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Register a new userOn some information-theoretic aspects of non-linear statistical inverse problems
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Added by
Richard Nickl
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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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.
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