arXiv subjects

Sourabh Bhattacharya

Publications and source records attributed to Sourabh Bhattacharya.

At least 55 records · Page 3Linked to original sources

arXiv2020

High-dimensional Asymptotic Theory of Bayesian Multiple Testing Procedures Under General Dependent Setup and Possible Misspecification

In this article, we investigate the asymptotic properties of Bayesian multiple testing procedures under general dependent setup, when the sample size and the number of hypotheses both tend to infinity. Specifically, we investigate strong consistency of the procedures and asymptotic properties of different versions of false discovery and false non-discovery rates under the high dimensional setup. We particularly focus on a novel Bayesian non-marginal multiple testing procedure and its associated error rates in this regard. Our results show that the asymptotic convergence rates of the error rates are directly associated with the Kullback-Leibler divergence from the true model, and the results hold even when the postulated class of models is misspecified. For illustration of our high-dimensional asymptotic theory, we consider a Bayesian variable selection problem in a time-varying covariate selection framework, with autoregressive response variables. We particularly focus on the setup where the number of hypotheses increases at a faster rate compared to the sample size, which is the so-called ultra-high dimensional situation.

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arXiv2018

Bayesian MISE convergence rates of Polya urn based density estimators: asymptotic comparisons and choice of prior parameters

Mixture models are well-known for their versatility, and the Bayesian paradigm is a suitable platform for mixture analysis, particularly when the number of components is unknown. Bhattacharya (2008) introduced a mixture model based on the Dirichlet process, where an upper bound on the unknown number of components is to be specified. Here we consider a Bayesian asymptotic framework for objectively specifying the upper bound, which we assume to depend on the sample size. In particular, we define a Bayesian analogue of the mean integrated squared error (Bayesian MISE), and select that form of the upper bound, and also that form of the precision parameter of the underlying Dirichlet process, for which Bayesian MISE of a specific density estimator, which is a suitable modification of the Polya-urn based prior predictive model, converges at a desired rate. As a byproduct of our approach, we investigate asymptotic choice of the precision parameter of the traditional Dirichlet process mixture model; the density estimator we consider here is a modification of the prior predictive distribution of Escobar & West (1995) associated with the Polya urn model. Various asymptotic issues related to the two aforementioned mixtures, including comparative performances, are also investigated.

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arXiv2018

On Classical and Bayesian Asymptotics in State Space Stochastic Differential Equations

In this article we investigate consistency and asymptotic normality of the maximum likelihood and the posterior distribution of the parameters in the context of state space stochastic differential equations (SDEs). We then extend our asymptotic theory to random effects models based on systems of state space SDEs, covering both independent and identical and independent but non-identical collections of state space SDEs. We also address asymptotic inference in the case of multidimensional linear random effects, and in situations where the data are available in discretized forms. It is important to note that asymptotic inference, either in the classical or in the Bayesian paradigm, has not been hitherto investigated in state space SDEs.

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arXiv2018

Tracking Mobile Intruders in an Art Gallery: Guard Deployment Strategies, Fundamental Limitations, and Performance Guarantees

This paper addresses the problem of tracking mobile intruders in a polygonal environment. We assume that a team of diagonal guards is deployed inside the polygon to provide mobile coverage. First, we formulate the problem of tracking a mobile intruder inside a polygonal environment as a multi-robot task allocation (MRTA) problem. Leveraging on guard deployment strategies in art gallery problems for mobile coverage, we show that the problem of finding the minimum speed of guards to persistently track a single mobile intruder is NP-hard. Next, for a given maximum speed of the intruder and the guards, we propose a technique to partition a polygon, and compute a feasible allocation of guards to the partitions. We prove the correctness of the proposed algorithm, and show its completeness for a specific class of inputs. We classify the guards based on the structural properties of the partitions allocated to them. Based on the classification, we propose motion strategy for the guards to track the mobile intruder when it is located in the partition allocated to the guard. Finally, we extend the proposed technique to address guard deployment and allocation strategies for non-simple polygons and multiple intruders.

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arXiv2018

Adaptive Target Tracking with a Mixed Team of Static and Mobile Guards: Deployment and Activation Strategies

This work explores a variation of the art gallery problem in which a team of static and mobile guards track a mobile intruder with unknown maximum speed. First, we present an algorithm to identify {\it candidate vertices} in a polygon at which either static guards can be placed or they can serve as endpoints of the trajectory of mobile guards. Based on the triangulation of the polygon and the deployment of the guards we propose an allocation technique for the guards, such that each one of them is assigned to guard a subregion of the environment when the intruder is inside it. The allocation strategy leads to a classification of the guards based on their task and coordination requirements. Next, we present an activation strategy for the static guards that is adaptive to the instantaneous speed of the intruder. The deployment and the activation technique guarantee that a variable speed intruder is successfully tracked. Simulation results are presented to validate the efficacy of the proposed techniques.

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arXiv2018

Non-marginal Decisions: A Novel Bayesian Multiple Testing Procedure

In this paper we consider the problem of multiple testing when the hypotheses are dependent. In most of the existing literature, either Bayesian or non-Bayesian, the decision rules mainly focus on the validity of the test procedure rather than actually utilizing the dependency to increase efficiency. Moreover, the decisions regarding different hypotheses are marginal in the sense that they do not depend upon each other directly. However, in realistic situations, the hypotheses are usually dependent, and hence it is desirable that the decisions regarding the dependent hypotheses are taken jointly. In this article we develop a novel Bayesian multiple testing procedure that coherently takes this requirement into consideration. Our method, which is based on new notions of error and non-error terms, substantially enhances efficiency by judicious exploitation of the dependence structure among the hypotheses. We prove that our method minimizes the posterior expected loss associated with a an additive "0-1" loss function, we also prove theoretical results on the relevant error probabilities, establishing the coherence and usefulness of our method. The optimal decision configuration is not available in closed form and we propose a novel and efficient simulated annealing algorithm for the purpose of optimization, which is also generically applicable to binary optimization problems. Numerical studies demonstrate that in dependent situations, our method performs significantly better than some existing popular conventional multiple testing methods, in terms of accuracy and power control. Moreover, application of our ideas to a real, spatial data set associated with radionuclide concentration in Rongelap islands yielded insightful results.

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arXiv2018

A Bayesian Semiparametric Approach to Learning About Gene-Gene Interactions in Case-Control Studies

Gene-gene interactions are often regarded as playing significant roles in influencing variabilities of complex traits. Although much research has been devoted to this area, to date a comprehensive statistical model that addresses the various sources of uncertainties, seem to be lacking. In this paper, we propose and develop a novel Bayesian semiparametric approach composed of finite mixtures based on Dirichlet processes and a hierarchical matrix-normal distribution that can comprehensively account for the unknown number of sub-populations and gene-gene interactions. Then, by formulating novel and suitable Bayesian tests of hypotheses we attempt to single out the roles of the genes, individually, and in interaction with other genes, in case-control studies. We also attempt to identify the significant loci associated with the disease. Our model facilitates a highly efficient parallel computing methodology, combining Gibbs sampling and Transformation based MCMC (TMCMC). Application of our ideas to biologically realistic data sets revealed quite encouraging performance. We also applied our ideas to a real, myocardial infarction dataset, and obtained interesting results that partly agree with, and also complement, the existing works in this area, to reveal the importance of sophisticated and realistic modeling of gene-gene interactions.

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arXiv2018

Asymptotic Theory of Bayes Factor in Stochastic Differential Equations: Part I

Research on asymptotic model selection in the context of stochastic differential equations (SDEs) is almost non-existent in the literature. In particular, when a collection of SDEs is considered, the problem of asymptotic model selection has not been hitherto investigated. Indeed, even though the diffusion coefficients may be considered known, questions on appropriate choice of the drift functions constitute a non-trivial model selection problem. In this article, we develop the asymptotic theory for comparisons between collections of SDEs with respect to the choice of drift functions using Bayes factors when the number of equations (individuals) in the collection of SDEs tend to infinity while the time domains remain bounded for each equation. Our asymptotic theory covers situations when the observed processes associated with the SDEs are independently and identically distributed (iid), as well as when they are independently but not identically distributed (non-iid). In particular, we allow incorporation of available time-dependent covariate information into each SDE through a multiplicative factor of the drift function; we also permit different initial values and domains of observations for the SDEs. Our model selection problem thus encompasses selection of a set of appropriate time-dependent covariates from a set of available time-dependent covariates, besides selection of the part of the drift function free of covariates. For both iid and non-iid set-ups we establish almost sure exponential convergence of the Bayes factor. Furthermore, we demonstrate with simulation studies that even in non-asymptotic scenarios Bayes factor successfully captures the right set of covariates.

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arXiv2018

Asymptotic Theory of Bayes Factor in Stochastic Differential Equations: Part II

The problem of model selection in the context of a system of stochastic differential equations (SDEs) has not been touched upon in the literature. Indeed, properties of Bayes factors have not been studied even in single SDE based model comparison problems. In this article, we first develop an asymptotic theory of Bayes factors when two SDEs are compared, assuming the time domain expands. Using this we then develop an asymptotic theory of Bayes factors when systems of SDEs are compared, assuming that the number of equations in each system, as well as the time domain, increase indefinitely. Our asymptotic theory covers situations when the observed processes associated with the SDEs are independently and identically distributed (iid), as well as when they are independently but not identically distributed (non-iid). Quite importantly, we allow inclusion of available time-dependent covariate information into each SDE through a multiplicative factor of the drift function in a random effects set-up; different initial values for the SDEs are also permitted. Thus, our general model-selection framework includes simultaneously the variable selection problem associated with time-varying covariates, as well as choice of the part of the drift function free of covariates. As we show, the Bayes factor is inconsistent for comparing individual SDEs, in the sense that the log-Bayes factor converges only in expectation, while the relevant variance does not converge to zero. Nevertheless, it has been possible to exploit this result to establish almost sure exponential convergence of the Bayes factor when, in addition, the number of individuals are also allowed to increase indefinitely. We carry out simulated and real data analyses to demonstrate that Bayes factor is a suitable candidate for covariate selection in our SDE models even in non-asymptotic situations.

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arXiv2018

A Short Note on Almost Sure Convergence of Bayes Factors in the General Set-Up

Although there is a significant literature on the asymptotic theory of Bayes factor, the set-ups considered are usually specialized and often involves independent and identically distributed data. Even in such specialized cases, mostly weak consistency results are available. In this article, for the first time ever, we derive the almost sure convergence theory of Bayes factor in the general set-up that includes even dependent data and misspecified models. Somewhat surprisingly, the key to the proof of such a general theory is a simple application of a result of Shalizi (2009) to a well-known identity satisfied by the Bayes factor.

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arXiv2018

Bayes meet Riemann- Bayesian Characterization of Infinite Series with Application to Riemann Hypothesis

In the classical literature on infinite series there are various tests to determine if a given infinite series converges, diverges, or oscillates. But unfortunately, for very many infinite series all the existing tests can fail to provide definitive answers. In this article we propose a novel Bayesian theory for assessment of convergence properties of any given infinite series. Remarkably, this theory attempts to provide conclusive answers to the question of convergence even where all the existing tests of convergence fail. We apply our ideas to seven different examples, obtaining very encouraging results. Importantly, we also apply our ideas to investigate the Riemann Hypothesis, and obtain results that do not completely support the conjecture. We also extend our ideas to develop a Bayesian theory on oscillating series, where we allow even infinite number of limit points. Analysis of Riemann Hypothesis using Bayesian multiple limit points theory yielded almost identical results as the Bayesian theory of convergence assessment.

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arXiv2018

A Novel Bayesian Multiple Testing Approach to Deregulated miRNA Discovery Harnessing Positional Clustering

MicroRNAs (miRNAs) are small non-coding RNAs that function as regulators of gene expression. In recent years, there has been a tremendous and growing interest among researchers to investigate the role of miRNAs in normal cellular as well as in disease processes. Thus to investigate the role of miRNAs in oral cancer, we analyse the expression levels of miRNAs to identify miRNAs with statistically significant differential expression in cancer tissues. In this article, we propose a novel Bayesian hierarchical model of miRNA expression data. Compelling evidences have demonstrated that the transcription process of miRNAs in human genome is a latent process instrumental for the observed expression levels. We take into account positional clustering of the miRNAs in the analysis and model the latent transcription phenomenon nonparametrically by an appropriate Gaussian process. For the testing purpose we employ a novel Bayesian multiple testing method where we mainly focus on utilizing the dependence structure between the hypotheses for better results, while also ensuring optimality in many respects. Indeed, our non-marginal method yielded results in accordance with the underlying scientific knowledge which are found to be missed by the very popular Benjamini-Hochberg method.

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arXiv2017

On Asymptotic Inference in Stochastic Differential Equations with Time-Varying Covariates

In this article, we introduce a system of stochastic differential equations (SDEs) consisting of time-dependent covariates and consider both fixed and random effects set-ups. We also allow the functional part associated with the drift function to depend upon unknown parameters. In this general set-up of SDE system we establish consistency and asymptotic normality of the M LE through verification of the regularity conditions required by existing relevant theorems. Besides, we consider the Bayesian approach to learning about the population parameters, and prove consistency and asymptotic normality of the corresponding posterior distribution. We supplement our theoretical investigation with simulated and real data analyses, obtaining encouraging results in each case.

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arXiv2017

Gaussian Random Functional Dynamic Spatio-Temporal Modeling of Discrete Time Spatial Time Series Data

Discrete time spatial time series data arise routinely in meteorological and environmental studies. Inference and prediction associated with them are mostly carried out using any of the several variants of the linear state space model that are collectively called linear dynamic spatio-temporal models (LDSTMs). However, real world environmental processes are highly complex and are seldom representable by models with such simple linear structure. Hence, nonlinear dynamic spatio-temporal models (NLDSTMs) based on the idea of nonlinear observational and evolutionary equation have been proposed as an alternative. However, in that case, the caveat lies in selecting the specific form of nonlinearity from a large class of potentially appropriate nonlinear functions. Moreover, modeling by NLDSTMs requires precise knowledge about the dynamics underlying the data. In this article, we address this problem by introducing the Gaussian random functional dynamic spatio-temporal model (GRFDSTM). Unlike the LDSTMs or NLDSTMs, in GRFDSTM both the functions governing the observational and evolutionary equations are composed of Gaussian random functions. We exhibit many interesting theoretical properties of the GRFDSTM and demonstrate how model fitting and prediction can be carried out coherently in a Bayesian framework. We also conduct an extensive simulation study and apply our model to a real, SO2 pollution data over Europe. The results are highly encouraging.

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arXiv2017

A Brief Review of Optimal Scaling of the Main MCMC Approaches and Optimal Scaling of Additive TMCMC Under Non-Regular Cases

Very recently, Transformation based Markov Chain Monte Carlo (TMCMC) was proposed by Dutta and Bhattcharya (2013) as a much efficient alternative to the Metropolis-Hastings algorithm, Random Walk Metropolis (RWM) algorithm, especially in high dimensions. The main advantage of this algorithm is that it simultaneously updates all components of a high dimensional parameter by some appropriate deterministic transformation of a single random variable, thereby reducing time complexity and enhancing the acceptance rate. The optimal scaling of the additive TMCMC approach has already been studied for the Gaussian proposal density by Dey and Bhattacharya(2013). In this paper, we discuss diffusion-based optimal scaling behavior for non-Gaussian proposal densities - in particular, uniform, Student's t and Cauchy proposals. We also consider diffusion based optimal scaling for non-Gaussian proposals when the target density is discontinuous. In the case of the Random Walk metropolis (RWM) algorithm these non-regular situations have been studied by Neal and Roberts (2011) in terms of expected squared jumping distance (ESJD), but the diffusion based approach has not been considered. Although we could not formally prove our diffusion result for the Cauchy proposal, simulation based results led us to a conjecture that the diffusion result still holds for the Cauchy case. We compare our diffusion based TMCMC approach with that of ESJD based RWM approach for the very challenging Cauchy proposal case, showing that our former approach clearly outperforms the latter.

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arXiv2017

Effects of Gene-Environment and Gene-Gene Interactions in Case-Control Studies: A Novel Bayesian Semiparametric Approach

Cognizance of gene-environment interactions may help prevent or detain the onset of complex diseases like cardiovascular disease, cancer, type2 diabetes, autism or asthma by adjustments to lifestyle. In this regard, we extend the Bayesian semiparametric gene-gene interaction model of Bhattacharya & Bhattacharya (2015) to include the possibility of influencing gene-gene interactions by environmental variables and possible mutations caused by the environment. Our model accounts for the unknown number of genetic sub-populations via finite mixtures composed of Dirichlet processes, which are related to each other through a hierarchical matrix normal structure responsible for inducing gene-gene interactions and possible mutations in association with environmental variables. We also extend the Bayesian hypotheses testing procedures of Bhattacharya & Bhattacharya (2015) to detect the roles of genes and their interactions, environment and the influence of environment on gene-gene interactions, in case-control studies. We develop an effective parallel computing methodology, which harnesses the power of parallel processing technology to the efficiencies of our conditionally independent Gibbs sampling and Transformation based MCMC (TMCMC) methods.Applications of our model and methods to simulation studies with biologically realistic case-control genotype datasets obtained under five distinct set-ups yield encouraging results in each case. We followed these up by application of our ideas to a real, case-control based genotype dataset on early onset of myocardial infarction. Beside being in broad agreement with the reported literature on this dataset, the results obtained give some interesting insights to the differential effect of gender on MI.

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arXiv2017

A Statistical Perspective on Inverse and Inverse Regression Problems

Inverse problems, where in broad sense the task is to learn from the noisy response about some unknown function, usually represented as the argument of some known functional form, has received wide attention in the general scientific disciplines. How- ever, in mainstream statistics such inverse problem paradigm does not seem to be as popular. In this article we provide a brief overview of such problems from a statistical, particularly Bayesian, perspective. We also compare and contrast the above class of problems with the perhaps more statistically familiar inverse regression problems, arguing that this class of problems contains the traditional class of inverse problems. In course of our review we point out that the statistical literature is very scarce with respect to both the inverse paradigms, and substantial research work is still necessary to develop the fields.

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arXiv2017

Transdimensional Transformation based Markov Chain Monte Carlo

In this article, we propose a novel and general dimension-hopping MCMC methodology that can update all the parameters as well as the number of parameters simultaneously using simple deterministic transformations of some low-dimensional (often one-dimensional) random variable. This methodology, which has been inspired by the recent Transformation based MCMC (TMCMC) for updating all the parameters simultaneously in general fixed-dimensional set-ups using low-dimensional random variables, facilitates great speed in terms of computation time and provides high acceptance rates, thanks to the low-dimensional random variables which effectively reduce the dimension dramatically. Quite importantly, our transformation based approach provides a natural way to automate the move-types in the variable dimensional problems. We refer to this methodology as Transdimensional Transformation based Markov Chain Monte Carlo (TTMCMC). We develop the theory of TTMCMC, illustrating it with gamma and normal mixtures with unknown number of components, for both simulated and real data sets. Comparisons with RJMCMC demonstrates far superior performance of TTMCMC in terms of mixing, acceptance rate, computational speed and automation. Furthermore, we demonstrate good performance of TTMCMC in multivariate normal mixtures, even for dimension as large as 20. To our knowledge, there exists no application of RJMCMC for such high-dimensional mixtures. Further, we propose a novel methodology to summarize the posterior, providing a way to obtain the mode of the posterior distribution of the densities and the associated highest posterior density credible regions. Based on our method we also propose a criterion to assess convergence of variable-dimensional algorithms. These methods of summarization and convergence assessment are applicable to general problems, not just to mixtures.

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