  In this paper we investigate the Bayesian approach to inverse Robin problems.
These are problems for certain elliptic boundary value problems of determining
a Robin coefficient on a hidden part of the boundary from Cauchy data on the
observable part. Such a nonlinear inverse problem arises naturally in the
initialisation of large-scale ice sheet models that are crucial in climate and
sea-level predictions. We motivate the Bayesian approach for a prototypical
Robin inverse problem by showing that the posterior mean converges in
probability to the data-generating ground truth as the number of observations
increases. Related to the stability theory for inverse Robin problems, we
establish a logarithmic convergence rate for Sobolev-regular Robin
coefficients, whereas for analytic coefficients we can attain an algebraic
rate. The use of rescaled analytic Gaussian priors in posterior consistency for
nonlinear inverse problems is new and may be of separate interest in other
inverse problems. Our numerical results illustrate the convergence property in
two observation settings.
