  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.
