Fast sampling with Gaussian scale-mixture priors in high-dimensional regression

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.

Two-Stage Metropolis-Hastings for Tall Data

This paper discusses the challenges presented by tall data problems associated with Bayesian classification (specifically binary classification) and the existing methods to handle them. Current methods include parallelizing the likelihood, subsampling, and consensus Monte Carlo. A new method based on the two-stage Metropolis-Hastings algorithm is also proposed. The purpose of this algorithm is to reduce the exact likelihood computational cost in the tall data situation. 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 can be adopted into the consensus Monte Carlo framework. The two-stage method is applied to logistic regression, hierarchical logistic regression, and Bayesian multivariate adaptive regression splines.

Bayesian Semiparametric Multivariate Density Deconvolution

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 of 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 which also automatically includes the case of multivariate multiplicative measurement errors. 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.

Bayesian Low Rank and Sparse Covariance Matrix Decomposition

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.