  We propose a kernel mixture of polynomials prior for Bayesian nonparametric
regression. The regression function is modeled by local averages of polynomials
with kernel mixture weights. We obtain the minimax-optimal rate of contraction
of the full posterior distribution up to a logarithmic factor that adapts to
the smoothness level of the true function by estimating metric entropies of
certain function classes. We also provide a frequentist sieve maximum
likelihood estimator with a near-optimal convergence rate. We further
investigate the application of the kernel mixture of polynomials to the partial
linear model and obtain both the near-optimal rate of contraction for the
nonparametric component and the Bernstein-von Mises limit (i.e., asymptotic
normality) of the parametric component. The proposed method is illustrated with
numerical examples and shows superior performance in terms of computational
efficiency, accuracy, and uncertainty quantification compared to the local
polynomial regression, DiceKriging, and the robust Gaussian stochastic process.
