Bayesian inverse problems with non-conjugate priors

TL;DR

This paper analyzes the convergence rates of nonparametric Bayesian methods with non-conjugate priors in linear inverse problems, providing theoretical guarantees and practical examples in ill-posed settings.

Contribution

It establishes posterior contraction rates for non-conjugate priors in inverse problems, extending results to sieve and wavelet priors, and demonstrates minimax optimality in various ill-posed scenarios.

Findings

Minimax optimal rates achieved in mildly ill-posed problems.

Oversmoothing priors yield minimax rates in severely ill-posed problems.

Results apply to deconvolution, heat equation, and Radon transform applications.

Abstract

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 a general Hilbert space setting under approximation-theoretic assumptions on the prior. The result is applied to non-conjugate priors, notably sieve and wavelet series priors, as well as in the conjugate setting. In the mildly ill-posed setting minimax optimal rates are obtained, with sieve priors being rate adaptive over Sobolev classes. In the severely ill-posed setting, oversmoothing the prior yields minimax rates. Previously established results in the conjugate setting are obtained using this method. Examples of applications include deconvolution, recovering the initial condition in the heat equation and the Radon transform.