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We propose a kernel mixture of polynomials prior for Bayesian nonparametric regression. The regression function is modeled by local average of polynomials with kernel weights. We obtain the minimax-optimal rate of contraction 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. These results are based on the development of convergence theory for the kernel mixture of polynomials. The proposed method is illustrated with numerical examples and shows superior performance in terms of accuracy and uncertainty quantification compared to the local polynomial regression, DiceKriging, and the robust Gaussian stochastic process.

Adaptive Bayesian Nonparametric Regression Using a Kernel Mixture of Polynomials with Application to Partial Linear Models