Open Access
February 2017 A Bayesian approach for envelope models
Kshitij Khare, Subhadip Pal, Zhihua Su
Ann. Statist. 45(1): 196-222 (February 2017). DOI: 10.1214/16-AOS1449


The envelope model is a new paradigm to address estimation and prediction in multivariate analysis. Using sufficient dimension reduction techniques, it has the potential to achieve substantial efficiency gains compared to standard models. This model was first introduced by [Statist. Sinica 20 (2010) 927–960] for multivariate linear regression, and has since been adapted to many other contexts. However, a Bayesian approach for analyzing envelope models has not yet been investigated in the literature. In this paper, we develop a comprehensive Bayesian framework for estimation and model selection in envelope models in the context of multivariate linear regression. Our framework has the following attractive features. First, we use the matrix Bingham distribution to construct a prior on the orthogonal basis matrix of the envelope subspace. This prior respects the manifold structure of the envelope model, and can directly incorporate prior information about the envelope subspace through the specification of hyperparamaters. This feature has potential applications in the broader Bayesian sufficient dimension reduction area. Second, sampling from the resulting posterior distribution can be achieved by using a block Gibbs sampler with standard associated conditionals. This in turn facilitates computationally efficient estimation and model selection. Third, unlike the current frequentist approach, our approach can accommodate situations where the sample size is smaller than the number of responses. Lastly, the Bayesian approach inherently offers comprehensive uncertainty characterization through the posterior distribution. We illustrate the utility of our approach on simulated and real datasets.


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Kshitij Khare. Subhadip Pal. Zhihua Su. "A Bayesian approach for envelope models." Ann. Statist. 45 (1) 196 - 222, February 2017.


Received: 1 July 2014; Revised: 1 January 2016; Published: February 2017
First available in Project Euclid: 21 February 2017

zbMATH: 1367.62174
MathSciNet: MR3611490
Digital Object Identifier: 10.1214/16-AOS1449

Primary: 62F15 , 62F30
Secondary: 60J20 , 62H12

Keywords: envelope model , Gibbs sampling , matrix Bingham distribution , Stiefel manifold , sufficient dimension reduction

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 1 • February 2017
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