Bayesian two-step 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, with theoretical guarantees on the posterior distribution.

Contribution

It introduces a Bayesian framework for two-step ODE parameter estimation using B-splines, proving the Bernstein-von Mises theorem for the posterior of the parameters.

Findings

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

The approach accommodates multidimensional responses and model misspecification.

Establishment of Bernstein-von Mises theorem for the Bayesian two-step estimation.

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 $\bm\theta$ 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 $\bm\theta$ is nevertheless estimated at $n^{-1/2}$ rate.