In nonparametric and high-dimensional statistical models, the classical Gauss-Fisher-Le Cam theory of the optimality of maximum likelihood and Bayesian posterior inference does not apply, and new foundations and ideas have been developed in the past several decades. This book gives a coherent account of the statistical theory in infinite-dimensional parameter spaces. The mathematical foundations include self-contained 'mini-courses' on the theory of Gaussian and empirical processes, on approximation and wavelet theory and on the basic theory of function spaces. The theory of statistical inference in such models-hypothesis testing, estimation and confidence sets-is then presented within the minimax paradigm of decision theory. This includes the basic theory of convolution kernel and projection estimation, as well as Bayesian nonparametrics and nonparametric maximum likelihood estimation. In the final chapter, the theory of adaptive inference in nonparametric models is developed, including Lepski's method, wavelet thresholding and adaptive confidence regions for self-similar functions. EVARIST GINÉ (1944-2015) was the head of the Department of Mathematics at the University of Connecticut. Giné was a distinguished mathematician who worked on mathematical statistics and probability in infinite dimensions. He was the author of two books and more than a hundred articles.

Given a sample from some unknown continuous density $f:\mathbb{R}\to\mathbb{R}$, we construct adaptive confidence bands that are honest for all densities in a "generic" subset of the union of $t$-H\"older balls, $0

We continue the investigation of Bernstein-von Mises theorems for nonparametric Bayes procedures from [Ann. Statist. 41 (2013-2028. We introduce multiscale spaces on which nonparametric priors and posteriors are naturally defined, and prove Bernstein-von Mises theorems for a variety of priors in the setting of Gaussian nonparametric regression and in the i.i.d. sampling model. From these results we deduce several applications where posterior-based inference coincides with efficient frequentist procedures, including Donskerand Kolmogorov-Smirnov theorems for the random posterior cumulative distribution functions. We also show that multiscale posterior credible bands for the regression or density function are optimal frequentist confidence bands.1. Introduction. The Bernstein-von Mises (BvM) theorem constitutes a powerful and precise tool to study Bayes procedures from a frequentist point of view. It gives universal conditions on the prior under which the posterior distribution has the approximate shape of a normal distribution. The theorem is well understood in finite-dimensional models (see [30] and [35]), but involves some delicate conceptual and mathematical issues in the infinite-dimensional setting. There exists a Donsker-type BvM theorem for the cumulative distribution function based on Dirichlet process priors, see Lo [31], and this carries over to a variety of closely related nonparametric situations, including quantile inference and censoring models, where Bernsteinvon Mises results are available: see [8,9,21,26,27] and [22]. The proofs of these results rely on a direct analysis of the posterior distribution, which is explicitly given in these settings (and typically of Dirichlet form).

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 − α frequentist coverage level and whose L 2 -diameter shrinks at the minimax rate of convergence (within logarithmic factors) over Hölder 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.

For M a simple surface, the nonlinear statistical inverse problem of recovering a matrix field normalΦ:M→frakturso()n from discrete, noisy measurements of the SO(n)‐valued scattering data CΦ of a solution of a matrix ODE is considered (n ≥ 2). Injectivity of the map Φ ↦ CΦ was established by Paternain, Salo, and Uhlmann in 2012. A statistical algorithm for the solution of this inverse problem based on Gaussian process priors is proposed, and it is shown how it can be implemented by infinite‐dimensional MCMC methods. It is further shown that as the number N of measurements of point evaluations of CΦ increases, the statistical error in the recovery of Φ converges to 0 in L2(M)‐distance at a rate that is algebraic in 1/N and approaches 1/N for smooth matrix fields Φ. The proof relies, among other things, on a new stability estimate for the inverse map CΦ → Φ. Key applications of our results are discussed in the case n = 3 to polarimetric neutron tomography. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC