A general approach to posterior contraction in nonparametric inverse problems

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 contraction 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 contraction 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.