Generalized modes in Bayesian inverse problems

Abstract

Uncertainty quantification requires efficient summarization of high- or even infinite-dimensional (i.e., non-parametric) distributions based on, e.g., suitable point estimates (modes) for posterior distributions arising from model-specific prior distributions. In this work, we consider non-parametric modes and MAP estimates for priors that do not admit continuous densities, for which previous approaches based on small ball probabilities fail. We propose a novel definition of generalized modes based on the concept of approximating sequences, which reduce to the classical mode in certain situations that include Gaussian priors but also exist for a more general class of priors. The latter includes the case of priors that impose strict bounds on the admissible parameters and in particular of uniform priors. For uniform priors defined by random series with uniformly distributed coefficients, we show that generalized MAP estimates -- but not classical MAP estimates -- can be characterized as minimizers of a suitable functional that plays the role of a generalized Onsager--Machlup functional. This is then used to show consistency of nonlinear Bayesian inverse problems with uniform priors and Gaussian noise.

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Cited by in corpus (8)

• Γ-convergence of Onsager-Machlup functionals. Part I: With applications to maximum a posteriori estimation in Bayesian inverse problems
• Γ-convergence of Onsager-Machlup functionals. Part II: Infinite product measures on Banach spaces
• Maximum a posteriori estimators in ${\ell }^p$ are well-defined for diagonal Gaussian priors
• An order-theoretic perspective on modes and maximum a posteriori estimation in Bayesian inverse problems
• Are minimizers of the Onsager-Machlup functional strong posterior modes?
• MAP estimators for nonparametric Bayesian inverse problems in Banach spaces
• Strong maximum a posteriori estimation in Banach spaces with Gaussian priors
• On the asymptotical regularization for linear inverse problems in presence of white noise