Efficient Bayesian estimation and uncertainty quantification in ordinary differential equation models

Abstract

Often the regression function is specified by a system of ordinary differential equations (ODEs) involving some unknown parameters. Typically analytical solution of the ODEs is not available, and hence likelihood evaluation at many parameter values by numerical solution of equations may be computationally prohibitive. Bhaumik and Ghosal (Electron. J. Stat. 9 (2015) 3124–3154) considered a Bayesian two-step approach by embedding the model in a larger nonparametric regression model, where a prior is put through a random series based on B-spline basis functions. A posterior on the parameter is induced from the regression function by minimizing an integrated weighted squared distance between the derivative of the regression function and the derivative suggested by the ODEs. Although this approach is computationally fast, the Bayes estimator is not asymptotically efficient. In this paper, we suggest a modification of the two-step method by directly considering the distance between the function in the nonparametric model and that obtained from a four stage Runge–Kutta (RK$4$) method. We also study the asymptotic behavior of the posterior distribution of $\mathbf{\theta}$ based on an approximate likelihood obtained from an RK$4$ numerical solution of the ODEs. We establish a Bernstein–von Mises theorem for both methods which assures that Bayesian uncertainty quantification matches with the frequentist one and the Bayes estimator is asymptotically efficient.