  The paper has two major themes. The first part of the paper establishes
certain general results for infinite-dimensional optimization problems on
Hilbert spaces. These results cover the classical representer theorem and many
of its variants as special cases and offer a wider scope of applications. The
second part of the paper then develops a systematic approach for learning the
drift function of a stochastic differential equation by integrating the results
of the first part with Bayesian hierarchical framework. Importantly, our
Baysian approach incorporates low-cost sparse learning through proper use of
shrinkage priors while allowing proper quantification of uncertainty through
posterior distributions. Several examples at the end illustrate the accuracy of
our learning scheme.
