Two-step Bayesian methods for generalized regression driven by partial differential equations

Abstract

In certain non-linear regression models, the functional form of the regression function is not explicitly available, but is only described by a set of differential equations. For regression models described by a set of ordinary differential equations (ODEs), both Bayesian and non-Bayesian methods for inference were developed in the literature. In this paper, we consider a Bayesian approach to non-linear regression with respect to a multidimensional predictor variable given by a set of partial differential equations (PDEs). We consider a computationally convenient two-step approach by first representing the functions nonparametrically, constructing a finite random series prior using tensor products of B-splines and directly inducing a posterior distribution on parameter space through an appropriate projection map. By considering three different choices of the projection map, we propose three different approaches with their merits. We allow generalized non-linear regression with the response variable following an exponential family of distributions, extending the method beyond regression with additive normal errors. We establish Bernstein-von Mises type theorems which show $\sqrt{n}$-consistency and asymptotically correct frequentist coverage of Bayesian credible regions. We also conduct a simulation study to evaluate the finite sample performances of the proposed methods.