax risk in the $l_2$ norm corresponding to a very general class of continuous shrinkage priors are studied in this work. The class of priors considered is rich enough to include a great variety of heavy tailed prior distributions, such as, the three parameter beta normal mixtures (including the horseshoe), the generalized double Pareto, the inverse gamma and the normal-exponential-gamma priors. Assuming that the number of non-zero components of the mean vector is known, we show that the Bayes estimators corresponding to this general class of priors attain the minimax risk in the $l_2$ norm (possibly up to a multiplicative constant) and the corresponding posterior distributions contract around the true mean vector at the minimax optimal rate for appropriate choice of the global shrinkage parameter. Moreover, we provide conditions for which these posterior distributions contract around the corresponding Bayes estimates at least as fast as the minimax risk in the $l_2$ norm. We also provide a lower bound to the total posterior variance for an important subclass of this general class of shrinkage priors that includes the generalized double Pareto priors with shape parameter $α=0.5$ and the three parameter beta normal mixtures with parameters $a=0.5$ and $b>0$ (including the horseshoe) in particular. The present work is inspired by the recent work of van der Pas et al. (2014) on the posterior contraction properties of the horseshoe prior under the present set-up. We extend their results for this general class of priors and come up with novel unifying proofs which work for a very broad class of one-group continuous shrinkage priors.

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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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arXiv2012

Asymptotic Bayes-optimality under sparsity of some multiple testing procedures

Within a Bayesian decision theoretic framework we investigate some asymptotic optimality properties of a large class of multiple testing rules. A parametric setup is considered, in which observations come from a normal scale mixture model and the total loss is assumed to be the sum of losses for individual tests. Our model can be used for testing point null hypotheses, as well as to distinguish large signals from a multitude of very small effects. A rule is defined to be asymptotically Bayes optimal under sparsity (ABOS), if within our chosen asymptotic framework the ratio of its Bayes risk and that of the Bayes oracle (a rule which minimizes the Bayes risk) converges to one. Our main interest is in the asymptotic scheme where the proportion p of "true" alternatives converges to zero. We fully characterize the class of fixed threshold multiple testing rules which are ABOS, and hence derive conditions for the asymptotic optimality of rules controlling the Bayesian False Discovery Rate (BFDR). We finally provide conditions under which the popular Benjamini-Hochberg (BH) and Bonferroni procedures are ABOS and show that for a wide class of sparsity levels, the threshold of the former can be approximated by a nonrandom threshold.

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arXiv2011

Asymptotic Bayes optimality under sparsity for generally distributed effect sizes under the alternative

Recent results concerning asymptotic Bayes-optimality under sparsity (ABOS) of multiple testing procedures are extended to fairly generally distributed effect sizes under the alternative. An asymptotic framework is considered where both the number of tests m and the sample size m go to infinity, while the fraction p of true alternatives converges to zero. It is shown that under mild restrictions on the loss function nontrivial asymptotic inference is possible only if n increases to infinity at least at the rate of log m. Based on this assumption precise conditions are given under which the Bonferroni correction with nominal Family Wise Error Rate (FWER) level alpha and the Benjamini- Hochberg procedure (BH) at FDR level alpha are asymptotically optimal. When n is proportional to log m then alpha can remain fixed, whereas when n increases to infinity at a quicker rate, then alpha has to converge to zero roughly like n^(-1/2). Under these conditions the Bonferroni correction is ABOS in case of extreme sparsity, while BH adapts well to the unknown level of sparsity. In the second part of this article these optimality results are carried over to model selection in the context of multiple regression with orthogonal regressors. Several modifications of Bayesian Information Criterion are considered, controlling either FWER or FDR, and conditions are provided under which these selection criteria are ABOS. Finally the performance of these criteria is examined in a brief simulation study.

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arXiv2008

Asymptotic optimality of a cross-validatory predictive approach to linear model selection

In this article we study the asymptotic predictive optimality of a model selection criterion based on the cross-validatory predictive density, already available in the literature. For a dependent variable and associated explanatory variables, we consider a class of linear models as approximations to the true regression function. One selects a model among these using the criterion under study and predicts a future replicate of the dependent variable by an optimal predictor under the chosen model. We show that for squared error prediction loss, this scheme of prediction performs asymptotically as well as an oracle, where the oracle here refers to a model selection rule which minimizes this loss if the true regression were known.

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