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Statistical regression models whose mean functions are represented byordinary differential equations (ODEs) can be used to describe phenomenonsdynamical in nature, which are abundant in areas such as biology, climatologyand genetics. The estimation of parameters of ODE based models is essential forunderstanding its dynamics, but the lack of an analytical solution of the ODEmakes the parameter estimation challenging. The aim of this paper is to proposea general and fast framework of statistical inference for ODE based models byrelaxation of the underlying ODE system. Relaxation is achieved by a properlychosen numerical procedure, such as the Runge-Kutta, and by introducingadditive Gaussian noises with small variances. Consequently, filtering methodscan be applied to obtain the posterior distribution of the parameters in theBayesian framework. The main advantage of the proposed method is computationspeed. In a simulation study, the proposed method was at least 14 times fasterthan the other methods. Theoretical results which guarantee the convergence ofthe posterior of the approximated dynamical system to the posterior of truemodel are presented. Explicit expressions are given that relate the order andthe mesh size of the Runge-Kutta procedure to the rate of convergence of theapproximated posterior as a function of sample size.

Inference for Differential Equation Models using Relaxation via Dynamical Systems