arXiv subjects

Sayantan Paul

Publications and source records attributed to Sayantan Paul.

4 recordsLinked to original sources

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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arXiv2024

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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