  This paper presents a method for calculating the smoothed state distribution
for Jump Markov Linear Systems. More specifically, the paper details a novel
two-filter smoother that provides closed-form expressions for the smoothed
hybrid state distribution. This distribution can be expressed as a Gaussian
mixture with a known, but exponentially increasing, number of Gaussian
components as the time index increases. This is accompanied by exponential
growth in memory and computational requirements, which rapidly becomes
intractable. To ameliorate this, we limit the number of allowed mixture terms
by employing a Gaussian mixture reduction strategy, which results in a
computationally tractable, but approximate smoothed distribution. The
approximation error can be balanced against computational complexity in order
to provide an accurate and practical smoothing algorithm that compares
favourably to existing state-of-the-art approaches.
