On well-posedness of Bayesian data assimilation and inverse problems in Hilbert space

TL;DR

This paper investigates the conditions under which Bayesian inverse problems in infinite-dimensional Hilbert spaces are well-posed, focusing on Gaussian measures, covariance properties, and the impact of data error covariance bounds.

Contribution

It establishes precise criteria for well-posedness of Bayesian inverse problems with Gaussian priors and data, especially regarding covariance bounds and commutation conditions.

Findings

Well-posedness holds when the prior and data error covariances are cylindric Gaussian measures with positive lower bounds.

If prior and data distributions are equivalent Gaussian measures, the Bayesian posterior is not well defined.

Commutation of covariances ensures well-defined posterior for all data vectors if the data error covariance has a positive lower bound.

Abstract

Bayesian inverse problem on an infinite dimensional separable Hilbert space with the whole state observed is well posed when the prior state distribution is a Gaussian probability measure and the data error covariance is a cylindric Gaussian measure whose covariance has positive lower bound. If the state distribution and the data distribution are equivalent Gaussian probability measures, then the Bayesian posterior measure is not well defined. If the state covariance and the data error covariance commute, then the Bayesian posterior measure is well defined for all data vectors if and only if the data error covariance has positive lower bound, and the set of data vectors for which the Bayesian posterior measure is not well defined is dense if the data error covariance does not have positive lower bound.