Abstract
The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a maximum a posteriori (MAP) estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager–Machlup (OM) functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the Γ-convergence of OM functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions.
Highlights
In diverse applications such as Bayesian inference and the transition path analysis of diffusion processes, it is important to be able to summarise a probability measure μ on a possibly infinitedimensional space X by a single distinguished point of X — a point of maximum probability under μ in some sense, i.e. a mode of μ
Standard definitions and results relating to Γ-convergence are collected in Appendix A and technical supporting results are given in Appendix B
In stochastic analysis and mathematical physics, the interpretation of the minimisers of Onsager– Machlup functionals over path spaces as most probable paths appears to be due to Durr and Bach (1978)
Summary
In diverse applications such as Bayesian inference and the transition path analysis of diffusion processes, it is important to be able to summarise a probability measure μ on a possibly infinitedimensional space X by a single distinguished point of X — a point of maximum probability under μ in some sense, i.e. a mode of μ. If X is an infinite-dimensional Banach space X, a Lebesgue density is not available In this case it has become common to define modes using the posterior probabilities of norm balls in the small-radius limit. One might argue that at least the cluster points as t → 0 of the modes of the measures μ(t) yield the two modes at x = ±r of the symmetric Gaussian mixture μ(0) Even this situation cannot be expected to hold true in general, as the example shows. Standard definitions and results relating to Γ-convergence are collected in Appendix A and technical supporting results are given in Appendix B
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