Inference for differential equation models using relaxation via dynamical systems

Statistical regression models whose mean functions are represented by ordinary differential equations (ODEs) can be used to describe phenomena which are dynamical in nature, and which are abundant in areas such as biology, climatology and genetics. The estimation of parameters of ODE based models is essential for understanding its dynamics, but the lack of an analytical solution of the ODE makes estimating its parameter challenging. The aim of this paper is to propose a general and fast framework of statistical inference for ODE based models by relaxation of the underlying ODE system. Relaxation is achieved by a properly chosen numerical procedure, such as the Runge-Kutta, and by introducing additive Gaussian noises with small variances. Consequently, filtering methods can be applied to obtain the posterior distribution of the parameters in the Bayesian framework. The main advantage of the proposed method is computational speed. In a simulation study, the proposed method was at least 35 times faster than the other Bayesian methods investigated. Theoretical results which guarantee the convergence of the posterior of the approximated dynamical system to the posterior of true model are presented. Explicit expressions are given that relate the order and the mesh size of the Runge-Kutta procedure to the rate of convergence of the approximated posterior as a function of sample size.