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1 - Bayesian Inverse Problems andWell-Posedness

Published online by Cambridge University Press:  27 July 2023

Daniel Sanz-Alonso
Affiliation:
University of Chicago
Andrew Stuart
Affiliation:
California Institute of Technology
Armeen Taeb
Affiliation:
University of Washington
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Summary

In this chapter we introduce the Bayesian approach to inverse problems in which the unknown parameter and the observed data are viewed as random variables. In this probabilistic formulation, the solution of the inverse problem is the posterior distribution on the parameter given the data. We will show that the Bayesian formulation leads to a form of well-posedness: small perturbations of the forward model or the observed data translate into small perturbations of the posterior distribution. Well-posedness requires a notion of distance between probability measures. We introduce the total variation and Hellinger distances, giving characterizations of them, and bounds relating them, that will be used throughout these notes. We prove well-posedness in the Hellinger distance.

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Publisher: Cambridge University Press
Print publication year: 2023

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