@article{wang_ghosal_2023, title={Posterior contraction and testing for multivariate isotonic regression}, volume={17}, ISSN={["1935-7524"]}, DOI={10.1214/23-EJS2115}, abstractNote={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.}, number={1}, journal={ELECTRONIC JOURNAL OF STATISTICS}, author={Wang, Kang and Ghosal, Subhashis}, year={2023}, pages={798–822} }
 @article{bhattacharya_ghosal_2022, title={BAYESIAN INFERENCE ON MULTIVARIATE MEDIANS AND QUANTILES}, volume={32}, ISSN={["1996-8507"]}, DOI={10.5705/ss.202020.0108}, abstractNote={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.}, number={1}, journal={STATISTICA SINICA}, author={Bhattacharya, Indrabati and Ghosal, Subhashis}, year={2022}, month={Jan}, pages={517–538} }
 @article{ghosal_2022, title={Discussion of "Confidence Intervals for Nonparametric Empirical Bayes Analysis" by Ignatiadis and Wager}, volume={117}, ISSN={["1537-274X"]}, DOI={10.1080/01621459.2022.2093726}, abstractNote={I congratulate the authors for a fantastic piece of work about confidence intervals in an inverse problem involving a mixing distribution under an empirical Bayes setup. We can regard the problem as inference on a functional of the distribution of latent variable with the observation coming from a specific parametric family of distributions driven by the latent variable. In mathematical notations, this can be written as}, number={539}, journal={JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION}, author={Ghosal, Subhashis}, year={2022}, month={Sep}, pages={1171–1174} }
 @article{chakraborty_ghosal_2022, title={Rates and coverage for monotone densities using projection-posterior}, volume={28}, ISSN={["1573-9759"]}, DOI={10.3150/21-BEJ1379}, abstractNote={We consider Bayesian inference for a monotone density on the unit interval and study the resulting asymptotic properties. We consider a “projection-posterior” approach, where we construct a prior on density functions through random histograms without imposing the monotonicity constraint, but induce a random distribution by projecting a sample from the posterior on the space of monotone functions. The approach allows us to retain posterior conjugacy, allowing explicit expressions extremely useful for studying asymptotic properties. We show that the projection-posterior contracts at the optimal n−1/3-rate. We then construct a consistent test based on the posterior distribution for testing the hypothesis of monotonicity. Finally, we obtain the limiting coverage of a projection-posterior credible interval for the value of the function at an interior point. Interestingly, the limiting coverage turns out to be higher than the nominal credibility level, the opposite of the undercoverage phenomenon observed in a smoothness regime. Moreover, we show that a recalibration method using a lower credibility level gives an intended limiting coverage. We also discuss extensions of the obtained results for densities on the half-line. We conduct a simulation study to demonstrate the accuracy of the asymptotic results in finite samples.}, number={2}, journal={BERNOULLI}, author={Chakraborty, Moumita and Ghosal, Subhashis}, year={2022}, month={May}, pages={1093–1119} }
 @article{bhaumik_shi_ghosal_2022, title={Two-step Bayesian methods for generalized regression driven by partial differential equations}, volume={28}, ISSN={["1573-9759"]}, DOI={10.3150/21-BEJ1363}, abstractNote={In certain non-linear regression models, the functional form of the regression function is not explicitly available, but is only described by set of differential equations. For regression models described by a set of ordinary differential equations (ODEs), both Bayesian and non-Bayesian methods for inference were developed in the literature. In this paper, we consider a Bayesian approach to non-linear regression with respect to a multidimensional predictor variable given by a set of partial differential equations (PDEs). We consider a computationally convenient two-step approach by first representing the functions nonparametrically, construct a finite random series prior using a tensor product of B-splines and directly inducing a posterior distribution on parameter space through an appropriate projection map. By considering three different choices of the projection map, we propose three different approaches with their merits. We allow generalized nonlinear regression with the response variable following an exponential family of distributions, extending the method beyond regression with additive normal errors. We establish Bernstein-von Mises type theorems which show √ n-consistency and asymptotically correct frequentist coverage of Bayesian credible regions. We also conduct a simulation study to evaluate finite sample performances of the proposed methods.}, number={3}, journal={BERNOULLI}, author={Bhaumik, Prithwish and Shi, Wenli and Ghosal, Subhashis}, year={2022}, month={Aug}, pages={1625–1647} }
 @article{shi_ghosal_martin_2021, title={Bayesian estimation of sparse precision matrices in the presence of Gaussian measurement error}, volume={15}, ISSN={["1935-7524"]}, DOI={10.1214/21-EJS1904}, abstractNote={Estimation of sparse, high-dimensional precision matrices is an important and challenging problem. Existing methods all assume that observations can be made precisely but, in practice, this often is not the case; for example, the instruments used to measure the response may have limited precision. The present paper incorporates measurement error in the context of estimating a sparse, high-dimensional precision matrix. In particular, for a Gaussian graphical model with data corrupted by Gaussian measurement error with unknown variance, we establish a general result which gives sufficient conditions under which the posterior contraction rates that hold in the no-measurement-error case carry over to the measurement-error case. Interestingly, this result does not require that the measurement error variance be small. We apply our general result to several cases with well-known prior distributions for sparse precision matrices and also to a case with a newly-constructed prior for precision matrices with a sparse factor-loading form. Two different simulation studies highlight the empirical benefits of accounting for the measurement error as opposed to ignoring it, even when that measurement error is relatively small.}, number={2}, journal={ELECTRONIC JOURNAL OF STATISTICS}, author={Shi, Wenli and Ghosal, Subhashis and Martin, Ryan}, year={2021}, pages={4545–4579} }
 @article{jeong_ghosal_2021, title={Posterior contraction in sparse generalized linear models}, volume={108}, ISSN={["1464-3510"]}, DOI={10.1093/biomet/asaa074}, abstractNote={Summary}, number={2}, journal={BIOMETRIKA}, author={Jeong, Seonghyun and Ghosal, Subhashis}, year={2021}, month={Jun}, pages={367–379} }
 @article{jeong_ghosal_2021, title={Unified Bayesian theory of sparse linear regression with nuisance parameters}, volume={15}, ISSN={["1935-7524"]}, DOI={10.1214/21-EJS1855}, abstractNote={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.}, number={1}, journal={ELECTRONIC JOURNAL OF STATISTICS}, author={Jeong, Seonghyun and Ghosal, Subhashis}, year={2021}, pages={3040–3111} }
 @article{ning_jeong_ghosal_2020, title={Bayesian linear regression for multivariate responses under group sparsity}, volume={26}, ISSN={["1573-9759"]}, DOI={10.3150/20-BEJ1198}, abstractNote={We study frequentist properties of a Bayesian high-dimensional multivariate linear regression model with correlated responses. The predictors are separated into many groups and the group structure is pre-determined. Two features of the model are unique: (i) group sparsity is imposed on the predictors. (ii) the covariance matrix is unknown and its dimensions can also be high. We choose a product of independent spike-and-slab priors on the regression coefficients and a new prior on the covariance matrix based on its eigendecomposition. Each spike-and-slab prior is a mixture of a point mass at zero and a multivariate density involving a $\ell_{2,1}$-norm. We first obtain the posterior contraction rate, the bounds on the effective dimension of the model with high posterior probabilities. We then show that the multivariate regression coefficients can be recovered under certain compatibility conditions. Finally, we quantify the uncertainty for the regression coefficients with frequentist validity through a Bernstein-von Mises type theorem. The result leads to selection consistency for the Bayesian method. We derive the posterior contraction rate using the general theory by constructing a suitable test from the first principle using moment bounds for certain likelihood ratios. This leads to posterior concentration around the truth with respect to the average Renyi divergence of order 1/2. This technique of obtaining the required tests for posterior contraction rate could be useful in many other problems.}, number={3}, journal={BERNOULLI}, author={Ning, Bo and Jeong, Seonghyun and Ghosal, Subhashis}, year={2020}, month={Aug}, pages={2353–2382} }
 @article{li_ghosal_2020, title={Posterior contraction and credible sets for filaments of regression functions}, volume={14}, ISSN={["1935-7524"]}, DOI={10.1214/20-EJS1705}, abstractNote={A filament consists of local maximizers of a smooth function $f$ when moving in a certain direction. Filamentary structures are important features of the shape of objects and are also considered as important lower dimensional characterization of multivariate data. There have been some recent theoretical studies of filaments in the nonparametric kernel density estimation context. This paper supplements the current literature in two ways. First, we provide a Bayesian approach to the filament estimation in regression context and study the posterior contraction rates using a finite random series of B-splines basis. Compared with the kernel-estimation method, this has theoretical advantage as the bias can be better controlled when the function is smoother, which allows obtaining better rates. Assuming that $f: \mathbb{R}^2 \mapsto \mathbb{R}$ belongs to an isotropic H\"{o}lder class of order $\alpha \geq 4$, with the optimal choice of smoothing parameters, the posterior contraction rates for the filament points on some appropriately defined integral curves and for the Hausdorff distance of the filament are both $(n/\log n)^{(2-\alpha)/(2(1+\alpha))}$. Secondly, we provide a way to construct a credible set with sufficient frequentist coverage for the filaments. Our valid credible region consists of posterior filaments that have frequentist interpretation. We demonstrate the success of our proposed method in simulations and application to earthquake data.}, number={1}, journal={ELECTRONIC JOURNAL OF STATISTICS}, author={Li, Wei and Ghosal, Subhashis}, year={2020}, pages={1707–1743} }
 @article{zhu_ghosal_2019, title={Bayesian Semiparametric ROC surface estimation under verification bias}, volume={133}, ISSN={0167-9473}, url={http://dx.doi.org/10.1016/J.CSDA.2018.09.003}, DOI={10.1016/J.CSDA.2018.09.003}, abstractNote={The Receiver Operating Characteristic (ROC) surface is a generalization of the ROC curve and is widely used for assessment of the accuracy of diagnostic tests on three categories. Verification bias occurs when not all subjects have their labels observed. This is a common problem in disease diagnosis since the gold standard test to get labels, i.e., the true disease status, can be invasive and expensive. The same situation happens in the evaluation of semi-supervised learning, where the unlabeled data are incorporated. A Bayesian approach for estimating the ROC surface is proposed based on continuous data under a semi-parametric trinormality assumption. The proposed method is then extended to situations in the presence of verification bias. The posterior distribution is computed under the trinormality assumption using a rank-based likelihood. The consistency of the posterior under a mild condition is also established. The proposed method is compared with existing methods for estimating an ROC surface. Simulation results show that it performs well in terms of accuracy. The method is applied to evaluate the performance of CA125 and HE4 in the diagnosis of epithelial ovarian cancer (EOC) as a demonstration.}, journal={Computational Statistics & Data Analysis}, publisher={Elsevier BV}, author={Zhu, Rui and Ghosal, Subhashis}, year={2019}, month={May}, pages={40–52} }
 @article{du_ghosal_2019, title={Multivariate Gaussian network structure learning}, volume={199}, ISSN={0378-3758}, url={http://dx.doi.org/10.1016/J.JSPI.2018.07.009}, DOI={10.1016/J.JSPI.2018.07.009}, abstractNote={We consider a graphical model where a multivariate normal vector is associated with each node of the underlying graph and estimate the graphical structure. We minimize a loss function obtained by regressing the vector at each node on those at the remaining ones under a group penalty. We show that the proposed estimator can be computed by a fast convex optimization algorithm. We show that as the sample size increases, the estimated regression coefficients and the correct graphical structure are correctly estimated with probability tending to one. By extensive simulations, we show the superiority of the proposed method over comparable procedures. We apply the technique on two real datasets. The first one is to identify gene and protein networks showing up in cancer cell lines, and the second one is to reveal the connections among different industries in the US.}, journal={Journal of Statistical Planning and Inference}, publisher={Elsevier BV}, author={Du, Xingqi and Ghosal, Subhashis}, year={2019}, month={Mar}, pages={327–342} }
 @article{du_ghosal_2018, title={Bayesian Discriminant Analysis Using a High Dimensional Predictor}, volume={80}, ISSN={0976-836X 0976-8378}, url={http://dx.doi.org/10.1007/S13171-018-0140-Z}, DOI={10.1007/S13171-018-0140-Z}, number={S1}, journal={Sankhya A}, publisher={Springer Science and Business Media LLC}, author={Du, Xingqi and Ghosal, Subhashis}, year={2018}, month={Aug}, pages={112–145} }
 @article{das_ghosal_2018, title={Bayesian non-parametric simultaneous quantile regression for complete and grid data}, volume={127}, ISSN={0167-9473}, url={http://dx.doi.org/10.1016/J.CSDA.2018.04.007}, DOI={10.1016/J.CSDA.2018.04.007}, abstractNote={Bayesian methods for non-parametric quantile regression have been considered with multiple continuous predictors ranging values in the unit interval. Two methods are proposed based on assuming that either the quantile function or the distribution function is smooth in the explanatory variables and is expanded in tensor product of B-spline basis functions. Unlike other existing methods of non-parametric quantile regressions, the proposed methods estimate the whole quantile function instead of estimating on a grid of quantiles. Priors on the coefficients of the B-spline expansion are put in such a way that the monotonicity of the estimated quantile levels are maintained unlike local polynomial quantile regression methods. The proposed methods are also modified for quantile grid data where only the percentile range of each response observations are known. A comparative simulation study of the performances of the proposed methods and some other existing methods are provided in terms of prediction mean squared errors and mean L1-errors over the quartiles. The proposed methods are used to estimate the quantiles of US household income data and North Atlantic hurricane intensity data.}, journal={Computational Statistics & Data Analysis}, publisher={Elsevier BV}, author={Das, Priyam and Ghosal, Subhashis}, year={2018}, month={Nov}, pages={172–186} }
 @article{das_ghosal_2017, title={Analyzing ozone concentration by Bayesian spatio-temporal quantile regression}, volume={28}, ISSN={1180-4009}, url={http://dx.doi.org/10.1002/ENV.2443}, DOI={10.1002/ENV.2443}, abstractNote={Ground‐level ozone is 1 of the 6 common air pollutants on which the Environmental Protection Agency has set national air quality standards. In order to capture the spatio‐temporal trend of 1‐ and 8‐hr average ozone concentration in the United States, we develop a method for spatio‐temporal simultaneous quantile regression. Unlike existing procedures, in the proposed method, smoothing across different sites is incorporated within modeling assumptions. This allows borrowing of information across locations, which is an essential step when the number of samples in each location is low. The quantile function has been assumed to be linear in time and smooth over space, and at any given site is given by a convex combination of 2 monotone increasing functions ξ1 and ξ2 not depending on time. A B‐spline basis expansion with increasing coefficients varying smoothly over the space is used to put a prior and a Bayesian analysis is performed. We analyze the average daily 1‐hr maximum and 8‐hr maximum ozone concentration level data of the United States and the state of California during 2006–2015 using the proposed method. It is noted that in the last 10 years, there is an overall decreasing trend in both 1‐hr maximum and 8‐hr maximum ozone concentration level over most parts of the US. In California, an overall a decreasing trend of 1‐hr maximum ozone level is observed whereas no particular overall trend has been observed in 8‐hr maximum ozone level.}, number={4}, journal={Environmetrics}, publisher={Wiley}, author={Das, P. and Ghosal, S.}, year={2017}, month={May}, pages={e2443} }
 @article{bhaumik_ghosal_2017, title={Bayesian inference for higher-order ordinary differential equation models}, volume={157}, ISSN={0047-259X}, url={http://dx.doi.org/10.1016/J.JMVA.2017.03.003}, DOI={10.1016/J.JMVA.2017.03.003}, abstractNote={Often the regression function appearing in fields like economics, engineering, and biomedical sciences obeys a system of higher-order ordinary differential equations (ODEs). The equations are usually not analytically solvable. We are interested in inferring on the unknown parameters appearing in such equations. Parameter estimation in first-order ODE models has been well investigated. Bhaumik and Ghosal (2015) considered a two-step Bayesian approach by putting a finite random series prior on the regression function using a B-spline basis. The posterior distribution of the parameter vector is induced from that of the regression function. Although this approach is computationally fast, the Bayes estimator is not asymptotically efficient. Bhaumik and Ghosal (2016) remedied this by directly considering the distance between the function in the nonparametric model and a Runge–Kutta (RK4) approximate solution of the ODE while inducing the posterior distribution on the parameter. They also studied the convergence properties of the Bayesian method based on the approximate likelihood obtained by the RK4 method. In this paper, we extend these ideas to the higher-order ODE model and establish Bernstein–von Mises theorems for the posterior distribution of the parameter vector for each method with n−1/2 contraction rate.}, journal={Journal of Multivariate Analysis}, publisher={Elsevier BV}, author={Bhaumik, Prithwish and Ghosal, Subhashis}, year={2017}, month={May}, pages={103–114} }
 @article{das_ghosal_2017, title={Bayesian quantile regression using random B-spline series prior}, volume={109}, ISSN={0167-9473}, url={http://dx.doi.org/10.1016/J.CSDA.2016.11.014}, DOI={10.1016/J.CSDA.2016.11.014}, abstractNote={A Bayesian method for simultaneous quantile regression on a real variable is considered. By monotone transformation, the response variable and the predictor variable are transformed into the unit interval. A representation of quantile function is given by a convex combination of two monotone increasing functions ξ1 and ξ2 not depending on the prediction variables. In a Bayesian approach, a prior is put on quantile functions by putting prior distributions on ξ1 and ξ2. The monotonicity constraint on the curves ξ1 and ξ2 are obtained through a spline basis expansion with coefficients increasing and lying in the unit interval. A Dirichlet prior distribution is put on the spacings of the coefficient vector. A finite random series based on splines obeys the shape restrictions. The proposed method is extended to multidimensional predictors such that the quantile regression depends on the predictors through an unknown linear combination only. In the simulation study, the proposed approach is compared with a Bayesian method using Gaussian process prior through an extensive simulation study and some other Bayesian approaches proposed in the literature. An application to a data on hurricane activities in the Atlantic region is given. The proposed method is also applied on region-wise population data of USA for the period 1985–2010.}, journal={Computational Statistics & Data Analysis}, publisher={Elsevier BV}, author={Das, Priyam and Ghosal, Subhashis}, year={2017}, month={May}, pages={121–143} }
 @article{sundaram_ma_ghoshal_2017, title={Median Analysis of Repeated Measures Associated with Recurrent Events in Presence of Terminal Event}, volume={13}, ISSN={["1557-4679"]}, DOI={10.1515/ijb-2016-0057}, abstractNote={Abstract}, number={1}, journal={INTERNATIONAL JOURNAL OF BIOSTATISTICS}, author={Sundaram, Rajeshwari and Ma, Ling and Ghoshal, Subhashis}, year={2017}, month={May} }
 @article{shen_ghosal_2017, title={Posterior Contraction Rates of Density Derivative Estimation}, volume={79}, ISSN={0976-836X 0976-8378}, url={http://dx.doi.org/10.1007/S13171-017-0105-7}, DOI={10.1007/S13171-017-0105-7}, number={2}, journal={Sankhya A}, publisher={Springer Science and Business Media LLC}, author={Shen, Weining and Ghosal, Subhashis}, year={2017}, month={Jun}, pages={336–354} }
 @article{luo_ghosal_2016, title={Forward selection and estimation in high dimensional single index models}, volume={33}, ISSN={1572-3127}, url={http://dx.doi.org/10.1016/J.STAMET.2016.09.002}, DOI={10.1016/J.STAMET.2016.09.002}, abstractNote={We propose a new variable selection and estimation technique for high dimensional single index models with unknown monotone smooth link function. Among many predictors, typically, only a small fraction of them have significant impact on prediction. In such a situation, more interpretable models with better prediction accuracy can be obtained by variable selection. In this article, we propose a new penalized forward selection technique which can reduce high dimensional optimization problems to several one dimensional optimization problems by choosing the best predictor and then iterating the selection steps until convergence. The advantage of optimizing in one dimension is that the location of optimum solution can be obtained with an intelligent search by exploiting smoothness of the criterion function. Moreover, these one dimensional optimization problems can be solved in parallel to reduce computing time nearly to the level of the one-predictor problem. Numerical comparison with the LASSO and the shrinkage sliced inverse regression shows very promising performance of our proposed method.}, journal={Statistical Methodology}, publisher={Elsevier BV}, author={Luo, Shikai and Ghosal, Subhashis}, year={2016}, month={Dec}, pages={172–179} }
 @article{ghosal_turnbull_zhang_hwang_2016, title={Sparse Penalized Forward Selection for Support Vector Classification}, volume={25}, ISSN={1061-8600 1537-2715}, url={http://dx.doi.org/10.1080/10618600.2015.1023395}, DOI={10.1080/10618600.2015.1023395}, abstractNote={We propose a new binary classification and variable selection technique especially designed for high-dimensional predictors. Among many predictors, typically, only a small fraction of them have significant impact on prediction. In such a situation, more interpretable models with better prediction accuracy can be obtained by variable selection along with classification. By adding an ℓ1-type penalty to the loss function, common classification methods such as logistic regression or support vector machines (SVM) can perform variable selection. Existing penalized SVM methods all attempt to jointly solve all the parameters involved in the penalization problem altogether. When data dimension is very high, the joint optimization problem is very complex and involves a lot of memory allocation. In this article, we propose a new penalized forward search technique that can reduce high-dimensional optimization problems to one-dimensional optimization by iterating the selection steps. The new algorithm can be regarded as a forward selection version of the penalized SVM and its variants. The advantage of optimizing in one dimension is that the location of the optimum solution can be obtained with intelligent search by exploiting convexity and a piecewise linear or quadratic structure of the criterion function. In each step, the predictor that is most able to predict the outcome is chosen in the model. The search is then repeatedly used in an iterative fashion until convergence occurs. Comparison of our new classification rule with ℓ1-SVM and other common methods show very promising performance, in that the proposed method leads to much leaner models without compromising misclassification rates, particularly for high-dimensional predictors.}, number={2}, journal={Journal of Computational and Graphical Statistics}, publisher={Informa UK Limited}, author={Ghosal, Subhashis and Turnbull, Bradley and Zhang, Hao Helen and Hwang, Wook Yeon}, year={2016}, month={Apr}, pages={493–514} }
 @article{banerjee_ghosal_2015, title={Bayesian structure learning in graphical models}, volume={136}, ISSN={0047-259X}, url={http://dx.doi.org/10.1016/J.JMVA.2015.01.015}, DOI={10.1016/J.JMVA.2015.01.015}, abstractNote={We consider the problem of estimating a sparse precision matrix of a multivariate Gaussian distribution, where the dimension p may be large. Gaussian graphical models provide an important tool in describing conditional independence through presence or absence of edges in the underlying graph. A popular non-Bayesian method of estimating a graphical structure is given by the graphical lasso. In this paper, we consider a Bayesian approach to the problem. We use priors which put a mixture of a point mass at zero and certain absolutely continuous distribution on off-diagonal elements of the precision matrix. Hence the resulting posterior distribution can be used for graphical structure learning. The posterior convergence rate of the precision matrix is obtained and is shown to match the oracle rate. The posterior distribution on the model space is extremely cumbersome to compute using the commonly used reversible jump Markov chain Monte Carlo methods. However, the posterior mode in each graph can be easily identified as the graphical lasso restricted to each model. We propose a fast computational method for approximating the posterior probabilities of various graphs using the Laplace approximation approach by expanding the posterior density around the posterior mode. We also provide estimates of the accuracy in the approximation.}, journal={Journal of Multivariate Analysis}, publisher={Elsevier BV}, author={Banerjee, Sayantan and Ghosal, Subhashis}, year={2015}, month={Apr}, pages={147–162} }
 @article{luo_ghosal_2015, title={Prediction consistency of forward iterated regression and selection technique}, volume={107}, ISSN={0167-7152}, url={http://dx.doi.org/10.1016/J.SPL.2015.08.005}, DOI={10.1016/J.SPL.2015.08.005}, abstractNote={Recently, Hwang et al. (2009) introduced a penalized forward selection technique for high dimensional linear regression which appears to possess excellent prediction and variable selection properties. In this article, we show that the procedure is prediction consistent.}, journal={Statistics & Probability Letters}, publisher={Elsevier BV}, author={Luo, Shikai and Ghosal, Subhashis}, year={2015}, month={Dec}, pages={79–83} }
 @article{ghoshal_kleijn_van der vaart_van zanten_2015, title={Special issue on Bayesian nonparametrics}, volume={166}, ISSN={0378-3758}, url={http://dx.doi.org/10.1016/J.JSPI.2015.04.008}, DOI={10.1016/J.JSPI.2015.04.008}, journal={Journal of Statistical Planning and Inference}, publisher={Elsevier BV}, author={Ghoshal, Subhashis and Kleijn, Bas and van der Vaart, Aad and van Zanten, Harry}, year={2015}, month={Nov}, pages={1} }
 @article{gu_ghosal_kleiner_2014, title={Bayesian ROC curve estimation under verification bias}, volume={33}, ISSN={0277-6715}, url={http://dx.doi.org/10.1002/SIM.6297}, DOI={10.1002/SIM.6297}, abstractNote={Receiver operating characteristic (ROC) curve has been widely used in medical science for its ability to measure the accuracy of diagnostic tests under the gold standard. However, in a complicated medical practice, a gold standard test can be invasive, expensive, and its result may not always be available for all the subjects under study. Thus, a gold standard test is implemented only when it is necessary and possible. This leads to the so‐called ‘verification bias’, meaning that subjects with verified disease status (also called label) are not selected in a completely random fashion. In this paper, we propose a new Bayesian approach for estimating an ROC curve based on continuous data following the popular semiparametric binormal model in the presence of verification bias. By using a rank‐based likelihood, and following Gibbs sampling techniques, we compute the posterior distribution of the binormal parameters intercept and slope, as well as the area under the curve by imputing the missing labels within Markov Chain Monte‐Carlo iterations. Consistency of the resulting posterior under mild conditions is also established. We compare the new method with other comparable methods and conclude that our estimator performs well in terms of accuracy. Copyright © 2014 John Wiley & Sons, Ltd.}, number={29}, journal={Statistics in Medicine}, publisher={Wiley}, author={Gu, Jiezhun and Ghosal, Subhashis and Kleiner, David E.}, year={2014}, month={Oct}, pages={5081–5096} }
 @article{banerjee_ghosal_2014, title={Bayesian variable selection in generalized additive partial linear models}, volume={3}, ISSN={2049-1573}, url={http://dx.doi.org/10.1002/STA4.70}, DOI={10.1002/STA4.70}, abstractNote={Variable selection in regression models has been well studied in the literature, with many non‐Bayesian and Bayesian methods available in this regard. An important class of regression models is generalized linear models, which involve situations where the response variable is discrete. To add more flexibility, generalized additive partial linear models can be considered, where some predictors can have a non‐linear effect while some predictors have a strictly linear effect. We consider Bayesian variable selection in these models. The functions in the non‐parametric additive part of the model are expanded in a B‐spline basis and multivariate Laplace prior put on the coefficients with point mass at zero. The coefficients corresponding to the strictly linear components are assigned a univariate Laplace prior with point mass at zero. The prior times the likelihood is mathematically intractable, but we find an approximation by expansion around the posterior mode, which is the group lasso solution in generalized linear model setting for the choice of prior. We thus completely avoid Markov chain Monte Carlo methods, which are extremely slow and unreliable in high‐dimensional models. We evaluate the performance of the Bayesian method by conducting simulation studies and real data analyses. Copyright © 2014 John Wiley & Sons, Ltd.}, number={1}, journal={Stat}, publisher={Wiley}, author={Banerjee, Sayantan and Ghosal, Subhashis}, year={2014}, month={Mar}, pages={363–378} }
 @article{mckay curtis_banerjee_ghosal_2014, title={Fast Bayesian model assessment for nonparametric additive regression}, volume={71}, ISSN={0167-9473}, url={http://dx.doi.org/10.1016/J.CSDA.2013.05.012}, DOI={10.1016/J.CSDA.2013.05.012}, abstractNote={Variable selection techniques for the classical linear regression model have been widely investigated. Variable selection in fully nonparametric and additive regression models has been studied more recently. A Bayesian approach for nonparametric additive regression models is considered, where the functions in the additive model are expanded in a B-spline basis and a multivariate Laplace prior is put on the coefficients. Posterior probabilities of models defined by selection of predictors in the working model are computed, using a Laplace approximation method. The prior times the likelihood is expanded around the posterior mode, which can be identified with the group LASSO, for which a fast computing algorithm exists. Thus Markov chain Monte-Carlo or any other time consuming sampling based methods are completely avoided, leading to quick assessment of various posterior model probabilities. This technique is applied to the high-dimensional situation where the number of parameters exceeds the number of observations.}, journal={Computational Statistics & Data Analysis}, publisher={Elsevier BV}, author={McKay Curtis, S. and Banerjee, Sayantan and Ghosal, Subhashis}, year={2014}, month={Mar}, pages={347–358} }
 @inbook{white_ghosal_2014, title={Multiple Testing Approaches for Removing Background Noise from Images}, ISBN={9781493905683 9781493905690}, ISSN={2194-1009 2194-1017}, url={http://dx.doi.org/10.1007/978-1-4939-0569-0_10}, DOI={10.1007/978-1-4939-0569-0_10}, abstractNote={Images arising from low-intensity settings such as in X-ray astronomy and computed tomography scan often show a relatively weak but constant background noise across the frame. The background noise can result from various uncontrollable sources. In such a situation, it has been observed that the performance of a denoising algorithm can be improved considerably if an additional thresholding procedure is performed on the processed image to set low intensity values to zero. The threshold is typically chosen by an ad-hoc method, such as 5% of the maximum intensity. In this article, we formalize the choice of thresholding through a multiple testing approach. At each pixel, the null hypothesis that the underlying intensity parameter equals the intensity of the background noise is tested, with due consideration of the multiplicity factor. Pixels where the null hypothesis is not rejected, the estimated intensity will be set to zero, thus creating a sharper contrast with the foreground. The main difference of the present context with the usual multiple testing applications is that in our setup, the null value in the hypotheses is not known, and must be estimated from the data itself. We employ a Gaussian mixture to estimate the unknown common null value of the background intensity level. We discuss three approaches to solve the problem and compare them through simulation studies. The methods are applied on noisy X-ray images of a supernova remnant.}, booktitle={Springer Proceedings in Mathematics & Statistics}, publisher={Springer New York}, author={White, John Thomas and Ghosal, Subhashis}, year={2014}, pages={95–104} }
 @article{thomas white_ghosal_2013, title={Denoising three-dimensional and colored images using a Bayesian multi-scale model for photon counts}, volume={93}, ISSN={0165-1684}, url={http://dx.doi.org/10.1016/J.SIGPRO.2013.04.003}, DOI={10.1016/J.SIGPRO.2013.04.003}, abstractNote={X-ray images of distant stars and galaxies are typically registered by low photon counts at the pixel level, for which the Poisson distribution is a sensible model description. The resulting count data can be represented in a multi-scale framework, where the likelihood function factorizes in functions of relative intensity parameters corresponding to different levels from the whole frame down to the pixel level. In a Bayesian approach, a prior is assigned on these relative intensity parameters independently across levels and the image is reconstructed using the posterior mean of intensity parameter of each pixel. A novel prior which allows ties in the values of relative intensity parameters of neighboring regions has been recently shown to be very successful in finding structures in images. We extend this idea to reconstruct colored images from noisy data. The proposed method is completely data-driven, since all smoothing parameters are automatically estimated from the data without any additional user input. In the context of astronomical X-ray images, color represents the energy level of photons, which are also typically recorded by telescopes. The energy level can be considered as the third dimension of images. In a more general sense, the technique we develop applies to all three dimensional images, and can be used to process medical images as well.}, number={11}, journal={Signal Processing}, publisher={Elsevier BV}, author={Thomas White, John and Ghosal, Subhashis}, year={2013}, month={Nov}, pages={2906–2914} }
 @article{bean_dimarco_mercer_thayer_roy_ghosal_2013, title={Finite skew-mixture models for estimation of positive false discovery rates}, volume={10}, ISSN={1572-3127}, url={http://dx.doi.org/10.1016/j.stamet.2012.05.005}, DOI={10.1016/j.stamet.2012.05.005}, abstractNote={We propose a mixture model framework for estimating positive false discovery rates in multiple-testing problems. The density of a transformed p-value is modeled by a finite mixture of skewed distributions. We argue that a mixture of skewed distributions like the skew-normal one is better for addressing some features in modeling than the more commonly used mixture of normal distributions. Using the fitted distributions, we estimate the proportion of true null hypotheses, the positive false discovery rate and other important functionals in multiple-testing problems. We investigate the performance of our methodology via simulation and illustrate the effectiveness of the proposed procedure using real data examples. We also discuss the role of an empirical null in place of the theoretical null distributions in the context of common biomedical applications.}, number={1}, journal={Statistical Methodology}, publisher={Elsevier BV}, author={Bean, Gordon J. and Dimarco, Elizabeth A. and Mercer, Laina D. and Thayer, Laura K. and Roy, Anindya and Ghosal, Subhashis}, year={2013}, month={Jan}, pages={46–57} }
 @article{turnbull_ghosal_zhang_2013, title={Iterative selection using orthogonal regression techniques}, volume={6}, ISSN={1932-1864}, url={http://dx.doi.org/10.1002/SAM.11212}, DOI={10.1002/SAM.11212}, abstractNote={Abstract}, number={6}, journal={Statistical Analysis and Data Mining}, publisher={Wiley}, author={Turnbull, Bradley and Ghosal, Subhashis and Zhang, Hao Helen}, year={2013}, month={Dec}, pages={557–564} }
 @article{belitser_ghosal_zanten_2012, title={OPTIMAL TWO-STAGE PROCEDURES FOR ESTIMATING LOCATION AND SIZE OF THE MAXIMUM OF A MULTIVARIATE REGRESSION FUNCTION}, volume={40}, ISSN={["0090-5364"]}, DOI={10.1214/12-AOS1053}, abstractNote={We propose a two-stage procedure for estimating the location μ and size M of the maximum of a smooth d -variate regression function f . In the first stage, a preliminary estimator of μ obtained from a standard nonparametric smoothing method is used. At the second stage, we "zoom-in" near the vicinity of the preliminary estimator and make further observations at some design points in that vicinity. We fit an appropriate polynomial regression model to estimate the location and size of the maximum. We establish that, under suitable smoothness conditions and appropriate choice of the zooming, the second stage estimators have better convergence rates than the corresponding first stage estimators of μ and M . More specifically, for α -smooth regression functions, the optimal nonparametric rates n −(α−1)/(2α+d) and n −α/(2α+d) at the first stage can be improved to n −(α−1)/(2α) and n −1/2 , respectively, for α>1+ 1+d/2 − − − − − − √ . These rates are optimal in the class of all possible sequential estimators. Interestingly, the two-stage procedure resolves "the curse of the dimensionality" problem to some extent, as the dimension d does not control the second stage convergence rates, provided that the function class is sufficiently smooth. We consider a multi-stage generalization of our procedure that attains the optimal rate for any smoothness level α>2 starting with a preliminary estimator with any power-law rate at the first stage.}, number={6}, journal={ANNALS OF STATISTICS}, author={Belitser, Eduard and Ghosal, Subhashis and Zanten, Harry}, year={2012}, month={Dec}, pages={2850–2876} }
 @article{wu_ghosal_2010, title={The <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math>-consistency of Dirichlet mixtures in multivariate Bayesian density estimation}, volume={101}, ISSN={0047-259X}, url={http://dx.doi.org/10.1016/j.jmva.2010.06.012}, DOI={10.1016/j.jmva.2010.06.012}, abstractNote={Density estimation, especially multivariate density estimation, is a fundamental problem in nonparametric inference. In the Bayesian approach, Dirichlet mixture priors are often used in practice for such problems. However, the asymptotic properties of such priors have only been studied in the univariate case. We extend the L1-consistency of Dirichlet mixutures in the multivariate density estimation setting. We obtain such a result by showing that the Kullback–Leibler property of the prior holds and that the size of the sieve in the parameter space in terms of L1-metric entropy is not larger than the order of n. However, it seems that the usual technique of choosing a sieve by controlling prior probabilities is unable to lead to a useful bound on the metric entropy required for the application of a general posterior consistency theorem for the multivariate case. We overcome this difficulty by using a structural property of Dirichlet mixtures. Our results apply to a multivariate normal kernel even when the multivariate normal kernel has a general variance–covariance matrix.}, number={10}, journal={Journal of Multivariate Analysis}, publisher={Elsevier BV}, author={Wu, Yuefeng and Ghosal, Subhashis}, year={2010}, month={Nov}, pages={2411–2419} }
 @article{gu_ghosal_2009, title={Bayesian ROC curve estimation under binormality using a rank likelihood}, volume={139}, ISSN={0378-3758}, url={http://dx.doi.org/10.1016/j.jspi.2008.09.014}, DOI={10.1016/j.jspi.2008.09.014}, abstractNote={There are various methods to estimate the parameters in the binormal model for the ROC curve. In this paper, we propose a conceptually simple and computationally feasible Bayesian estimation method using a rank-based likelihood. Posterior consistency is also established. We compare the new method with other estimation methods and conclude that our estimator generally performs better than its competitors.}, number={6}, journal={Journal of Statistical Planning and Inference}, publisher={Elsevier BV}, author={Gu, Jiezhun and Ghosal, Subhashis}, year={2009}, month={Jun}, pages={2076–2083} }
 @article{roy_ghosal_rosenberger_2009, title={Convergence properties of sequential Bayesian D-optimal designs}, volume={139}, ISSN={0378-3758}, url={http://dx.doi.org/10.1016/j.jspi.2008.04.025}, DOI={10.1016/j.jspi.2008.04.025}, abstractNote={We establish convergence properties of sequential Bayesian optimal designs. In particular, for sequential D-optimality under a general nonlinear location-scale model for binary experiments, we establish posterior consistency, consistency of the design measure, and the asymptotic normality of posterior following the design. We illustrate our results in the context of a particular application in the design of phase I clinical trials, namely a sequential design of Haines et al. [2003. Bayesian optimal designs for phase I clinical trials. Biometrics 59, 591--600] that incorporates an ethical constraint on overdosing.}, number={2}, journal={Journal of Statistical Planning and Inference}, publisher={Elsevier BV}, author={Roy, Anindya and Ghosal, Subhashis and Rosenberger, William F.}, year={2009}, month={Feb}, pages={425–440} }
 @article{gu_ghosal_roy_2008, title={Bayesian bootstrap estimation of ROC curve}, volume={27}, ISSN={0277-6715 1097-0258}, url={http://dx.doi.org/10.1002/sim.3366}, DOI={10.1002/sim.3366}, abstractNote={Abstract}, number={26}, journal={Statistics in Medicine}, publisher={Wiley}, author={Gu, Jiezhun and Ghosal, Subhashis and Roy, Anindya}, year={2008}, month={Nov}, pages={5407–5420} }
 @article{gu_ghosal_2008, title={Strong approximations for resample quantile processes and application to ROC methodology}, volume={20}, ISSN={1048-5252 1029-0311}, url={http://dx.doi.org/10.1080/10485250801954128}, DOI={10.1080/10485250801954128}, abstractNote={Abstract The receiver operating characteristic (ROC) curve is defined as true positive rate versus false positive rate obtained by varying a decision threshold criterion. It has been widely used in medical sciences for its ability to measure the accuracy of diagnostic or prognostic tests. Mathematically speaking, ROC curve is the composition of survival function of one population with the quantile function of another population. In this paper, we study strong approximation for the quantile processes of the Bayesian bootstrap (BB) resampling distributions, and use this result to study strong approximations for the BB version of the ROC process in terms of two independent Kiefer processes. The results imply asymptotically accurate coverage probabilities for the confidence bands for the ROC curve and confidence intervals for the area under the curve functional of the ROC constructed using the BB method. Similar results follow for the bootstrap resampling distribution.}, number={3}, journal={Journal of Nonparametric Statistics}, publisher={Informa UK Limited}, author={Gu, Jiezhun and Ghosal, Subhashis}, year={2008}, month={Apr}, pages={229–240} }
 @article{tang_ghosal_2007, title={A consistent nonparametric Bayesian procedure for estimating autoregressive conditional densities}, volume={51}, ISSN={0167-9473}, url={http://dx.doi.org/10.1016/j.csda.2006.06.020}, DOI={10.1016/j.csda.2006.06.020}, abstractNote={This article proposes a Bayesian infinite mixture model for the estimation of the conditional density of an ergodic time series. A nonparametric prior on the conditional density is described through the Dirichlet process. In the mixture model, a kernel is used leading to a dynamic nonlinear autoregressive model. This model can approximate any linear autoregressive model arbitrarily closely while imposing no constraint on parameters to ensure stationarity. We establish sufficient conditions for posterior consistency in two different topologies. The proposed method is compared with the mixture of autoregressive model [Wong and Li, 2000. On a mixture autoregressive model. J. Roy. Statist. Soc. Ser. B 62(1), 91–115] and the double-kernel local linear approach [Fan et al., 1996. Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika 83, 189–206] by simulations and real examples. Our method shows excellent performances in these studies.}, number={9}, journal={Computational Statistics & Data Analysis}, publisher={Elsevier BV}, author={Tang, Yongqiang and Ghosal, Subhashis}, year={2007}, month={May}, pages={4424–4437} }
 @article{choudhuri_ghosal_roy_2007, title={Nonparametric binary regression using a Gaussian process prior}, volume={4}, ISSN={1572-3127}, url={http://dx.doi.org/10.1016/j.stamet.2006.07.003}, DOI={10.1016/j.stamet.2006.07.003}, abstractNote={The article describes a nonparametric Bayesian approach to estimating the regression function for binary response data measured with multiple covariates. A multiparameter Gaussian process, after some transformation, is used as a prior on the regression function. Such a prior does not require any assumptions like monotonicity or additivity of the covariate effects. However, additivity, if desired, may be imposed through the selection of appropriate parameters of the prior. By introducing some latent variables, the conditional distributions in the posterior may be shown to be conjugate, and thus an efficient Gibbs sampler to compute the posterior distribution may be developed. A hierarchical scheme to construct a prior around a parametric family is described. A robustification technique to protect the resulting Bayes estimator against miscoded observations is also designed. A detailed simulation study is conducted to investigate the performance of the proposed methods. We also analyze some real data using the methods developed in this article.}, number={2}, journal={Statistical Methodology}, publisher={Elsevier BV}, author={Choudhuri, Nidhan and Ghosal, Subhashis and Roy, Anindya}, year={2007}, month={Apr}, pages={227–243} }
 @article{tang_ghosal_2007, title={Posterior consistency of Dirichlet mixtures for estimating a transition density}, volume={137}, ISSN={0378-3758}, url={http://dx.doi.org/10.1016/j.jspi.2006.03.007}, DOI={10.1016/j.jspi.2006.03.007}, abstractNote={The Dirichlet process mixture of normal densities has been successfully used as a prior for Bayesian density estimation for independent and identically distributed (i.i.d.) observations. A Markov model, which generalizes the i.i.d. set up, may be thought of as a suitable framework for observations arising over time. The predictive density of the future observation is then given by the posterior expectation of the transition density given the observations. We consider a Dirichlet process mixture prior for the transition density and study posterior consistency. Like the i.i.d. case, posterior consistency is obtained if the Kullback–Leibler neighborhoods of the true transition density receive positive prior probabilities and uniformly exponentially consistent tests exist for testing the true density against the complement of its neighborhoods. We show that under reasonable conditions, the Kullback–Leibler property holds for the Dirichlet mixture prior. For certain topologies on the space of transition densities, we show consistency holds under appropriate conditions by constructing the required tests. This approach, however, may not always lead to the best possible results. By modifying a recent approach of Walker [2004. New approaches to Bayesian consistency. Ann. Statist. 32, 2028–2043] for the i.i.d. case, we also show that better conditions for consistency can be given for certain weaker topologies.}, number={6}, journal={Journal of Statistical Planning and Inference}, publisher={Elsevier BV}, author={Tang, Yongqiang and Ghosal, Subhashis}, year={2007}, month={Jun}, pages={1711–1726} }