Some results on contraction rates for Bayesian inverse problems

TL;DR

This paper develops a general lemma to derive contraction rates for Bayesian inverse problems, applying it to linear and semilinear cases with Gaussian priors, achieving minimax optimal rates under various conditions.

Contribution

Introduces a new lemma for contraction rates in Bayesian inverse problems and applies it to both linear and semilinear cases with Gaussian priors, matching minimax rates.

Findings

Contraction rates match minimax rates in severely ill-posed problems.

Contraction rates match minimax rates in mildly ill-posed problems when the true solution is not too smooth.

Derived contraction rates for inversion of semilinear operators with Gaussian priors.

Abstract

We prove a general lemma for deriving contraction rates for linear inverse problems with non parametric nonconjugate priors. We then apply it to get contraction rates for both mildly and severely ill posed linear inverse problems with Gaussian priors in non conjugate cases. In the severely illposed case, our rates match the minimax rates using scalable priors with scales which do not depend upon the smoothness of true solution. In the mildly illposed case, our rates match the minimax rates using scalable priors when the true solution is not too smooth. Further, using the lemma, we find contraction rates for inversion of a semilinear operator with Gaussian priors. We find the contraction rates for a compactly supported prior. We also discuss the minimax rates applicable to our examples when the Sobolev balls in which the true solution lies, are different from the usual Sobolev balls defined via the basis of forward operator.