Title: On the Bernstein-Von Mises Theorem for High Dimensional Nonlinear Bayesian Inverse Problems,
Abstract: We prove a Bernstein-von Mises theorem for a general class of high
dimensional nonlinear Bayesian inverse problems in the vanishing noise limit.
We propose a sufficient condition on the growth rate of the number of unknown
parameters under which the posterior distribution is asymptotically normal.
This growth condition is expressed explicitly in terms of the model dimension,
the degree of ill-posedness of the inverse problem and the noise parameter. The
theoretical results are applied to a Bayesian estimation of the medium
parameter in an elliptic problem.