Showing papers by "Anirban Bhattacharya published in 2018"

Scalable Bayesian Variable Selection Using Nonlocal Prior Densities in Ultrahigh-Dimensional Settings

Abstract: Bayesian model selection procedures based on nonlocal alternative prior densities are extended to ultrahigh dimensional settings and compared to other variable selection procedures using precision-recall curves. Variable selection procedures included in these comparisons include methods based on g-priors, reciprocal lasso, adaptive lasso, scad, and minimax concave penalty criteria. The use of precision-recall curves eliminates the sensitivity of our conclusions to the choice of tuning parameters. We find that Bayesian variable selection procedures based on nonlocal priors are competitive to all other procedures in a range of simulation scenarios, and we subsequently explain this favorable performance through a theoretical examination of their consistency properties. When certain regularity conditions apply, we demonstrate that the nonlocal procedures are consistent for linear models even when the number of covariates p increases sub-exponentially with the sample size n. A model selection procedure based on Zellner's g-prior is also found to be competitive with penalized likelihood methods in identifying the true model, but the posterior distribution on the model space induced by this method is much more dispersed than the posterior distribution induced on the model space by the nonlocal prior methods. We investigate the asymptotic form of the marginal likelihood based on the nonlocal priors and show that it attains a unique term that cannot be derived from the other Bayesian model selection procedures. We also propose a scalable and efficient algorithm called Simplified Shotgun Stochastic Search with Screening (S5) to explore the enormous model space, and we show that S5 dramatically reduces the computing time without losing the capacity to search the interesting region in the model space, at least in the simulation settings considered. The S5 algorithm is available in an R package BayesS5 on CRAN.

On Statistical Optimality of Variational Bayes

Abstract: The article addresses a long-standing open problem on the justification of using variational Bayes methods for parameter estimation. We provide general conditions for obtaining optimal risk bounds for point estimates acquired from mean-field variational Bayesian inference. The conditions pertain to the existence of certain test functions for the distance metric on the parameter space and minimal assumptions on the prior. A general recipe for verification of the conditions is outlined which is broadly applicable to existing Bayesian models with or without latent variables. As illustrations, specific applications to Latent Dirichlet Allocation and Gaussian mixture models are discussed.

Signal Adaptive Variable Selector for the Horseshoe Prior

Abstract: In this article, we propose a simple method to perform variable selection as a post model-fitting exercise using continuous shrinkage priors such as the popular horseshoe prior. The proposed Signal Adaptive Variable Selector (SAVS) approach post-processes a point estimate such as the posterior mean to group the variables into signals and nulls. The approach is completely automated and does not require specification of any tuning parameters. We carried out a comprehensive simulation study to compare the performance of the proposed SAVS approach to frequentist penalization procedures and Bayesian model selection procedures. SAVS was found to be highly competitive across all the settings considered, and was particularly found to be robust to correlated designs. We also applied SAVS to a genomic dataset with more than 20,000 covariates to illustrate its scalability.

Heteroscedastic Bandits with Reneging.

Abstract: Although shown to be useful in many areas as models for solving sequential decision problems with side observations (contexts), contextual bandits are subject to two major limitations. First, they neglect user "reneging" that occurs in real-world applications. That is, users unsatisfied with an interaction quit future interactions forever. Second, they assume that the reward distribution is homoscedastic, which is often invalidated by real-world datasets, e.g., datasets from finance. We propose a novel model of "heteroscedastic contextual bandits with reneging" to overcome the two limitations. Our model allows each user to have a distinct "acceptance level," with any interaction falling short of that level resulting in that user reneging. It also allows the variance to be a function of context. We develop a UCB-type of policy, called HR-UCB, and prove that with high probability it achieves $\mathcal{O}\Big(\sqrt{{T}}\big(\log({T})\big)^{3/2}\Big)$ regret.

A Modified Sequential Probability Ratio Test

Abstract: We describe a modified sequential probability ratio test that can be used to reduce the average sample size required to perform statistical hypothesis tests at specified levels of significance and power. Examples are provided for $z$ tests, $t$ tests, and tests of binomial success probabilities. A description of a software package to implement the test designs is provided. We compare the sample sizes required in fixed design tests conducted at 5$\%$ significance levels to the average sample sizes required in sequential tests conducted at 0.5$\%$ significance levels, and we find that the two sample sizes are approximately equal.

Higher significance with smaller samples: A modified Sequential Probability Ratio Test

Abstract: We describe a modified sequential probability ratio test that can be used to reduce the average sample size required to perform statistical hypothesis tests at specified levels of significance and power. Examples are provided for Z-tests, T-Tests, and tests of binomial success probabilities. A description of a software package to implement the tests is provided. We also compare the sample sizes required in fixed design tests conducted at 5% significance levels to the average sample sizes required in sequential tests conducted at 0.5% significance levels, and find that the two sample sizes are approximately the same. This illustrates that the proposed sequential tests can provide higher levels of significance using smaller sample sizes.

Stay With Me: Lifetime Maximization Through Heteroscedastic Linear Bandits With Reneging

Abstract: Sequential decision making for lifetime maximization is a critical problem in many real-world applications, such as medical treatment and portfolio selection. In these applications, a "reneging" phenomenon, where participants may disengage from future interactions after observing an unsatisfiable outcome, is rather prevalent. To address the above issue, this paper proposes a model of heteroscedastic linear bandits with reneging, which allows each participant to have a distinct "satisfaction level," with any interaction outcome falling short of that level resulting in that participant reneging. Moreover, it allows the variance of the outcome to be context-dependent. Based on this model, we develop a UCB-type policy, namely HR-UCB, and prove that it achieves $\mathcal{O}\big(\sqrt{{T}(\log({T}))^{3}}\big)$ regret. Finally, we validate the performance of HR-UCB via simulations.

Process modeling for slope and aspect with application to elevation data maps

Abstract: Learning about the behavior of land surface gradients and, in particular, slope and aspect over a region from a dataset of levels obtained at a set of (possibly) irregularly spaced locations assumes importance in a variety of applications. A primary example considers digital terrain models for exploring roughness of land surfaces. In a geographic information system software package, gradient information is typically extracted from a digital elevation/terrain model (DEM/DTM), which usually presents the topography of the surface in terms of a set of pre-specified regular grid points, each with an assigned elevation value. That is, the DEM arises from preprocessing of an originally irregularly spaced set of elevation observations. Slope in one dimension is defined as “rise over run”. However, in two dimensions, at a given location, there is a rise over run in every direction. Then, the slope at the location is customarily taken as the maximum slope over all directions. It can be expressed as an angle whose tangent is the ratio of the rise to the run at the maximum. In practice, at each point of the grid, rise/run is obtained through comparison of the elevation at the point to that of a set of neighboring grid points, usually the eight compass neighbors, to find the maximum. Aspect is defined as the angular direction of maximum slope over the compass neighbors. We present a fully model-based approach for inference regarding slope and aspect. In particular, we define process versions of the slope and aspect over a continuous spatial domain. Modeling slopes takes us to directional derivative processes; modeling angles takes us to spatial processes for angular data. Using a stationary Gaussian process model for the elevation data, we obtain distribution theory for slope and associated aspect as well as covariance structure. Hierarchical models emerge; fitting in a Bayesian framework enables attachment of uncertainty. We illustrate with both a simulation example and a real data example using elevations from a collection of monitoring station locations in South Africa.

The Soft Multivariate Truncated Normal Distribution

Abstract: We propose a new distribution, called the soft tMVN distribution, which provides a smooth approximation to the truncated multivariate normal (tMVN) distribution with linear constraints. An efficient blocked Gibbs sampler is developed to sample from the soft tMVN distribution in high dimensions. We provide theoretical support to the approximation capability of the soft tMVN and provide further empirical evidence thereof. The soft tMVN distribution can be used to approximate simulations from a multivariate truncated normal distribution with linear constraints, or itself as a prior in shape-constrained problems.

The Soft Multivariate Truncated Normal Distribution with Applications to Bayesian Constrained Estimation

Abstract: We propose a new distribution, called the soft tMVN distribution, which provides a smooth approximation to the truncated multivariate normal (tMVN) distribution with linear constraints. An efficient blocked Gibbs sampler is developed to sample from the soft tMVN distribution in high dimensions. We provide theoretical support to the approximation capability of the soft tMVN and provide further empirical evidence thereof. The soft tMVN distribution can be used to approximate simulations from a multivariate truncated normal distribution with linear constraints, or itself as a prior in shape-constrained problems.

Bayesian Hierarchical Modeling on Covariance Valued Data

Abstract: Analysis of structural and functional connectivity (FC) of human brains is of pivotal importance for diagnosis of cognitive ability. The Human Connectome Project (HCP) provides an excellent source of neural data across different regions of interest (ROIs) of the living human brain. Individual specific data were available from an existing analysis (Dai et al., 2017) in the form of time varying covariance matrices representing the brain activity as the subjects perform a specific task. As a preliminary objective of studying the heterogeneity of brain connectomics across the population, we develop a probabilistic model for a sample of covariance matrices using a scaled Wishart distribution. We stress here that our data units are available in the form of covariance matrices, and we use the Wishart distribution to create our likelihood function rather than its more common usage as a prior on covariance matrices. Based on empirical explorations suggesting the data matrices to have low effective rank, we further model the center of the Wishart distribution using an orthogonal factor model type decomposition. We encourage shrinkage towards a low rank structure through a novel shrinkage prior and discuss strategies to sample from the posterior distribution using a combination of Gibbs and slice sampling. We extend our modeling framework to a dynamic setting to detect change points. The efficacy of the approach is explored in various simulation settings and exemplified on several case studies including our motivating HCP data. We extend our modeling framework to a dynamic setting to detect change points.