arXiv subjects

Subhashis Ghosal

Publications and source records attributed to Subhashis Ghosal.

At least 19 recordsLinked to original sources

arXiv2025

Bayesian Graphical High-Dimensional Time Series Models for Detecting Structural Changes

We study the structural changes in multivariate time-series by estimating and comparing stationary graphs for macroeconomic time series before and after an economic crisis such as the Great Recession. Building on a latent time series framework called Orthogonally-rotated Univariate Time-series (OUT), we propose a shared-parameter framework-the spOUT autoregressive model (spOUTAR)-that jointly models two related multivariate time series and enables coherent Bayesian estimation of their corresponding stationary precision matrices. This framework provides a principled mechanism to detect and quantify which conditional relationships among the variables changed, or formed following the crisis. Specifically, we study the impact of the Great Recession (December 2007-June 2009) that substantially disrupted global and national economies, prompting long-lasting shifts in macroeconomic indicators and their interrelationships. While many studies document its economic consequences, far less is known about how the underlying conditional dependency structure among economic variables changed as economies moved from pre-crisis stability through the shock and back to normalcy. Using the proposed approach to analyze U.S. and OECD macroeconomic data, we demonstrate that spOUTAR effectively captures recession-induced changes in stationary graphical structure, offering a flexible and interpretable tool for studying structural shifts in economic systems.

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arXiv2025

Bayesian Inference for High-dimensional Time Series with a Stationary Directed Acyclic Graphical Structure

In multivariate time series analysis, understanding the underlying causal relationships among variables is often of interest for various applications. Directed acyclic graphs (DAGs) provide a powerful framework for representing causal dependencies. This paper proposes a novel Bayesian approach for modeling multivariate time series where conditional independencies and causal structure are encoded by a DAG. The proposed model allows structural properties such as stationarity to be easily accommodated. Given the application, we further extend the model for matrix-variate time series. We take a Bayesian approach to inference, and a ``immersion-posterior'' based efficient computational algorithm is developed. The posterior convergence properties of the proposed method are established along with two identifiability results for the unrestricted structural equation models. The utility of the proposed method is demonstrated through simulation studies and real data analysis.

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arXiv2024

Bayesian High-dimensional Grouped-regression using Sparse Projection-posterior

We present a novel Bayesian approach for high-dimensional grouped regression under sparsity. We leverage a sparse projection method that uses a sparsity-inducing map to derive an induced posterior on a lower-dimensional parameter space. Our method introduces three distinct projection maps based on popular penalty functions: the Group LASSO Projection Posterior, Group SCAD Projection Posterior, and Adaptive Group LASSO Projection Posterior. Each projection map is constructed to immerse dense posterior samples into a structured, sparse space, allowing for effective group selection and estimation in high-dimensional settings. We derive optimal posterior contraction rates for estimation and prediction, proving that the methods are model selection consistent. Additionally, we propose a Debiased Group LASSO Projection Map, which ensures exact coverage of credible sets. Our methodology is particularly suited for applications in nonparametric additive models, where we apply it with B-spline expansions to capture complex relationships between covariates and response. Extensive simulations validate our theoretical findings, demonstrating the robustness of our approach across different settings. Finally, we illustrate the practical utility of our method with an application to brain MRI volume data from the Alzheimer's Disease Neuroimaging Initiative (ADNI), where our model identifies key brain regions associated with Alzheimer's progression.

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arXiv2024

Relational Graph in Vector Autoregression: A Case Study on the Effect of the Great Recession on Connectivity of Economic Indicators

Under a high-dimensional vector autoregressive (VAR) model, we propose a way of efficiently estimating both the stationary graph structure between the nodal time series and their temporal dynamics. The framework is then used to make inferences on the change in interdependencies between several economic indicators due to the impact of the Great Recession, the financial crisis that lasted from 2007 through 2009. There are several key advantages of the proposed framework; (1) it develops a reparametrized VAR likelihood that can be used in general high-dimensional VAR problems, (2) it strictly maintains causality of the estimated process, making inference on stationary features more meaningful and (3) it is computationally efficient due to the reduced rank structure of the parameterization. We apply the methodology to the seasonally adjusted quarterly economic indicators available in the FRED-QD database of the Federal Reserve. The analysis essentially confirms much of the prevailing knowledge about the impact of the Great Recession on different economic indicators. At the same time, it provides deeper insight into the nature and extent of the impact on the interplay of the different indicators. We also contribute to the theory of Bayesian VAR by showing the consistency of the posterior under sparse priors for the parameters of the reduced rank formulation of the VAR process.

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arXiv2024

Coverage of Credible Sets for Regression under Variable Selection

We study the asymptotic frequentist coverage of credible sets based on a novel Bayesian approach for a multiple linear regression model under variable selection. We initially ignore the issue of variable selection, which allows us to put a conjugate normal prior on the coefficient vector. The variable selection step is incorporated directly in the posterior through a sparsity-inducing map and uses the induced prior for making an inference instead of the natural conjugate posterior. The sparsity-inducing map minimizes the sum of the squared l2-distance weighted by the data matrix and a suitably scaled l1-penalty term. We obtain the limiting coverage of various credible regions and demonstrate that a modified credible interval for a component has the exact asymptotic frequentist coverage if the corresponding predictor is asymptotically uncorrelated with other predictors. Through extensive simulation, we provide a guideline for choosing the penalty parameter as a function of the credibility level appropriate for the corresponding coverage. We also show finite-sample numerical results that support the conclusions from the asymptotic theory. We also provide the credInt package that implements the method in R to obtain the credible intervals along with the posterior samples.

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arXiv2024

Bayesian Learning of Relational Graph in Semiparametric High-dimensional Time Series

Time series data arising in many applications nowadays are high-dimensional. A large number of parameters describe features of these time series. We propose a novel approach to modeling a high-dimensional time series through several independent univariate time series, which are then orthogonally rotated and sparsely linearly transformed. With this approach, any specified intrinsic relations among component time series given by a graphical structure can be maintained at all time snapshots. We call the resulting process an Orthogonally-rotated Univariate Time series (OUT). Key structural properties of time series such as stationarity and causality can be easily accommodated in the OUT model. For Bayesian inference, we put suitable prior distributions on the spectral densities of the independent latent times series, the orthogonal rotation matrix, and the common precision matrix of the component times series at every time point. A likelihood is constructed using the Whittle approximation for univariate latent time series. An efficient Markov Chain Monte Carlo (MCMC) algorithm is developed for posterior computation. We study the convergence of the pseudo-posterior distribution based on the Whittle likelihood for the model's parameters upon developing a new general posterior convergence theorem for pseudo-posteriors. We find that the posterior contraction rate for independent observations essentially prevails in the OUT model under very mild conditions on the temporal dependence described in terms of the smoothness of the corresponding spectral densities. Through a simulation study, we compare the accuracy of estimating the parameters and identifying the graphical structure with other approaches. We apply the proposed methodology to analyze a dataset on different industrial components of the US gross domestic product between 2010 and 2019 and predict future observations.

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arXiv2023

Bayesian Inference for $k$-Monotone Densities with Applications to Multiple Testing

Shape restriction, like monotonicity or convexity, imposed on a function of interest, such as a regression or density function, allows for its estimation without smoothness assumptions. The concept of $k$-monotonicity encompasses a family of shape restrictions, including decreasing and convex decreasing as special cases corresponding to $k=1$ and $k=2$. We consider Bayesian approaches to estimate a $k$-monotone density. By utilizing a kernel mixture representation and putting a Dirichlet process or a finite mixture prior on the mixing distribution, we show that the posterior contraction rate in the Hellinger distance is $(n/\log n)^{- k/(2k + 1)}$ for a $k$-monotone density, which is minimax optimal up to a polylogarithmic factor. When the true $k$-monotone density is a finite $J_0$-component mixture of the kernel, the contraction rate improves to the nearly parametric rate $\sqrt{(J_0 \log n)/n}$. Moreover, by putting a prior on $k$, we show that the same rates hold even when the best value of $k$ is unknown. A specific application in modeling the density of $p$-values in a large-scale multiple testing problem is considered. Simulation studies are conducted to evaluate the performance of the proposed method.

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arXiv2023

Bayesian Inference for Multivariate Monotone Densities

We consider a nonparametric Bayesian approach to estimation and testing for a multivariate monotone density. Instead of following the conventional Bayesian route of putting a prior distribution complying with the monotonicity restriction, we put a prior on the step heights through binning and a Dirichlet distribution. An arbitrary piece-wise constant probability density is converted to a monotone one by a projection map, taking its $\mathbb{L}_1$-projection onto the space of monotone functions, which is subsequently normalized to integrate to one. We construct consistent Bayesian tests to test multivariate monotonicity of a probability density based on the $\mathbb{L}_1$-distance to the class of monotone functions. The test is shown to have a size going to zero and high power against alternatives sufficiently separated from the null hypothesis. To obtain a Bayesian credible interval for the value of the density function at an interior point with guaranteed asymptotic frequentist coverage, we consider a posterior quantile interval of an induced map transforming the function value to its value optimized over certain blocks. The limiting coverage is explicitly calculated and is seen to be higher than the credibility level used in the construction. By exploring the asymptotic relationship between the coverage and the credibility, we show that a desired asymptomatic coverage can be obtained exactly by starting with an appropriate credibility level.

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arXiv2022

Coverage of Credible Intervals in Bayesian Multivariate Isotonic Regression

We consider the nonparametric multivariate isotonic regression problem, where the regression function is assumed to be nondecreasing with respect to each predictor. Our goal is to construct a Bayesian credible interval for the function value at a given interior point with assured limiting frequentist coverage. We put a prior on unrestricted step-functions, but make inference using the induced posterior measure by an "immersion map" from the space of unrestricted functions to that of multivariate monotone functions. This allows maintaining the natural conjugacy for posterior sampling. A natural immersion map to use is a projection via a distance, but in the present context, a block isotonization map is found to be more useful. The approach of using the induced "immersion posterior" measure instead of the original posterior to make inference provides a useful extension of the Bayesian paradigm, particularly helpful when the model space is restricted by some complex relations. We establish a key weak convergence result for the posterior distribution of the function at a point in terms of some functional of a multi-indexed Gaussian process that leads to an expression for the limiting coverage of the Bayesian credible interval. Analogous to a recent result for univariate monotone functions, we find that the limiting coverage is slightly higher than the credibility, the opposite of a phenomenon observed in smoothing problems. Interestingly, the relation between credibility and limiting coverage does not involve any unknown parameter. Hence by a recalibration procedure, we can get a predetermined asymptotic coverage by choosing a suitable credibility level smaller than the targeted coverage, and thus also shorten the credible intervals.

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arXiv2022

Posterior Contraction and Testing for Multivariate Isotonic Regression

We consider the nonparametric regression problem with multiple predictors and an additive error, where the regression function is assumed to be coordinatewise nondecreasing. We propose a Bayesian approach to make an inference on the multivariate monotone regression function, obtain the posterior contraction rate, and construct a universally consistent Bayesian testing procedure for multivariate monotonicity. To facilitate posterior analysis, we set aside the shape restrictions temporarily, and endow a prior on blockwise constant regression functions with heights independently normally distributed. The unknown variance of the error term is either estimated by the marginal maximum likelihood estimate or is equipped with an inverse-gamma prior. Then the unrestricted block heights are a posteriori also independently normally distributed given the error variance, by conjugacy. To comply with the shape restrictions, we project samples from the unrestricted posterior onto the class of multivariate monotone functions, inducing the "projection-posterior distribution", to be used for making an inference. Under an $\mathbb{L}_1$-metric, we show that the projection-posterior based on $n$ independent samples contracts around the true monotone regression function at the optimal rate $n^{-1/(2+d)}$. Then we construct a Bayesian test for multivariate monotonicity based on the posterior probability of a shrinking neighborhood of the class of multivariate monotone functions. We show that the test is universally consistent, that is, the level of the Bayesian test goes to zero, and the power at any fixed alternative goes to one. Moreover, we show that for a smooth alternative function, power goes to one as long as its distance to the class of multivariate monotone functions is at least of the order of the estimation error for a smooth function.

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arXiv2021

Bayesian inference in high-dimensional models

Models with dimension more than the available sample size are now commonly used in various applications. A sensible inference is possible using a lower-dimensional structure. In regression problems with a large number of predictors, the model is often assumed to be sparse, with only a few predictors active. Interdependence between a large number of variables is succinctly described by a graphical model, where variables are represented by nodes on a graph and an edge between two nodes is used to indicate their conditional dependence given other variables. Many procedures for making inferences in the high-dimensional setting, typically using penalty functions to induce sparsity in the solution obtained by minimizing a loss function, were developed. Bayesian methods have been proposed for such problems more recently, where the prior takes care of the sparsity structure. These methods have the natural ability to also automatically quantify the uncertainty of the inference through the posterior distribution. Theoretical studies of Bayesian procedures in high-dimension have been carried out recently. Questions that arise are, whether the posterior distribution contracts near the true value of the parameter at the minimax optimal rate, whether the correct lower-dimensional structure is discovered with high posterior probability, and whether a credible region has adequate frequentist coverage. In this paper, we review these properties of Bayesian and related methods for several high-dimensional models such as many normal means problem, linear regression, generalized linear models, Gaussian and non-Gaussian graphical models. Effective computational approaches are also discussed.

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arXiv2020

Unified Bayesian theory of sparse linear regression with nuisance parameters

We study frequentist asymptotic properties of Bayesian procedures for high-dimensional Gaussian sparse regression when unknown nuisance parameters are involved. Nuisance parameters can be finite-, high-, or infinite-dimensional. A mixture of point masses at zero and continuous distributions is used for the prior distribution on sparse regression coefficients, and appropriate prior distributions are used for nuisance parameters. The optimal posterior contraction of sparse regression coefficients, hampered by the presence of nuisance parameters, is also examined and discussed. It is shown that the procedure yields strong model selection consistency. A Bernstein-von Mises-type theorem for sparse regression coefficients is also obtained for uncertainty quantification through credible sets with guaranteed frequentist coverage. Asymptotic properties of numerous examples are investigated using the theories developed in this study.

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arXiv2020

Convergence Rates for Bayesian Estimation and Testing in Monotone Regression

Shape restrictions such as monotonicity on functions often arise naturally in statistical modeling. We consider a Bayesian approach to the problem of estimation of a monotone regression function and testing for monotonicity. We construct a prior distribution using piecewise constant functions. For estimation, a prior imposing monotonicity of the heights of these steps is sensible, but the resulting posterior is harder to analyze theoretically. We consider a ``projection-posterior'' approach, where a conjugate normal prior is used, but the monotonicity constraint is imposed on posterior samples by a projection map on the space of monotone functions. We show that the resulting posterior contracts at the optimal rate $n^{-1/3}$ under the $L_1$-metric and at a nearly optimal rate under the empirical $L_p$-metrics for $0<p\le 2$. The projection-posterior approach is also computationally more convenient. We also construct a Bayesian test for the hypothesis of monotonicity using the posterior probability of a shrinking neighborhood of the set of monotone functions. We show that the resulting test has a universal consistency property and obtain the separation rate which ensures that the resulting power function approaches one.

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arXiv2020

Bayesian nonparametric tests for multivariate locations

In this paper, we propose novel, fully Bayesian non-parametric tests for one-sample and two-sample multivariate location problems. We model the underlying distribution using a Dirichlet process prior, and develop a testing procedure based on the posterior credible region for the spatial median functional of the distribution. For the one-sample problem, we fail to reject the null hypothesis if the credible set contains the null value. For the two-sample problem, we form a credible set for the difference of the spatial medians of the two samples and we fail to reject the null hypothesis of equality if the credible set contains zero. We derive the local asymptotic power of the tests under shrinking alternatives, and also present a simulation study to compare the finite-sample performance of our testing procedures with existing parametric and non-parametric tests.

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arXiv2020

Bayesian Multivariate Quantile Regression Using Dependent Dirichlet Process Prior

In this article, we consider a non-parametric Bayesian approach to multivariate quantile regression. The collection of related conditional distributions of a response vector Y given a univariate covariate X is modeled using a Dependent Dirichlet Process (DDP) prior. The DDP is used to introduce dependence across x. As the realizations from a Dirichlet process prior are almost surely discrete, we need to convolve it with a kernel. To model the error distribution as flexibly as possible, we use a countable mixture of multidimensional normal distributions as our kernel. For posterior computations, we use a truncated stick-breaking representation of the DDP. This approximation enables us to deal with only a finitely number of parameters. We use a Block Gibbs sampler for estimating the model parameters. We illustrate our method with simulation studies and real data applications. Finally, we provide a theoretical justification for the proposed method through posterior consistency. Our proposed procedure is new even when the response is univariate.

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arXiv2020

Bayesian Semi-supervised Multi-category Classification under Nonparanormality

Semi-supervised learning is a model training method that uses both labeled and unlabeled data. This paper proposes a fully Bayes semi-supervised learning algorithm that can be applied to any multi-category classification problem. We assume the labels are missing at random when using unlabeled data in a semi-supervised setting. Suppose we have $K$ classes in the data. We assume that the observations follow $K$ multivariate normal distributions depending on their true class labels after some common unknown transformation is applied to each component of the observation vector. The function is expanded in a B-splines series, and a prior is added to the coefficients. We consider a normal prior on the coefficients and constrain the values to meet the normality and identifiability constraints requirement. The precision matrices of the Gaussian distributions are given a conjugate Wishart prior, while the means are given the improper uniform prior. The resulting posterior is still conditionally conjugate, and the Gibbs sampler aided by a data-augmentation technique can thus be adopted. An extensive simulation study compares the proposed method with several other available methods. The proposed method is also applied to real datasets on diagnosing breast cancer and classification of signals. We conclude that the proposed method has a better prediction accuracy in various cases.

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arXiv2019

Bayesian Inference on Multivariate Medians and Quantiles

In this paper, we consider Bayesian inference on a class of multivariate median and the multivariate quantile functionals of a joint distribution using a Dirichlet process prior. Since, unlike univariate quantiles, the exact posterior distribution of multivariate median and multivariate quantiles are not obtainable explicitly, we study these distributions asymptotically. We derive a Bernstein-von Mises theorem for the multivariate $\ell_1$-median with respect to general $\ell_p$-norm, which in particular shows that its posterior concentrates around its true value at $n^{-1/2}$-rate and its credible sets have asymptotically correct frequentist coverage. In particular, asymptotic normality results for the empirical multivariate median with general $\ell_p$-norm is also derived in the course of the proof which extends the results from the case $p=2$ in the literature to a general $p$. The technique involves approximating the posterior Dirichlet process by a Bayesian bootstrap process and deriving a conditional Donsker theorem. We also obtain analogous results for an affine equivariant version of the multivariate $\ell_1$-median based on an adaptive transformation and re-transformation technique. The results are extended to a joint distribution of multivariate quantiles. The accuracy of the asymptotic result is confirmed by a simulation study. We also use the results to obtain Bayesian credible regions for multivariate medians for Fisher's iris data, which consists of four features measured for each of three plant species.

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arXiv2018

Regression-Based Bayesian Estimation and Structure Learning for Nonparanormal Graphical Models

A nonparanormal graphical model is a semiparametric generalization of a Gaussian graphical model for continuous variables in which it is assumed that the variables follow a Gaussian graphical model only after some unknown smooth monotone transformations. We consider a Bayesian approach to inference in a nonparanormal graphical model in which we put priors on the unknown transformations through a random series based on B-splines. We use a regression formulation to construct the likelihood through the Cholesky decomposition on the underlying precision matrix of the transformed variables and put shrinkage priors on the regression coefficients. We apply a plug-in variational Bayesian algorithm for learning the sparse precision matrix and compare the performance to a posterior Gibbs sampling scheme in a simulation study. We finally apply the proposed methods to a real data set. KEYWORDS:

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