Adaptive Bayesian Inference in Nonlinear Regression Models

Abstract

There has been continuing interest in Bayesian regressions without imposing any parametric assumption on the error distribution, but the asymptotic efficiency of such procedures has not been fully understood yet. In this article, we consider semiparametric Bayesian nonlinear regression models. We do not impose a parametric form for the likelihood function; rather, we treat the true density function of error terms as an infinite dimensional nuisance parameter and estimate it nonparametrically. Thereafter, we conduct conventional parametric Bayesian inference using MCMC methods. We derive the asymptotic properties of the resulting estimator and identify conditions of adaptive estimation, under which our two-step Bayes estimator enjoys the same asymptotic normality as if we knew the true density. We compare accuracy and coverage of the adaptive Bayesian estimator with the maximum likelihood estimator in empirical studies on simulated and real data. In particular, we observe that the Bayesian inference may be superior in numerical stability for small sample sizes.