Abstract
We investigate approximate Bayesian inference
techniques for nonlinear systems described by ordinary
differential equation (ODE) models. In particular,
the approximations will be based on set-valued
reachability analysis approaches, yielding approximate
models for the posterior distribution. Nonlinear
ODEs are widely used to mathematically describe
physical and biological models. However, these
models are often described by parameters that are not
directly measurable and have an impact on the system
behaviors. Often, noisy measurement data combined
with physical/biological intuition serve as the means
for finding appropriate values of these parameters.
Our approach operates under a Bayesian framework,
given prior distribution over the parameter space
and noisy observations under a known sampling
distribution. We explore subsets of the space of model
parameters, computing bounds on the likelihood
for each subset. This is performed using nonlinear
set-valued reachability analysis that is made faster by
means of linearization around a reference trajectory.
The tiling of the parameter space can be adaptively
refined to make bounds on the likelihood tighter. We
evaluate our approach on a variety of nonlinear benchmarks and compare our results with Markov Chain
Monte Carlo and Sequential Monte Carlo approaches.