  Results by van der Vaart (1991) from semi-parametric statistics about the
existence of a non-zero Fisher information are reviewed in an
infinite-dimensional non-linear Gaussian regression setting.
Information-theoretically optimal inference on aspects of the unknown parameter
is possible if and only if the adjoint of the linearisation of the regression
map satisfies a certain range condition. It is shown that this range condition
may fail in a commonly studied elliptic inverse problem with a divergence form
equation, and that a large class of smooth linear functionals of the
conductivity parameter cannot be estimated efficiently in this case. In
particular, Gaussian `Bernstein von Mises'-type approximations for Bayesian
posterior distributions do not hold in this setting.
