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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.
The general problem of constructing confidence regions is unsolved in the sense that there is no algorithm that provides such a region with guaranteed coverage for an arbitrary parameter $psiinPsi.$ Moreover, even when such a region exists, it may be
Bayesian methods are actively used for parameter identification and uncertainty quantification when solving nonlinear inverse problems with random noise. However, there are only few theoretical results justifying the Bayesian approach. Recent papers,
The features of a logically sound approach to a theory of statistical reasoning are discussed. A particular approach that satisfies these criteria is reviewed. This is seen to involve selection of a model, model checking, elicitation of a prior, chec
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
We study the statistical properties of stochastic evolution equations driven by space-only noise, either additive or multiplicative. While forward problems, such as existence, uniqueness, and regularity of the solution, for such equations have been s