On the approximation of the quotient of two Gaussian densities for multiple-model smoothing

The quotient of two multivariate Gaussian densities can be written as an unnormalized Gaussian density, which has been applied in some recently developed multiple-model fixed-interval smoothing algorithms. However, this expression is invalid if instead of being positive definite, the covariance of the unnormalized Gaussian density is indefinite (i.e., it has both positive and negative eigenvalues) or undefined (i.e., computing it requires inverting a singular matrix). This paper considers approximating the quotient of two Gaussian densities in this case using two different approaches to mitigate the caused numerical problems. The first approach directly replaces the indefinite covariance of the unnormalized Gaussian density with a positive definite matrix nearest to it. The second approach computes the approximation through solving, using the natural gradient, an optimization problem with a Kullback-Leibler divergence-based cost function. This paper illustrates the application of the theoretical results by incorporating them into an existing smoothing method for jump Markov systems and utilizing the obtained smoothers to track a maneuvering target.