Permutation invariance in Bayesian estimation of two targets that maneuver in and out formation flight

In theory, a good particle filter allows to approximate the exact Bayesian filter solution arbitrarily well. This has motivated a strong and successful development of particle filtering approaches towards target tracking. Literature also shows that successful multiple maneuvering target tracking seems to depend on the adoption of some suitable heuristics in handling permutation invariance. According to the exact Bayesian joint conditional density however, there is no requirement to introduce any of these heuristics. In order to improve the insight for this kind of problem, this paper studies which role permutation invariance plays in the exact joint conditional density of two targets that may maneuver in and out formation flight given potentially noisy, missing and false measurements.