Corpus ID: 88522141

# On the Bernstein-Von Mises Theorem for High Dimensional Nonlinear Bayesian Inverse Problems

@article{Lu2017OnTB,
title={On the Bernstein-Von Mises Theorem for High Dimensional Nonlinear Bayesian Inverse Problems},
author={Y. Lu},
journal={arXiv: Statistics Theory},
year={2017}
}
• Y. Lu
• Published 2017
• Mathematics
• arXiv: Statistics Theory
We prove a Bernstein-von Mises theorem for a general class of high dimensional nonlinear Bayesian inverse problems in the vanishing noise limit. We propose a sufficient condition on the growth rate of the number of unknown parameters under which the posterior distribution is asymptotically normal. This growth condition is expressed explicitly in terms of the model dimension, the degree of ill-posedness of the inverse problem and the noise parameter. The theoretical results are applied to a… Expand
Bayesian inference for nonlinear inverse problems
Bayesian methods are actively used for parameter identification and uncertainty quantification when solving nonlinear inverse problems with random noise. However, there are only few theoreticalExpand
Gaussian Approximations for Probability Measures on Rd
• Mathematics, Computer Science
• SIAM/ASA J. Uncertain. Quantification
• 2017
The theory developed is then applied to understand the frequentist consistency of Bayesian inverse problems in finite dimensions and proves a Bernstein-Von Mises type result for the posterior measure in the small noise limit. Expand
Gaussian Approximations for Probability Measures on $\mathbf{R}^d$
• Mathematics
• 2016
This paper concerns the approximation of probability measures on $\mathbf{R}^d$ with respect to the Kullback-Leibler divergence. Given an admissible target measure, we show the existence of the bestExpand
Asymptotic analysis and computations of probability measures
This thesis is devoted to asymptotic analysis and computations of probability measures. We are concerned with the probability measures arising from two classes of problems: Bayesian inverse problemsExpand
Bernstein–von Mises theorems for statistical inverse problems I: Schrödinger equation
The inverse problem of determining the unknown potential $f>0$ in the partial differential equation \frac{\Delta}{2} u - fu =0 \text{ on } \mathcal O ~~\text{s.t. } u = g \text { on } \partialExpand
Frequentist Consistency of Variational Bayes
• Computer Science, Mathematics
• ArXiv
• 2017
It is proved that the VB posterior converges to the Kullback–Leibler (KL) minimizer of a normal distribution, centered at the truth and the corresponding variational expectation of the parameter is consistent and asymptotically normal. Expand
Frequentist Consistency of Generalized Variational Inference
This paper investigates Frequentist consistency properties of the posterior distributions constructed via Generalized Variational Inference (GVI). A number of generic and novel strategies are givenExpand
The nonparametric LAN expansion for discretely observed diffusions
Consider a scalar reflected diffusion $(X_t)_{t\geq 0}$, where the unknown drift function $b$ is modelled nonparametrically. We show that in the low frequency sampling case, when the sample consistsExpand
• Computer Science, Mathematics
• ICML
• 2015
It is shown that under standard assumptions, getting one sample from a posterior distribution is differentially private "for free"; and this sample as a statistical estimator is often consistent, near optimal, and computationally tractable; and this observations lead to an "anytime" algorithm for Bayesian learning under privacy constraint. Expand
Bayesian Generative Models for Knowledge Transfer in MRI Semantic Segmentation Problems
• Computer Science, Engineering
• Front. Neurosci.
• 2019
This work proposes a knowledge transfer method between diseases via the Generative Bayesian Prior network, compared to a pre-train approach and random initialization and obtains the best results in terms of Dice Similarity Coefficient metric for the small subsets of the Brain Tumor Segmentation 2018 database (BRATS2018). Expand

#### References

SHOWING 1-10 OF 38 REFERENCES
On the Bernstein-von Mises Theorem with Infinite Dimensional Parameters
If there are many independent, identically distributed observations governed by a smooth, finite-dimensional statistical model, the Bayes estimate and the maximum likelihood estimate will be close.Expand
Bayesian posterior contraction rates for linear severely ill-posed inverse problems
• Mathematics
• 2013
Abstract. We consider a class of linear ill-posed inverse problems arising from inversion of a compact operator with singular values which decay exponentially to zero. We adopt a Bayesian approach,Expand
Bayesian inverse problems with non-conjugate priors
We investigate the frequentist posterior contraction rate of nonparametric Bayesian procedures in linear inverse problems in both the mildly and severely ill-posed cases. A theorem is proved in aExpand
BERNSTEIN-VON MISES THEOREM FOR LINEAR FUNCTIONALS OF THE DENSITY
• Mathematics
• 2009
In this paper, we study the asymptotic posterior distribution of linear functionals of the density by deriving general conditions to obtain a semi-parametric version of the Bernstein–von MisesExpand
Nonparametric Bernstein–von Mises theorems in Gaussian white noise
• Mathematics
• 2013
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 inferenceExpand
Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors
This paper brings a contribution to the Bayesian theory of nonparametric and semiparametric estimation. We are interested in the asymptotic normality of the posterior distribution in Gaussian linearExpand
A semiparametric Bernstein–von Mises theorem for Gaussian process priors
This paper is a contribution to the Bayesian theory of semiparametric estimation. We are interested in the so-called Bernstein–von Mises theorem, in a semiparametric framework where the unknownExpand
POSTERIOR CONSISTENCY OF THE BAYESIAN APPROACH TO LINEAR ILL-POSED INVERSE PROBLEMS
• Mathematics
• 2012
We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting, with Gaussian prior and noise distribution. A method of identifying theExpand
BAYESIAN INVERSE PROBLEMS WITH GAUSSIAN PRIORS
• Mathematics
• 2011
The posterior distribution in a nonparametric inverse problem is shown to contract to the true parameter at a rate that depends on the smoothness of the parameter, and the smoothness and scale of theExpand
A Bernstein–von Mises theorem for smooth functionals in semiparametric models
• Mathematics
• 2015
A Bernstein–von Mises theorem is derived for general semiparametric functionals. The result is applied to a variety of semiparametric problems in i.i.d. and non-i.i.d. situations. In particular, newExpand