Sourabh Bhattacharya

A Bayesian approach to modeling dynamic effective connectivity with fMRI data.

A state-space modeling approach for examining dynamic relationship between multiple brain regions was proposed in Ho, Ombao and Shumway (Ho, M.R., Ombao, H., Shumway, R., 2005. A State-Space Approach to Modelling Brain Dynamics to Appear in Statistica Sinica). Their approach assumed that the quantity representing the influence of one neuronal system over another, or effective connectivity, is time-invariant. However, more and more empirical evidence suggests that the connectivity between brain areas may be dynamic which calls for temporal modeling of effective connectivity. A Bayesian approach is proposed to solve this problem in this paper. Our approach first decomposes the observed time series into measurement error and the BOLD (blood oxygenation level-dependent) signals. To capture the complexities of the dynamic processes in the brain, region-specific activations are subsequently modeled, as a linear function of the BOLD signals history at other brain regions. The coefficients in these linear functions represent effective connectivity between the regions under consideration. They are further assumed to follow a random walk process so to characterize the dynamic nature of brain connectivity. We also consider the temporal dependence that may be present in the measurement errors. ML-II method (Berger, J.O., 1985. Statistical Decision Theory and Bayesian Analysis (2nd ed.). Springer, New York) was employed to estimate the hyperparameters in the model and Bayes factor was used to compare among competing models. Statistical inference of the effective connectivity coefficients was based on their posterior distributions and the corresponding Bayesian credible regions (Carlin, B.P., Louis, T.A., 2000. Bayes and Empirical Bayes Methods for Data Analysis (2nd ed.). Chapman and Hall, Boca Raton). The proposed method was applied to a functional magnetic resonance imaging data set and results support the theory of attentional control network and demonstrate that this network is dynamic in nature.

On geometric ergodicity of additive and multiplicative transformation-based Markov Chain Monte Carlo in high dimensions

Recently Dutta and Bhattacharya (2013) introduced a novel Markov Chain Monte Carlo methodology that can simultaneously update all the components of high dimensional parameters using simple deterministic transformations of a one-dimensional random variable drawn from any arbitrary distribution defined on a relevant support. The methodology, which the authors refer to as Transformation-based Markov Chain Monte Carlo (TMCMC), greatly enhances computational speed and acceptance rate in high-dimensional problems. Two significant transformations associated with TMCMC are additive and multiplicative transformations. Combinations of additive and multiplicative transformations are also of much interest. In this work we investigate geometric ergodicity associated with additive and multiplicative TMCMC, along with their combinations, and illustrate their efficiency in practice with simulation studies.

A novel Bayesian semiparametric algorithm for inferring population structure and adjusting for case-control association tests.

While the population-based case-control approach is the popular study design for association mapping of complex genetic traits because of ease of data collection and statistical analyses, it suffers from the inherent problem of population stratification. There have been methodological developments for adjusting these studies for population substructure, but efficient estimation of the number of subpopulations (K), which has evolutionary significance, remains a statistical challenge. In this article, we propose a Bayesian semiparametric approach to estimate population substructure under the assumption that K is random. Using extensive simulations, we find that our proposed method is not only computationally much faster than an existing Bayesian approach Structure, but also estimates the number of subpopulations more accurately, and thus, yields more power in detecting association in case-control studies.

A brief tutorial on transformation based Markov Chain Monte Carlo and optimal scaling of the additive transformation

We consider the recently introduced Transformation-based Markov Chain Monte Carlo (TMCMC) (Dutta and Bhattacharya (2014)), a methodology that is designed to update all the parameters simultaneously using some simple deterministic transformation of a onedimensional random variable drawn from some arbitrary distribution on a relevant support. The additive transformation based TMCMC is similar in spirit to random walk Metropolis, except the fact that unlike the latter, additive TMCMC uses a single draw from a onedimensional proposal distribution to update the high-dimensional parameter. In this paper, we first provide a brief tutorial on TMCMC, exploring its connections and contrasts with various available MCMC methods. Then we study the diffusion limits of additive TMCMC under various set-ups ranging from the product structure of the target density to the case where the target is absolutely continuous with respect to a Gaussian measure; we also consider the additive TMCMC within Gibbs approach for all the above set-ups. These investigations lead to appropriate scaling of the one-dimensional proposal density. We also show that the optimal acceptance rate of additive TMCMC is 0.439 under all the aforementioned set-ups, in contrast with the well-established 0.234 acceptance rate associated with optimal random walk Metropolis algorithms under the same set-ups. We also elucidate the ramifications of our results and clear advantages of additive TMCMC over random walk Metropolis with ample simulation studies and Bayesian analysis of a real, spatial dataset with which 160 unknowns are associated.

Perfect Simulation for Mixtures with Known and Unknown Number of Components

We propose and develop a novel and effective perfect sampling methodology for simulating from posteriors corresponding to mixtures with either known (fixed) or unknown number of components. For the latter we consider the Dirichlet process-based mixture model developed by these authors, and show that our ideas are applicable to conjugate, and importantly, to non-conjugate cases. As to be expected, and, as we show, perfect sampling for mixtures with known number of components can be achieved with much less effort with a simplified version of our general methodology, whether or not conjugate or non-conjugate priors are used. While no special assumption is necessary in the conjugate set-up for our theory to work, we require the assumption of bounded parameter space in the non-conjugate set-up. However, we argue, with appropriate analytical, simulation, and real data studies as support, that such boundedness assumption is not unrealistic and is not an impediment in practice. Not only do we validate our ideas theoretically and with simulation studies, but we also consider application of our proposal to three real data sets used by several authors in the past in connection with mixture models. The results we achieved in each of our experiments with either simulation study or real data application, are quite encouraging.

An efficient Bayesian meta-analysis approach for studying cross-phenotype genetic associations

Simultaneous analysis of genetic associations with multiple phenotypes may reveal shared genetic susceptibility across traits (pleiotropy). For a locus exhibiting overall pleiotropy, it is important to identify which specific traits underlie this association. We propose a Bayesian meta-analysis approach (termed CPBayes) that uses summary-level data across multiple phenotypes to simultaneously measure the evidence of aggregate-level pleiotropic association and estimate an optimal subset of traits associated with the risk locus. This method uses a unified Bayesian statistical framework based on a spike and slab prior. CPBayes performs a fully Bayesian analysis by employing the Markov Chain Monte Carlo (MCMC) technique Gibbs sampling. It takes into account heterogeneity in the size and direction of the genetic effects across traits. It can be applied to both cohort data and separate studies of multiple traits having overlapping or non-overlapping subjects. Simulations show that CPBayes can produce higher accuracy in the selection of associated traits underlying a pleiotropic signal than the subset-based meta-analysis ASSET. We used CPBayes to undertake a genome-wide pleiotropic association study of 22 traits in the large Kaiser GERA cohort and detected six independent pleiotropic loci associated with at least two phenotypes. This includes a locus at chromosomal region 1q24.2 which exhibits an association simultaneously with the risk of five different diseases: Dermatophytosis, Hemorrhoids, Iron Deficiency, Osteoporosis and Peripheral Vascular Disease. We provide an R-package ‘CPBayes’ implementing the proposed method.

Model fitting and inference under latent equilibrium processes

This paper presents a methodology for model fitting and inference in the context of Bayesian models of the type f(Y|X,θ)f(X|θ)f(θ), where Y is the (set of) observed data, θ is a set of model parameters and X is an unobserved (latent) stationary stochastic process induced by the first order transition model f(X(t+1)|X(t),θ), where X(t) denotes the state of the process at time (or generation) t. The crucial feature of the above type of model is that, given θ, the transition model f(X(t+1)|X(t),θ) is known but the distribution of the stochastic process in equilibrium, that is f(X|θ), is, except in very special cases, intractable, hence unknown. A further point to note is that the data Y has been assumed to be observed when the underlying process is in equilibrium. In other words, the data is not collected dynamically over time.We refer to such specification as a latent equilibrium process (LEP) model. It is motivated by problems in population genetics (though other applications are discussed), where it is of interest to learn about parameters such as mutation and migration rates and population sizes, given a sample of allele frequencies at one or more loci. In such problems it is natural to assume that the distribution of the observed allele frequencies depends on the true (unobserved) population allele frequencies, whereas the distribution of the true allele frequencies is only indirectly specified through a transition model.As a hierarchical specification, it is natural to fit the LEP within a Bayesian framework. Fitting such models is usually done via Markov chain Monte Carlo (MCMC). However, we demonstrate that, in the case of LEP models, implementation of MCMC is far from straightforward. The main contribution of this paper is to provide a methodology to implement MCMC for LEP models. We demonstrate our approach in population genetics problems with both simulated and real data sets. The resultant model fitting is computationally intensive and thus, we also discuss parallel implementation of the procedure in special cases.

Bayesian nonparametric dynamic state space modeling with circular latent states

State space models are well-known for their versatility in modeling dynamic systems that arise in various scientific disciplines. Although parametric state space models are well studied, nonparametric approaches are much less explored in comparison. In this article we propose a novel Bayesian nonparametric approach to state space modeling, assuming that both the observational and evolutionary functions are unknown and are varying with time; crucially, we assume that the unknown evolutionary equation describes dynamic evolution of some latent circular random variable. Based on appropriate kernel convolution of the standard Weiner process, we model the time-varying observational and evolutionary functions as suitable Gaussian processes that take both linear and circular variables as arguments. Additionally, for the time-varying evolutionary function, we wrap the Gaussian process thus constructed around the unit circle to form an appropriate circular Gaussian process. We show that our process thus created satisfies desirable properties.For the purpose of inference we develop a Markov-chain Monte Carlo (MCMC)-based methodology combining Gibbs sampling and Metropolis-Hastings algorithms. Applications to a simulated data set, a real wind speed data set, and a real ozone data set demonstrated quite encouraging performances of our model and methodologies.

A Bayesian semiparametric approach to learning about gene–gene interactions in case-control studies

ABSTRACT 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.

A novel bayesian multiple testing approach to deregulated miRNA discovery harnessing positional clustering: Multiple Testing for Deregulated miRNA Discovery

MicroRNAs (miRNAs) are small non-coding RNAs that function as regulators of gene expression. In recent years, there has been a tremendous interest among researchers to investigate the role of miRNAs in normal as well as in disease processes. To investigate the role of miRNAs in oral cancer, we analyse 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 evidence has demonstrated that the transcription process of miRNAs in the 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 purpose of testing, 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.