A general approach to posterior contraction in nonparametric inverse problems

TL;DR

This paper introduces a versatile method for establishing posterior contraction rates in nonparametric inverse problems, applicable to various priors and models, and demonstrates its effectiveness through multiple examples.

Contribution

A general theorem that derives contraction rates from related direct problems, extending applicability beyond priors tied to singular value decomposition.

Findings

Derived minimax adaptive contraction rates for multiple inverse problem models

Applicable to priors not related to singular value decomposition

Validated approach on linear inverse problems in different statistical models

Abstract

In this paper we propose a general method to derive an upper bound for the contraction rate of the posterior distribution for nonparametric inverse problems. We present a general theorem that allows us to derive con- traction rates for the parameter of interest from contraction rates of the related direct problem of estimating transformed parameter of interest. An interesting aspect of this approach is that it allows us to derive con- traction rates for priors that are not related to the singular value decomposition of the operator. We apply our result to several examples of linear inverse problems, both in the white noise sequence model and the nonparametric regression model, using priors based on the singular value decomposition of the operator, location-mixture priors and splines prior, and recover minimax adaptive contraction rates.