Title: Adaptation in Bayesian inverse problems using an empirical Bayes approach
Authors: Natalia Bochkina - University of Edinburgh (United Kingdom) [presenting]
Jenovah Rodrigues - University of Edinburgh (United Kingdom)
Abstract: A sequence space formulation of Bayesian inverse problems with a Gaussian prior is considered where the prior scale parameter is estimated using the Empirical Bayes approach. We show that for oversmoothing priors, i.e. when a function is assumed to be a priori smoother that the true unknown function, the posterior distribution of the unknown function with plugged in empirical Bayes estimator of the prior scale parameter achieves the minimax rate of contraction in the considered cases, namely mildly and severely ill-posed inverse problems with true function in Sobolev class, and severely ill-posed problems with analytic true functions. Considered error models are white noise and fractional Brownian motion. We will illustrate behaviour of the empirical Bayes estimator and the empirical Bayes posterior distribution on simulated data.