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In the Bayesian approach, the a priori knowledge about the input of amathematical model is described via a probability measure. The jointdistribution of the unknown input and the data is then conditioned, usingBayes' formula, giving rise to the posterior distribution on the unknown input.In this setting we prove posterior consistency for nonlinear inverse problems:a sequence of data is considered, with diminishing fluctuations around a singletruth and it is then of interest to show that the resulting sequence ofposterior measures arising from this sequence of data concentrates around thetruth used to generate the data. Posterior consistency justifies the use of theBayesian approach very much in the same way as error bounds and convergenceresults for regularisation techniques do. As a guiding example, we consider theinverse problem of reconstructing the diffusion coefficient from noisyobservations of the solution to an elliptic PDE in divergence form. Thisproblem is approached by splitting the forward operator into the underlyingcontinuum model and a simpler observation operator based on the output of themodel.  In general, these splittings allow us to conclude posterior consistencyprovided a deterministic stability result for the underlying inverse problemand a posterior consistency result for the Bayesian regression problem with thepush-forward prior.  Moreover, we prove posterior consistency for the Bayesian regression problembased on the regularity, the tail behaviour and the small ball probabilities ofthe prior.

Posterior Consistency for Bayesian Inverse Problems through Stability and Regression Results