  The Bayesian solution to a statistical inverse problem can be summarised by a
mode of the posterior distribution, i.e. a MAP estimator. The MAP estimator
essentially coincides with the (regularised) variational solution to the
inverse problem, seen as minimisation of the Onsager-Machlup 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 $\Gamma$-convergence of Onsager-Machlup 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.
