A Comprehensive Bayesian Framework for Envelope Models

The envelope model aims to increase efficiency in multivariate analysis by using dimension reduction techniques. It has been used in many contexts including linear regression, generalized linear models, matrix/tensor variate regression, reduced rank regression, and quantile regression, and has shown the potential to provide substantial efficiency gains. Virtually all of these advances, however, have been made from a frequentist perspective, and the literature addressing envelope models from a Bayesian point of view is sparse. The objective of this article is to propose a Bayesian framework that is applicable across various envelope model contexts. The proposed framework aids straightforward interpretation of model parameters and allows easy incorporation of prior information. We provide a simple block Metropolis-within-Gibbs MCMC sampler for practical implementations of our method. Simulations and data examples are included for illustration. Supplementary materials for this article are available online.

Automated summary

The envelope model aims to increase efficiency in multivariate analysis by using dimension reduction techniques. However, most of the existing research on envelope models has been done from a frequentist perspective, with limited literature from a Bayesian point of view. This article proposes a Bayesian framework that can be applied to various envelope model contexts, allowing for straightforward interpretation of model parameters and easy incorporation of prior information. A practical MCMC sampler is provided for implementing the proposed method, and simulations and data examples are included for illustration.