A Bayesian approach for envelope models

Kshitij Khare, Subhadip Pal, Zhihua Su

Research output: Contribution to journalArticlepeer-review

27 Citations (Scopus)


The envelope model is a new paradigm to address estimation and prediction inmultivariate 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.

Original languageEnglish
Pages (from-to)196-222
Number of pages27
JournalAnnals of Statistics
Issue number1
Publication statusPublished - Feb 2017
Externally publishedYes


  • Envelope model
  • Gibbs sampling
  • Matrix Bingham distribution
  • Stiefel manifold
  • Sufficient dimension reduction

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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