We consider the problem of recovering an unknown function from noisy and indirect observations. We adopt a Bayesian approach and consider a class of product exponential priors with tails between Gaussian and Laplace. We discuss some results on characterisation of the modes of the posterior measure which lead to some weak consistency results. We then show a contraction theory for the posterior measure.
This is based on joint works with S. Agapiou, M. Burger and T. Helin.