Bayesian inverse problems with unknown operators

Abstract

We consider the Bayesian approach to linear inverse problems when the underlying operator depends on an unknown parameter. Allowing for finite dimensional as well as infinite dimensional parameters, the theory covers several models with different levels of uncertainty in the operator. Using product priors, we prove contraction rates for the posterior distribution which coincide with the optimal convergence rates up to logarithmic factors. In order to adapt to the unknown smoothness, an empirical Bayes procedure is constructed based on Lepski's method. The procedure is illustrated in numerical examples.

24 pages, 2 figures

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• Deconvolution with unknown error distribution
• Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions
• Nonlinear estimation for linear inverse problems with error in the operator
• Rates of contraction for posterior distributions in $\bolds{L^r}$-metrics, $\bolds{1\le r\le\infty}$
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Cited by in corpus (4)

• Convergence Rates for Learning Linear Operators from Noisy Data
• Bayesian linear inverse problems in regularity scales
• Bayesian inference for nonlinear inverse problems
• Deep learning for inverse problems with unknown operator