Are minimizers of the Onsager-Machlup functional strong posterior modes?

TL;DR

This paper establishes that in infinite-dimensional Bayesian inverse problems, the modes of the posterior distribution coincide with the minimizers of the Onsager-Machlup functional, linking MAP estimation to regularization techniques.

Contribution

It proves the equivalence of Bayesian posterior modes and Onsager-Machlup minimizers in Hilbert spaces under mild conditions, extending previous results to more general likelihoods.

Findings

Posterior modes exist and match Onsager-Machlup minimizers.

MAP estimation corresponds to Tikhonov-Phillips regularization in infinite dimensions.

Results apply to inverse problems with Gaussian or Laplacian noise, including ill-posed linear problems.

Abstract

In this work we connect two notions: That of the nonparametric mode of a probability measure, defined by asymptotic small ball probabilities, and that of the Onsager-Machlup functional, a generalized density also defined via asymptotic small ball probabilities. We show that in a separable Hilbert space setting and under mild conditions on the likelihood, modes of a Bayesian posterior distribution based upon a Gaussian prior exist and agree with the minimizers of its Onsager-Machlup functional and thus also with weak posterior modes. We apply this result to inverse problems and derive conditions on the forward mapping under which this variational characterization of posterior modes holds. Our results show rigorously that in the limit case of infinite-dimensional data corrupted by additive Gaussian or Laplacian noise, nonparametric maximum a posteriori estimation is equivalent to Tikhonov-Phillips regularization. In comparison with the work of Dashti, Law, Stuart, and Voss (2013), the assumptions on the likelihood are relaxed so that they cover in particular the important case of white Gaussian process noise. We illustrate our results by applying them to a severely ill-posed linear problem with Laplacian noise, where we express the maximum a posteriori estimator analytically and study its rate of convergence in the small noise limit.