High-Dimensional Inference via Adaptive Bayes

• Activity Number: 471 - Advances in High-Dimensional Inference and Multiple Testing
• Type: Contributed
• Date/Time: Wednesday, July 31, 2019 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Learning and Data Science
• Author(s): Jiapeng Liu* and Yixuan Qiu and Xiao Wang
• Companies: Purdue Unversity and Carnegie Mellon University and Purdue University
• Keywords: High Dimensional statistics; Inference; Deep learning; Dimension Reduction

Abstract

Inference on a high dimensional regression problem is challenging. The Bayesian framework needs to specify sparsity priors for the regression coefficients while the frequentist framework studies the post selection inference. In this paper, we develop a new framework, called Adaptive Bayes, for inference on high dimensional linear regression. In particular, we first transform the regression problem into the Many-Normal-Means problem. We then propose a novel deep neural network architecture to adaptively learn both the prior and the posterior distributions. This framework allows to specify flexible and proper prior distributions. Finally, inference on the high dimensional regression coefficients can be completed by solving an L-0 inverse regularization problem. Numerical analysis demonstrates that our approach outperforms other methods for both point estimation and uncertainty quantification.