  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.
