Bayesian Posterior Contraction Rates for Linear Severely Ill-posed Inverse Problems

TL;DR

This paper analyzes the rate at which Bayesian posterior distributions concentrate around the true solution in linear severely ill-posed inverse problems with Gaussian priors, providing explicit logarithmic contraction rates.

Contribution

It derives explicit logarithmic contraction rates for Bayesian posteriors in severely ill-posed inverse problems, optimizing prior scale parameters based on smoothness assumptions.

Findings

Explicit logarithmic contraction rates derived

Optimal prior scale parameters identified

Posterior consistency established in small noise limit

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, assuming a Gaussian prior on the unknown function. If the observational noise is assumed to be Gaussian then this prior is conjugate to the likelihood so that the posterior distribution is also Gaussian. We study Bayesian posterior consistency in the small observational noise limit. We assume that the forward operator and the prior and noise covariance operators commute with one another. We show how, for given smoothness assumptions on the truth, the scale parameter of the prior can be adjusted to optimize the rate of posterior contraction to the truth, and we explicitly compute the logarithmic rate.