Regularised posteriors in linear ill-posed inverse problems
Seminar Room 1, Newton Institute
The aim of the paper is to obtain a solution for a signal-noise problem, namely we want to make inference on an unobserved infinite dimensional parameter through a noisy indirect observation of this quantity. The parameter of interest appears as the solution of an ill-posed inverse problem. We place us in a Bayesian framework so that the parameter of interest is a stochastic process and the solution to the inference problem is the posterior distribution of such a parameter. We define and propose an easy way to identify the posterior distribution on a functional space, but due to the infinite dimension of our problem it is only possible to compute a regularized version of it. Furthermore, under some regularity condition of the true value of the parameter, we prove "frequentist" consistency of the regularized posterior distribution, but we find that the prior distribution is not able to generate the true value of the parameter satisfying this regularity condition. It perfectly agrees with previous literature and confirms once again the possible prior inconsistency in infinite-dimensional Bayesian experiments already stressed by Diaconis and Freedman (1986). However, the prior distribution that we specify is able to generate trajectories of the parameter of interest very closed to the true value. We also compute sufficient statistics for infinite dimensional parameters. A Monte Carlo simulation confirms goods properties of the proposed estimator.