Posterior concentration for a misspecified Bayesian regression model with functional covariates

Abstract

We address a Bayesian regression model with a functional covariate and a scalar response. The model is misspecified in two ways: the posterior distribution is calculated by using normal errors and, mostly, by restraining the functional regression coefficient to a possibly small subset of L2([0,1]). A first general concentration result shows that the posterior distribution concentrates within Kullback–Leibler type neighborhoods of a set called the asymptotic carrier. This result is valid whether the asymptotic carrier is empty or not and includes the misspecified and the well specified cases. We focus on the particular misspecified case in which the functional regression parameter is restrained to a union of finite dimensional linear spaces. This restriction is motivated by the practical situation in which the functional regression parameter is expressed by a finite combination of B-splines with free knots. We provide sufficient conditions for the posterior concentration with respect to the norm of the parameter space along with a precise description of the asymptotic carrier.