  We consider the problem of simultaneously inferring the heterogeneous
coefficient field for a Robin boundary condition on an inaccessible part of the
boundary along with the shape of the boundary for the Poisson problem. Such a
problem arises in, for example, corrosion detection, and thermal parameter
estimation. We carry out both linearised uncertainty quantification, based on a
local Gaussian approximation, and full exploration of the joint posterior using
Markov chain Monte Carlo (MCMC) sampling. By exploiting a known invariance
property of the Poisson problem, we are able to circumvent the need to re-mesh
as the shape of the boundary changes. The linearised uncertainty analysis
presented here relies on a local linearisation of the parameter-to-observable
map, with respect to both the Robin coefficient and the boundary shape,
evaluated at the maximum a posteriori (MAP) estimates. Computation of the MAP
estimate is carried out using the Gauss-Newton method. On the other hand, to
explore the full joint posterior we use the Metropolis-adjusted Langevin
algorithm (MALA), which requires the gradient of the log-posterior. We thus
derive both the Fr\'{e}chet derivative of the solution to the Poisson problem
with respect to the Robin coefficient and the boundary shape, and the gradient
of the log-posterior, which is efficiently computed using the so-called adjoint
approach. The performance of the approach is demonstrated via several numerical
experiments with simulated data.
