A hierarchical Bayesian regression model for the uncertain functional constraint using screened scale mixtures of Gaussian distributions

Abstract

This paper considers a hierarchical Bayesian analysis of regression models using a class of Gaussian scale mixtures. This class provides a robust alternative to the common use of the Gaussian distribution as a prior distribution in particular for estimating the regression function subject to uncertainty about the constraint. For this purpose, we use a family of rectangular screened multivariate scale mixtures of Gaussian distribution as a prior for the regression function, which is flexible enough to reflect the degrees of uncertainty about the functional constraint. Specifically, we propose a hierarchical Bayesian regression model for the constrained regression function with uncertainty on the basis of three stages of a prior hierarchy with Gaussian scale mixtures, referred to as a hierarchical screened scale mixture of Gaussian regression models (HSMGRM). We describe distributional properties of HSMGRM and an efficient Markov chain Monte Carlo algorithm for posterior inference, and apply the proposed model to real applications with constrained regression models subject to uncertainty.

Keywords:

• elliptically contoured distribution
• hierarchical Bayesian model
• Markov chain Monte Carlo
• rectangular screened scale mixtures
• uncertain constraint

AMS Subject Classifications:

• 62F15
• 62G05
• 62J02

Acknowledgements

We greatly appreciate all the comments and suggestions from the Editor, the Associate Editor and two referees.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Research of Hea-Jung Kim was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT, and Future Planning (NRF-2013R1A2A2A01004790). Research of Taeryon Choi was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2012R1A1A2041370).