Bayesian Estimation and Uncertainty Quantification in Differential Equation Models

• Activity Number: 183 - SPEED: Bayesian Methods Student Awards
• Type: Contributed
• Date/Time: Monday, July 31, 2017 : 10:30 AM to 11:15 AM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Prithwish Bhaumik*
• Companies: Quantifind Inc.
• Keywords: Bayesian inference ; differential equation ; nonparametric regression ; approximate likelihood ; Bernstein-von Mises theorem ; Runge-Kutta method

Abstract

In engineering, physics, biomedical sciences and many other fields the regression function is often a solution of ordinary differential equations (ODEs). We want to estimate the unknown parameters involved in the ODE. A two-step approach to solve this problem consists of the first fitting the data nonparametrically and then estimating the parameter by minimizing the distance between the nonparametrically estimated derivative and the derivative suggested by the system of ODEs. We consider a Bayesian analog of the two step approach by putting a finite random series prior on the regression function using B-spline basis. We establish a Bernstein-von Mises theorem for the posterior distribution of the parameter of interest induced from that on the regression function with the n^{-0.5} contraction rate. Although this approach is computationally fast, the Bayes estimator is not asymptotically efficient. This can be remedied by directly considering the distance between the function in the nonparametric model and a Runge-Kutta (RK4) approximate solution of the ODE while inducing the posterior distribution on the parameter. We also extend these methods for higher order ODEs and PDEs.