  We propose a new framework for imposing monotonicity constraints in a
Bayesian nonparametric setting based on numerical solutions of stochastic
differential equations. We derive a nonparametric model of monotonic functions
that allows for interpretable priors and principled quantification of
hierarchical uncertainty. We demonstrate the efficacy of the proposed model by
providing competitive results to other probabilistic monotonic models on a
number of benchmark functions. In addition, we consider the utility of a
monotonic random process as a part of a hierarchical probabilistic model; we
examine the task of temporal alignment of time-series data where it is
beneficial to use a monotonic random process in order to preserve the
uncertainty in the temporal warpings.
