Bayesian inference for nonlinear inverse problems
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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, see e.g. <cit.> and references therein, illustrate the main difficulties and challenges in studying the properties of the posterior distribution in the nonparametric setup. This paper offers a new approach for study the frequentist properties of the nonparametric Bayes procedures. The idea of the approach is to introduce an auxiliary functional parameter and to replace the structural equation with a penalty. This leads to a new model with an extended parameter set but the corresponding stochastic term is linear w.r.t. the unknown parameter. This allows to state sharp bounds on posterior concentration and on the accuracy of Gaussian approximation of the posterior, and a number of further results. All the bounds are given in terms of effective dimension, and we show that the proposed calming device does not significantly affect this value.
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