Posterior Consistency for Bayesian Inverse Problems through Stability and Regression Results

Abstract

In the Bayesian approach, the a priori knowledge about the input of a mathematical model is described via a probability measure. The joint distribution of the unknown input and the data is then conditioned, using Bayes' formula, giving rise to the posterior distribution on the unknown input. In this setting we prove posterior consistency for nonlinear inverse problems: a sequence of data is considered, with diminishing fluctuations around a single truth and it is then of interest to show that the resulting sequence of posterior measures arising from this sequence of data concentrates around the truth used to generate the data. Posterior consistency justifies the use of the Bayesian approach very much in the same way as error bounds and convergence results for regularisation techniques do. As a guiding example, we consider the inverse problem of reconstructing the diffusion coefficient from noisy observations of the solution to an elliptic PDE in divergence form. This problem is approached by splitting the forward operator into the underlying continuum model and a simpler observation operator based on the output of the model. In general, these splittings allow us to conclude posterior consistency provided a deterministic stability result for the underlying inverse problem and a posterior consistency result for the Bayesian regression problem with the push-forward prior. Moreover, we prove posterior consistency for the Bayesian regression problem based on the regularity, the tail behaviour and the small ball probabilities of the prior.

38 pages, 1 figure

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• Consistency of Bayesian inference with Gaussian process priors for a parabolic inverse problem
• On the Bernstein-Von Mises Theorem for High Dimensional Nonlinear Bayesian Inverse Problems
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• Bayesian inverse problems with unknown operators
• Convergence rates for Penalised Least Squares Estimators in PDE-constrained regression problems
• Stabilities of Shape Identification Inverse Problems in a Bayesian Framework
• Gaussian Approximations for Probability Measures on ${\mathbf{R}}^d$
• Strong maximum a posteriori estimation in Banach spaces with Gaussian priors
• Sequential Monte Carlo Methods for Bayesian Elliptic Inverse Problems
• Spatiotemporal Besov Priors for Bayesian Inverse Problems
• Posterior Convergence of Gaussian and General Stochastic Process Regression Under Possible Misspecifications
• Optimization Based Methods for Partially Observed Chaotic Systems
• A Statistical Perspective on Inverse and Inverse Regression Problems