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

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 cylindrical 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.

Γ -convergence of Onsager–Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems

“…Such potentials cannot be interpreted as (non-negative) misfit functionals, as discussed by e.g. Stuart (2010), (remark 3.8) and Kasanický and Mandel (2017). Note also that a standing assumption of Dashti et al (2013) is that Φ is locally Lipschitz continuous, which is stronger than the local uniform continuity assumed in theorem 6.1, and that boundedness of Φ from below is also assumed by Dashti et al (2013), (theorem 3.5), just as in the hypothesis of theorem 6.1(d).…”

Γ-convergence of Onsager-Machlup functionals. Part I: With applications to maximum a posteriori estimation in Bayesian inverse problems

“…Such potentials cannot be interpreted as (non-negative) misfit functionals, as discussed by e.g. Stuart (2010, Remark 3.8) and Kasanický and Mandel (2017). Note also that a standing assumption of Dashti et al (2013) is that Φ is locally Lipschitz continuous, which is stronger than the local uniform continuity assumed in Theorem 6.1, and that boundedness of Φ from below is also assumed by Dashti et al (2013, Theorem 3.5), just as in the hypothesis of Theorem 6.1(d).…”