Bayesian estimation in differential equation models

TL;DR

This paper develops a Bayesian two-step estimation method for parameters in differential equation models, allowing for multidimensional responses and model misspecification, and proves asymptotic normality of the parameter estimates.

Contribution

It introduces a Bayesian framework for differential equation parameter estimation using spline priors and establishes the Bernstein-von Mises theorem for the posterior distribution of parameters.

Findings

Posterior distribution of parameters converges at the parametric rate of n^{-1/2}.

The Bayesian approach accommodates multidimensional responses and model misspecification.

The method extends existing frequentist two-step estimation to a Bayesian setting.

Abstract

Ordinary differential equations (ODEs) are used to model dynamic systems appearing in engineering, physics, biomedical sciences and many other fields. These equations contain unknown parameters, say $\theta$ of physical significance which have to be estimated from the noisy data. Often there is no closed form analytic solution of the equations and hence we cannot use the usual non-linear least squares technique to estimate the unknown parameters. There is a two step approach to solve this problem, where the first step involves fitting the data nonparametrically. In the second step the parameter is estimated by minimizing the distance between the nonparametrically estimated derivative and the derivative suggested by the system of ODEs. The statistical aspects of this approach have been studied under the frequentist framework. We consider this two step estimation under the Bayesian framework. The response variable is allowed to be multidimensional and the true mean function of it is not assumed to be in the model. We induce a prior on the regression function using a random series based on the B-spline basis functions. We establish the Bernstein-von Mises theorem for the posterior distribution of the parameter of interest. Interestingly, even though the posterior distribution of the regression function based on splines converges at a rate slower than $n^{-1/2}$, the parameter vector $\theta$ is nevertheless estimated at $n^{-1/2}$ rate.