  In this paper, the problem of state estimation, in the context of both
filtering and smoothing, for nonlinear state-space models is considered. Due to
the nonlinear nature of the models, the state estimation problem is generally
intractable as it involves integrals of general nonlinear functions and the
filtered and smoothed state distributions lack closed-form solutions. As such,
it is common to approximate the state estimation problem. In this paper, we
develop an assumed Gaussian solution based on variational inference, which
offers the key advantage of a flexible, but principled, mechanism for
approximating the required distributions. Our main contribution lies in a new
formulation of the state estimation problem as an optimisation problem, which
can then be solved using standard optimisation routines that employ exact
first- and second-order derivatives. The resulting state estimation approach
involves a minimal number of assumptions and applies directly to nonlinear
systems with both Gaussian and non-Gaussian probabilistic models. The
performance of our approach is demonstrated on several examples; a challenging
scalar system, a model of a simple robotic system, and a target tracking
problem using a von Mises-Fisher distribution and outperforms alternative
assumed Gaussian approaches to state estimation.
