The Bayesian smoothing equations are generally intractable for systemsdescribed by nonlinear stochastic differential equations and discrete-timemeasurements. Gaussian approximations are a computationally efficient way toapproximate the true smoothing distribution. In this work, we present acomparison between two Gaussian approximation methods. The Gaussian filteringbased Gaussian smoother uses a Gaussian approximation for the filteringdistribution to form an approximation for the smoothing distribution. Thevariational Gaussian smoother is based on minimizing the Kullback-Leiblerdivergence of the approximate smoothing distribution with respect to the truedistribution. The results suggest that for highly nonlinear systems, thevariational Gaussian smoother can be used to iteratively improve the Gaussianfiltering based smoothing solution. We also present linearization andsigma-point methods to approximate the intractable Gaussian expectations in theVariational Gaussian smoothing equations. In addition, we extend thevariational Gaussian smoother for certain class of systems with singulardiffusion matrix.
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