On the Convergence of Bayesian Regression Models

Abstract: We consider heteroscedastic nonparametric regression models, when both the mean function and variance function are unknown and to be estimated with nonparametric approaches. We derive convergence rates of posterior distributions for this model with different priors, including splines and Gaussian process priors. The results are based on the general ones on the rates of convergence of posterior distributions for independent, non-identically distributed observations, and are established for both of the cases with random covariates, and deterministic covariates. We also illustrate that the results can be achieved for all levels of regularity, which means they are adaptive.