MAP estimators and their consistency in Bayesian inverse problems for functions

Masoumeh Dashti (The University of Sussex)

Frank Adams 1,

We consider the inverse problem of estimating an unknown function from noisy measurements of a known, possibly nonlinear map of the unknown function. Under certain conditions, the Bayesian approach to this problem results in a well-defined posterior measure. Under these conditions we show that the maximum a posteriori (MAP) estimator is characterised as the minimiser of an Onsager-Machlup functional defined on the set of admissible shifts of the prior.  We then, in the case of vanishing observational noise, establish a form of Bayesian posterior consistency for the MAP estimator.

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