Nonparametric Bernstein-von Mises Phenomenon: A Tuning Prior Perspective

Statistical inference on infinite-dimensional parameters in Bayesian framework is investigated. The main contribution of this paper is to demonstrate that nonparametric Bernstein-von Mises theorem can be established in a very general class of nonparametric regression models under a novel tuning prior (indexed by a non-random hyper-parameter). Surprisingly, this type of prior connects two important classes of statistical methods: nonparametric Bayes and smoothing spline at a fundamental level. The intrinsic connection with smoothing spline greatly facilitates both theoretical analysis and practical implementation for nonparametric Bayesian inference. For example, we can apply smoothing spline theory in our Bayesian setting, and can also employ generalized cross validation to select a proper tuning prior, under which credible regions/intervals are frequentist valid. A collection of probabilistic tools such as Cameron-Martin theorem and Gaussian correlation inequality are employed in this paper.