On Bayesian estimation of regression models subject to uncertainty about functional constraints

In this paper, we provide a Bayesian estimation procedure for the regression models when the constraint of the regression function needs to be incorporated in modeling but such a restriction is uncertain. For this purpose, we consider a family of rectangle screened multivariate Gaussian prior distributions in order to reflect uncertainty about the functional constraint, and propose the Bayesian estimation procedure of the regression models based on two stages of a prior hierarchy of the functional constraint, referred to as hierarchical screened Gaussian regression models (HSGRM). Specifically, we explore theoretical properties of the proposed estimation procedure by deriving the posterior distribution and predictive distribution of the unknown parameters under HSGRM in analytic forms, and discuss specific applications to regression models with uncertain functional constraints that can be explained in the context of HSGRM.