  Ordinary differential equations are arguably the most popular and useful
mathematical tool for describing physical and biological processes in the real
world. Often, these physical and biological processes are observed with errors,
in which case the most natural way to model such data is via regression where
the mean function is defined by an ordinary differential equation believed to
provide an understanding of the underlying process. These regression based
dynamical models are called differential equation models. Parameter inference
from differential equation models poses computational challenges mainly due to
the fact that analytic solutions to most differential equations are not
available. In this paper, we propose an approximation method for obtaining the
posterior distribution of parameters in differential equation models. The
approximation is done in two steps. In the first step, the solution of a
differential equation is approximated by the general one-step method which is a
class of numerical methods for ordinary differential equations including the
Euler and the Runge-Kutta procedures; in the second step, nuisance parameters
are marginalized using Laplace approximation. The proposed Laplace approximated
posterior gives a computationally fast alternative to the full Bayesian
computational scheme (such as Markov Chain Monte Carlo) and produces more
accurate and stable estimators than the popular smoothing methods (called
collocation methods) based on frequentist procedures. For a theoretical support
of the proposed method, we prove that the Laplace approximated posterior
converges to the actual posterior under certain conditions and analyze the
relation between the order of numerical error and its Laplace approximation.
The proposed method is tested on simulated data sets and compared with the
other existing methods.
