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MAP estimators and their consistency in Bayesian nonparametric inverse problems

journal contribution
posted on 2023-06-08, 15:50 authored by Masoumeh DashtiMasoumeh Dashti, K J H Law, A M Stuart, J Voss
We consider the inverse problem of estimating an unknown function u from noisy measurements y of a known, possibly nonlinear, map $\mathcal {G}$ applied to u. We adopt a Bayesian approach to the problem and work in a setting where the prior measure is specified as a Gaussian random field µ0. We work under a natural set of conditions on the likelihood which implies the existence of a well-posed posterior measure, µy. Under these conditions, we show that the maximum a posteriori (MAP) estimator is well defined as the minimizer of an Onsager–Machlup functional defined on the Cameron–Martin space of the prior; thus, we link a problem in probability with a problem in the calculus of variations. We then consider the case where the observational noise vanishes and establish a form of Bayesian posterior consistency for the MAP estimator. We also prove a similar result for the case where the observation of $\mathcal {G}(u)$ can be repeated as many times as desired with independent identically distributed noise. The theory is illustrated with examples from an inverse problem for the Navier–Stokes equation, motivated by problems arising in weather forecasting, and from the theory of conditioned diffusions, motivated by problems arising in molecular dynamics.

History

Publication status

  • Published

File Version

  • Published version

Journal

Inverse Problems

ISSN

0266-5611

Publisher

Institute of Physics

Issue

9

Volume

29

Page range

095017

Department affiliated with

  • Mathematics Publications

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2013-09-19

First Compliant Deposit (FCD) Date

2013-09-19

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