A Bayesian approach to estimate parameters of ordinary differential equation

We develop a Bayesian approach to estimate the parameters of ordinary differential equations (ODE) from the observed noisy data. Our method does not need to solve ODE directly. We replace the ODE constraint with a probability expression and combine it with the nonparametric data fitting procedure into a joint likelihood framework. One advantage of the proposed method is that for some ODE systems, one can obtain closed form conditional posterior distributions for all variables which substantially reduce the computational cost and facilitate the convergence process. An efficient Riemann manifold based hybrid Monte Carlo scheme is implemented to generate samples for variables whose conditional posterior distribution cannot be written in terms of closed form. Our approach can be applied to situations where the state variables are only partially observed. The usefulness of the proposed method is demonstrated through applications to both simulated and real data.