Bayesian Objective Functions for Estimating Parameters in Nonlinear Stochastic Differential Equation Models with Limited Data

Abstract

An Approximate Bayesian Expectation Maximization (ABEM) methodology and a Laplace Approximation Bayesian (LAB) methodology are developed for estimating parameters in nonlinear stochastic differential equation (SDE) models of chemical processes. These new methodologies are more powerful than previous maximum-likelihood methodologies for SDEs because they enable modelers to account for prior information about unknown parameters and initial conditions. The ABEM methodology is suitable for situations in which the modeler can assume that measurement noise variances are well-known, whereas LAB includes measurement noise variances among the parameters that require estimation. Both techniques estimate the magnitude of stochastic terms included in the differential equations to account for model mismatch and unknown process disturbances. The proposed ABEM and LAB methodologies are illustrated using a nonlinear continuous stirred tank reactor (CSTR) case study, with simulated data sets generated using a variety of s...