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Search Results: 1 - 10 of 139219 matches for " Bani K. Mallick "
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Nonparametric Bayesian Approaches to Non-homogeneous Hidden Markov Models
Abhra Sarkar,Anindya Bhadra,Bani K. Mallick
Statistics , 2012,
Abstract: In this article a flexible Bayesian non-parametric model is proposed for non-homogeneous hidden Markov models. The model is developed through the amalgamation of the ideas of hidden Markov models and predictor dependent stick-breaking processes. Computation is carried out using auxiliary variable representation of the model which enable us to perform exact MCMC sampling from the posterior. Furthermore, the model is extended to the situation when the predictors can simultaneously in influence the transition dynamics of the hidden states as well as the emission distribution. Estimates of few steps ahead conditional predictive distributions of the response have been used as performance diagnostics for these models. The proposed methodology is illustrated through simulation experiments as well as analysis of a real data set concerned with the prediction of rainfall induced malaria epidemics.
Bayes Regularized Graphical Model Estimation in High Dimensions
Suprateek Kundu,Veera Baladandayuthapani,Bani K. Mallick
Statistics , 2013,
Abstract: There has been an intense development of Bayes graphical model estimation approaches over the past decade - however, most of the existing methods are restricted to moderate dimensions. We propose a novel approach suitable for high dimensional settings, by decoupling model fitting and covariance selection. First, a full model based on a complete graph is fit under novel class of continuous shrinkage priors on the precision matrix elements, which induces shrinkage under an equivalence with Cholesky-based regularization while enabling conjugate updates of entire precision matrices. Subsequently, we propose a post-fitting graphical model estimation step which proceeds using penalized joint credible regions to perform neighborhood selection sequentially for each node. The posterior computation proceeds using straightforward fully Gibbs sampling, and the approach is scalable to high dimensions. The proposed approach is shown to be asymptotically consistent in estimating the graph structure for fixed $p$ when the truth is a Gaussian graphical model. Simulations show that our approach compares favorably with Bayesian competitors both in terms of graphical model estimation and computational efficiency. We apply our methods to high dimensional gene expression and microRNA datasets in cancer genomics.
Fast sampling with Gaussian scale-mixture priors in high-dimensional regression
Anirban Bhattacharya,Antik Chakraborty,Bani K. Mallick
Statistics , 2015,
Abstract: We propose an efficient way to sample from a class of structured multivariate Gaussian distributions which routinely arise as conditional posteriors of model parameters that are assigned a conditionally Gaussian prior. The proposed algorithm only requires matrix operations in the form of matrix multiplications and linear system solutions. We exhibit that the computational complexity of the proposed algorithm grows linearly with the dimension unlike existing algorithms relying on Cholesky factorizations with cubic orders of complexity. The algorithm should be broadly applicable in settings where Gaussian scale mixture priors are used on high dimensional model parameters. We provide an illustration through posterior sampling in a high dimensional regression setting with a horseshoe prior on the vector of regression coefficients.
Bayesian Big Data Classification: A Review with Complements
Richard D. Payne,Bani K. Mallick
Statistics , 2014,
Abstract: This paper focuses on the specific problem of big data classification in a Bayesian setup using Markov Chain Monte Carlo methods. It discusses the challenges presented by the big data problems associated with classification and the existing methods to handle them. Next, a new method based on two-stage Metropolis-Hastings (MH) algorithm is proposed in this context. The purpose of of this algorithm is to reduce the exact likelihood computational cost in the big data context. In the first stage, a new proposal is tested by the approximate likelihood based model. The full likelihood based posterior computation will be conducted only if the proposal passes the first stage screening. Furthermore, this method is adopted into the consensus Monte Carlo framework. Methods are illustrated on two large datasets.
Bayesian Low Rank and Sparse Covariance Matrix Decomposition
Lin Zhang,Abhra Sarkar,Bani K. Mallick
Statistics , 2013,
Abstract: We consider the problem of estimating high-dimensional covariance matrices of a particular structure, which is a summation of low rank and sparse matrices. This covariance structure has a wide range of applications including factor analysis and random effects models. We propose a Bayesian method of estimating the covariance matrices by representing the covariance model in the form of a factor model with unknown number of latent factors. We introduce binary indicators for factor selection and rank estimation for the low rank component combined with a Bayesian lasso method for the sparse component estimation. Simulation studies show that our method can recover the rank as well as the sparsity of the two components respectively. We further extend our method to a graphical factor model where the graphical model of the residuals as well as selecting the number of factors is of interest. We employ a hyper-inverse Wishart prior for modeling decomposable graphs of the residuals, and a Bayesian graphical lasso selection method for unrestricted graphs. We show through simulations that the extended models can recover both the number of latent factors and the graphical model of the residuals successfully when the sample size is sufficient relative to the dimension.
Bayesian sparse graphical models and their mixtures using lasso selection priors
Rajesh Talluri,Veerabhadran Baladandayuthapani,Bani K. Mallick
Statistics , 2013,
Abstract: We propose Bayesian methods for Gaussian graphical models that lead to sparse and adaptively shrunk estimators of the precision (inverse covariance) matrix. Our methods are based on lasso-type regularization priors leading to parsimonious parameterization of the precision matrix, which is essential in several applications involving learning relationships among the variables. In this context, we introduce a novel type of selection prior that develops a sparse structure on the precision matrix by making most of the elements exactly zero, in addition to ensuring positive definiteness -- thus conducting model selection and estimation simultaneously. We extend these methods to finite and infinite mixtures of Gaussian graphical models for clustered data using Dirichlet process priors. We discuss appropriate posterior simulation schemes to implement posterior inference in the proposed models, including the evaluation of normalizing constants that are functions of parameters of interest which result from the restrictions on the correlation matrix. We evaluate the operating characteristics of our method via several simulations and in application to real data sets.
Bayesian Semiparametric Multivariate Density Deconvolution
Abhra Sarkar,Debdeep Pati,Bani K. Mallick,Raymond J. Carroll
Statistics , 2014,
Abstract: We consider the problem of multivariate density deconvolution when the interest lies in estimating the distribution of a vector valued random variable but precise measurements on the variable of interest are not available, observations being contaminated with additive measurement errors. The existing sparse literature on the problem assumes the density of the measurement errors to be completely known. We propose robust Bayesian semiparametric multivariate deconvolution approaches when the measurement error density is not known but replicated proxies are available for each unobserved value of the random vector. Additionally, we allow the variability of the measurement errors to depend on the associated unobserved value of the vector of interest through unknown relationships. Basic properties of finite mixture models, multivariate normal kernels and exchangeable priors are exploited in many novel ways to meet the modeling and computational challenges. Theoretical results that show the flexibility of the proposed methods are provided. We illustrate the efficiency of the proposed methods in recovering the true density of interest through simulation experiments. The methodology is applied to estimate the joint consumption pattern of different dietary components from contaminated 24 hour recalls.
Adaptive Posterior Convergence Rates in Bayesian Density Deconvolution with Supersmooth Errors
Abhra Sarkar,Debdeep Pati,Bani K. Mallick,Raymond J. Carroll
Statistics , 2013,
Abstract: Bayesian density deconvolution using nonparametric prior distributions is a useful alternative to the frequentist kernel based deconvolution estimators due to its potentially wide range of applicability, straightforward uncertainty quantification and generalizability to more sophisticated models. This article is the first substantive effort to theoretically quantify the behavior of the posterior in this recent line of research. In particular, assuming a known supersmooth error density, a Dirichlet process mixture of Normals on the true density leads to a posterior convergence rate same as the minimax rate $(\log n)^{-\eta/\beta}$ adaptively over the smoothness $\eta$ of an appropriate H\"{o}lder space of densities, where $\beta$ is the degree of smoothness of the error distribution. Our main contribution is achieving adaptive minimax rates with respect to the $L_p$ norm for $2 \leq p \leq \infty$ under mild regularity conditions on the true density. En route, we develop tight concentration bounds for a class of kernel based deconvolution estimators which might be of independent interest.
Investigating international new product diffusion speed: A semiparametric approach
Brian M. Hartman,Bani K. Mallick,Debabrata Talukdar
Statistics , 2012, DOI: 10.1214/11-AOAS519
Abstract: Global marketing managers are interested in understanding the speed of the new product diffusion process and how the speed has changed in our ever more technologically advanced and global marketplace. Understanding the process allows firms to forecast the expected rate of return on their new products and develop effective marketing strategies. The most recent major study on this topic [Marketing Science 21 (2002) 97--114] investigated new product diffusions in the United States. We expand upon that study in three important ways. (1) Van den Bulte notes that a similar study is needed in the international context, especially in developing countries. Our study covers four new product diffusions across 31 developed and developing nations from 1980--2004. Our sample accounts for about 80% of the global economic output and 60% of the global population, allowing us to examine more general phenomena. (2) His model contains the implicit assumption that the diffusion speed parameter is constant throughout the diffusion life cycle of a product. Recognizing the likely effects on the speed parameter of recent changes in the marketplace, we model the parameter as a semiparametric function, allowing it the flexibility to change over time. (3) We perform a variable selection to determine that the number of internet users and the consumer price index are strongly associated with the speed of diffusion.
Bayesian Variable Selection with Structure Learning: Applications in Integrative Genomics
Suprateek Kundu,Minsuk Shin,Yichen Cheng,Ganiraju Manyam,Bani K. Mallick,Veera Baladandayuthapani
Statistics , 2015,
Abstract: Significant advances in biotechnology have allowed for simultaneous measurement of molecular data points across multiple genomic and transcriptomic levels from a single tumor/cancer sample. This has motivated systematic approaches to integrate multi-dimensional structured datasets since cancer development and progression is driven by numerous co-ordinated molecular alterations and the interactions between them. We propose a novel two-step Bayesian approach that combines a variable selection framework with integrative structure learning between multiple sources of data. The structure learning in the first step is accomplished through novel joint graphical models for heterogeneous (mixed scale) data allowing for flexible incorporation of prior knowledge. This structure learning subsequently informs the variable selection in the second step to identify groups of molecular features within and across platforms associated with outcomes of cancer progression. The variable selection strategy adjusts for collinearity and multiplicity, and also has theoretical justifications. We evaluate our methods through simulations and apply them to a motivating genomic (DNA copy number and methylation) and transcriptomic (mRNA expression) data for assessing important markers associated with Glioblastoma progression.
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