Approximating the Bayesian inverse for nonlinear dynamical systems
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.
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.