The New England Journal of Statistics in Data Science

As a prominent dimension reduction method for multivariate linear regression, the envelope model has received increased attention over the past decade due to its modeling flexibility and success in enhancing estimation and prediction efficiencies. Several enveloping approaches have been proposed in the literature; among these, the partial response envelope model [57] that focuses on only enveloping the coefficients for predictors of interest, and the simultaneous envelope model [14] that combines the predictor and the response envelope models within a unified modeling framework, are noteworthy. In this article we incorporate these two approaches within a Bayesian framework, and propose a novel Bayesian simultaneous partial envelope model that generalizes and addresses some limitations of the two approaches. Our method offers the flexibility of incorporating prior information if available, and aids coherent quantification of all modeling uncertainty through the posterior distribution of model parameters. A block Metropolis-within-Gibbs algorithm for Markov chain Monte Carlo (MCMC) sampling from the posterior is developed. The utility of our model is corroborated by theoretical results, comprehensive simulations, and a real imaging genetics data application for the Alzheimer’s Disease Neuroimaging Initiative (ADNI) study.