Bayesian Estimation in Differential Equation Models

• Activity Number: 398
• Type: Contributed
• Date/Time: Tuesday, August 5, 2014 : 2:00 PM to 3:50 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Prithwish Bhaumik*+ and Subhashis Ghosal
• Companies: North Carolina State University and North Carolina State University
• Keywords: Ordinary differential equation ; Bayesian inference ; spline smoothing ; Bernstein-von Mises theorem

Abstract

Ordinary differential equations (ODEs), used to model dynamic systems appearing in various applied fields, contain unknown parameters of physical significance which have to be estimated from the noisy data. When the equations are not analytically solvable, there is a two step approach for the estimation problem. The first step involves fitting the data nonparametrically and the second step estimates the parameter by minimizing the distance between the nonparametrically estimated derivative and the derivative suggested by the system of ODEs. We consider this two step estimation under the Bayesian framework without assuming the true mean function of the response to be in the model. A prior is induced 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 is nevertheless estimated at $n^{-1/2}$ rate.