Bartek Knapik (VU Amsterdam)

Convergence rates of posterior distributions in nonparametric inverse problems

Since the seminal works of Ghosal, Ghosh and van der Vaart (2000) and Shen and Wasserman (2001), posterior contraction has attracted much attention, resulting in the rich literature on this subject. However, these results are not suitable to deal with trully ill-posed inverse problems, where one is interested in the parameter of interest f, given noisy observations of its transformed version Kf. General theorems yield contraction results in some metric measuring the distance between Kf and Kf_0, whereas the interest lies in the distance betweend f and f_0, and these two metrics are not equivalent. In this talk we review (a part of) the existing literature on Bayesian approach to nonparametric inverse problems, and present a general contraction theorem. We show that in mildly ill-posed problems with conjugate priors, both empirical and hierarchical Bayes approaches lead to (nearly) optimal recovery of an infinite-dimensional parameter of interest. Moreover, using our general result, we obtain minimax adaptive concentration rates in two examples in the fixed-design regression setting.