Efficient Bayesian estimation and uncertainty quantification in ordinary differential equation models

Abstract: In engineering, physics, biomedical sciences and many other fields the regression function is known to satisfy a system of ordinary differential equations (ODEs). Our interest lies in the unknown parameters involved in the ODEs. When the analytical solution of the ODEs is not available, one approach is to use numerical methods to solve the system. A four stage Runge-Kutta (RK4) method is one such method. The approximate solution can be used to construct an approximate likelihood. We assign a prior on the parameters and then draw posterior samples, but this method may be computationally expensive. Bhaumik and Ghosal (2014) considered a two-step approach of parameter estimation based on integrated squared error, where a posterior is induced on the parameters using a random series based on the B-spline basis functions. The parameter is estimated by minimizing the distance between the nonparametrically estimated derivative and the derivative suggested by the ODE. Although this approach is computationally fast, the Bayes estimator is not asymptotically efficient. In this paper we also suggest a modification of the two-step method by directly considering the distance between the function in the nonparametric model and that obtained from RK4 method. We study the asymptotic behavior of the posterior distribution of θ in both RK-4 approximate likelihood based and modified two-step approaches and establish a Bernstein-von Mises theorem which assures that Bayesian uncertainty quantification matches with the frequentist one. We allow the model to be misspecified in that the true regression function may lie outside the ODE model. Unlike in the original twostep procedure, the precision matrix matches with the Fisher information matrix.