Approximating the Bayesian inverse for nonlinear dynamical systems

Abstract

We are given a sequence of measurement vectors and a possibly nonlinear relation to a corresponding sequence of state vectors. We are also given a possibly nonlinear model for the dynamics of the state vectors. The goal is to estimate (invert) for the state vectors when there is noise in both the measurements and the dynamics of the state. In general, obtaining the minimum variance (Bayesian) estimate of the state vectors is difficult because it requires evaluations of high dimensional integrals with no closed analytic form. We use a block tridiagonal Cholesky algorithm to simulate from the Laplace approximation for the posterior of the state vectors. These simulations are used as a proposal density for the random-walk Metropolis algorithm to obtain samples from the actual posterior. This provides a means to approximate the minimum variance estimate, as well as confidence intervals, for the state vector sequence. Simulations of a fed-batch bio-reactor model are used to demonstrate that this approach obtains better estimates and confidence intervals than the iterated Kalman smoother.