Bayesian Inverse Problems with Heterogeneous Variance

TL;DR

This paper develops a Bayesian framework for inverse problems with correlated Gaussian noise, introducing a wavelet-based method to analyze posterior contraction rates and adaptivity in mildly ill-posed problems.

Contribution

It introduces a novel wavelet-based approach called vaguelette-vaguelette for analyzing Bayesian inverse problems with heterogeneous noise, extending sequence space methods beyond diagonalizable operators.

Findings

Posterior contraction rates are derived and compared to minimax rates.

Plugging in variance estimators affects contraction rates, especially with repeated observations.

Adaptive empirical Bayes methods achieve minimax optimal contraction rates.

Abstract

We consider inverse problems in Hilbert spaces under correlated Gaussian noise and use a Bayesian approach to find their regularised solution. We focus on mildly ill-posed inverse problems with the noise being generalised derivative of fractional Brownian motion, using a novel wavelet - based approach we call vaguelette-vaguelette. It allows us to apply sequence space methods without assuming that all operators are simultaneously diagonalisable. The results are proved for more general bases and covariance operators. Our primary aim is to study the posterior contraction rate in such inverse problems over Sobolev classes of true functions, comparing it to the derived minimax rate. Secondly, we study the effect of plugging in a consistent estimator of variances in sequence space on the posterior contraction rate, for instance where there are repeated observations. This result is also applied to the problem where the forward operator is observed with error. Thirdly, we show that an adaptive empirical Bayes posterior distribution contracts at the optimal rate, in the minimax sense, under a condition on prior smoothness, with a plugged in maximum marginal likelihood estimator of the prior scale. These theoretical results are illustrated on simulated data.