On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures

By I. Castillo, Richard Nickl. 2013

137 citations recorded by the source corpus.

We continue the investigation of Bernstein-von Mises theorems for nonparametric Bayes procedures from [Ann. Statist. 41 (2013) 1999-2028]. 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 Donsker- and 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.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/29a981d19f1348d395bdb5e9ab999d5d2bcd5217

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Citations in the corpus

On Bayesian supremum norm contraction rates (2013, methodology): “In [7], we use the results of the present paper to derive nonparametric Bernstein–von Mises theorems in density estimation, as well as Donsker-type results for the posterior distribution function.”
Modern regularization methods for inverse problems (2018, methodology): “an approach to inverse problems, which does not only deal with point estimates, but analogous question for the full posterior distributions. This topic is beyond the scope of this survey, we refer to [234, 296, 344, 248, 105, 106, 239, 73, 297] for further details. 19 5 Variational Regularization Me”
Uncertainty Quantification for the Horseshoe (with Discussion) (2016, methodology): “…Bayesian methods for uncertainty quantification for the estimation of functions has been previously considered in Szabó et al. (2015a,b); Serra and Krivobokova (2017); Castillo and Nickl (2014); Ray (2014); Sniekers and van der Vaart (2015a,c,b); Belitser (2017); Rousseau and Szabo (2016).”
Frequentist coverage and sup-norm convergence rate in Gaussian process regression (2017, background): “A similar result for the stronger L ∞ -norm can be found in [11].”
Bayesian Probabilistic Numerical Methods (2017, background): “The consistency and robustness of Bayesian inference procedures — particularly with respect to perturbations of the prior such as those arising from numerical approximations — in such settings is a subtle topic, with both positive (Castillo and Nickl, 2014; Doob, 1949; Kleijn and van der Vaart, 2012”
Adaptive Bernstein–von Mises theorems in Gaussian white noise (2014, methodology): “See Giné and Nickl [21] and also Bull [6] for further discussion about this condition.”
Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions (2015, unknown): “[7, 8] for the diffusion model considered here.”
Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors (2020, methodology): “We follow ideas from [7, 9, 41, 43] and prove a Bernstein-von Mises theorem by proving convergence of the moment generating functions (Laplace transforms) of √ N ( 〈θ,ψ〉L2(Z)|DN − Ψ̂N )”
A Bernstein–von Mises theorem for smooth functionals in semiparametric models (2013, methodology): “Finally, although there is no immediate analogue of the BvM property for infinite dimensional parameters, as pointed out by Cox [16] and Freedman [19], some recent contributions have introduced possible notions of nonparametric BvM; see Castillo and Nickl [13] and also Leahu [29].”
Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression (2014, unknown): “A similar result for the stronger L∞-norm using this weak notion of BvM theorem is considered in [4].”
Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions (2018, methodology): “4 Bernstein-von Mises theorems for b We now adopt the framework of nonparametric Bernstein-von Mises theorems from [11, 12], see also the recent contributions [10, 41, 32, 33, 31].”