  The Bayesian smoothing equations are generally intractable for systems
described by nonlinear stochastic differential equations and discrete-time
measurements. Gaussian approximations are a computationally efficient way to
approximate the true smoothing distribution. In this work, we present a
comparison between two Gaussian approximation methods. The Gaussian filtering
based Gaussian smoother uses a Gaussian approximation for the filtering
distribution to form an approximation for the smoothing distribution. The
variational Gaussian smoother is based on minimizing the Kullback-Leibler
divergence of the approximate smoothing distribution with respect to the true
distribution. The results suggest that for highly nonlinear systems, the
variational Gaussian smoother can be used to iteratively improve the Gaussian
filtering based smoothing solution. We also present linearization and
sigma-point methods to approximate the intractable Gaussian expectations in the
Variational Gaussian smoothing equations. In addition, we extend the
variational Gaussian smoother for certain class of systems with singular
diffusion matrix.
