1Two-filter Gaussian mixture smoothing with

Abstract—In this paper, we address the problem of smoothing on Gaussian mixture (GM) posterior densities using the two-filter smoothing (TFS) strategy. The structure of the likelihoods in the backward filter of the TFS is analysed in detail. These likelihoods look similar to GMs, but are not proper density functions in the state-space since they may have constant value in a subspace of the state space. We present how the traditional GM reduction techniques can be extended to this kind of GMs. We also propose a posterior-based pruning strategy, where the filtering density can be used to make further approximations of the likelihood in the backward filter. Compared to the forward–backward smoothing (FBS) method based on N−scan pruning approximations, the proposed algorithm is shown to perform better in terms of track loss, normalized estimation error squared (NEES), computational complexity and root mean squared error (RMSE). Index Terms—filtering, smoothing, Gaussian mixtures, two-filter smoothing, backward likelihood, data associatio