Adaptive Bernstein–von Mises theorems in Gaussian white noise

By Kolyan Ray. 2014

56 citations recorded by the source corpus.

We investigate Bernstein-von Mises theorems for adaptive nonparametric Bayesian procedures in the canonical Gaussian white noise model. We consider both a Hilbert space and multiscale setting with applications in $L^2$ and $L^\infty$ respectively. This provides a theoretical justification for plug-in procedures, for example the use of certain credible sets for sufficiently smooth linear functionals. We use this general approach to construct optimal frequentist confidence sets based on the posterior distribution. We also provide simulations to numerically illustrate our approach and obtain a visual representation of the geometries involved.

Subfields: Artificial Intelligence, Ocean Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/230d5c866bf13f39da40f94991a0501fb4af8ead

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Kolyan Ray
2. Richard Nickl
3. Botond Szab'o
4. A. van der Vaart
5. Subhashis Ghosal
6. Stéphanie van der Pas
7. Yun Yang
8. Judith Rousseau

In Ocean Engineering

1. Richard Nickl
2. Subhashis Ghosal
3. William Weimin Yoo
4. Anirban Bhattacharya
5. Yun Yang
6. A. van der Vaart
7. Debdeep Pati
8. Botond Szab'o

In Statistics and Probability

1. Kolyan Ray
2. Botond Szab'o
3. Richard Nickl
4. A. van der Vaart
5. Stéphanie van der Pas
6. Judith Rousseau
7. Subhashis Ghosal
8. Yun Yang

Citations in the corpus

• 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).”
• Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors (2020, background): “In more complex settings with infinite-dimensional parameter spaces and inverse problems, such results are more recent and the present article contributes to the programme developed in [6, 7, 9, 53, 41, 39, 43, 22, 42, 8, 10].”
• Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression (2014, methodology): “In the setting of the Gaussian white noise model, Ray [23] constructed adaptive L2-credible sets using a weak BvM theorem, and also adaptive L∞-credible band using a spike and slab prior.”
• Frequentist coverage and sup-norm convergence rate in Gaussian process regression (2017, methodology): “[43] also investigated the adaptivity of the credible sets to unknown smoothness using an empirical Bayes approach; see also [33, 37].”
• 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].”