Bayesian Inference for High-Dimensional Models: Convergence Properties and Computational Issues

• Activity Number: 235
• Type: Topic Contributed
• Date/Time: Monday, August 10, 2015 : 2:00 PM to 3:50 PM
• Sponsor: International Society for Bayesian Analysis (ISBA)
• Author(s): Sayantan Banerjee* and Subhashis Ghoshal
• Companies: MD Anderson Cancer Center and North Carolina State University
• Keywords: Precision matrix ; Posterior convergence rate ; Laplace approximation ; Graphical models

Abstract

We consider Bayesian estimation of a sparse precision matrix and discuss two approaches for estimating the same in a high dimensional Gaussian model, one based on the so called graphical Wishart prior, and the other based on a Bayesian analog of the graphical lasso. In the former case, exploiting the conjugacy of the graphical Wishart prior, we can compute the posterior mean and study its performance by simulation. For nearly banded true precision matrices, we obtain the convergence rate of the Bayesian procedure under an operator norm. In the latter case, lack of a conjugacy structure poses high challenge for Bayesian computation, since the standard approach using MCMC is not practically feasible in high dimensions. We derive the posterior convergence rate at a sparse true precision matrix under the Frobenius norm, and show that it agrees with the oracle convergence rate. We device an approximate computing technique based on Laplace approximation avoiding MCMC methods completely. We also discuss the computing technique based on Laplace approximations for learning relations with predictors explaining the mean in case of generalized additive partial linear models.