Nonparametric Bernstein-von Mises theorems in Gaussian white noise

TL;DR

This paper establishes nonparametric Bernstein-von Mises theorems in Gaussian white noise models, showing that Bayesian credible sets can serve as efficient frequentist confidence sets with optimal convergence properties.

Contribution

It proves Bernstein-von Mises theorems for a broad class of nonparametric priors, justifying Bayesian methods as asymptotically optimal frequentist procedures in Gaussian white noise models.

Findings

Bayesian credible sets achieve asymptotic $1-eta$ coverage.

Credible sets have $L^2$-diameter shrinking at minimax rate.

Results apply to nonconjugate priors on general orthonormal bases.

Abstract

Bernstein-von Mises theorems for nonparametric Bayes priors in the Gaussian white noise model are proved. It is demonstrated how such results justify Bayes methods as efficient frequentist inference procedures in a variety of concrete nonparametric problems. Particularly Bayesian credible sets are constructed that have asymptotically exact $1-\alpha$ frequentist coverage level and whose $L^2$-diameter shrinks at the minimax rate of convergence (within logarithmic factors) over H\"{o}lder balls. Other applications include general classes of linear and nonlinear functionals and credible bands for auto-convolutions. The assumptions cover nonconjugate product priors defined on general orthonormal bases of $L^2$ satisfying weak conditions.