A Bayesian analysis of the mixed labelling phenomenon in two-target tracking

In mulit-target tracking and labelling (MTTL), mixed labelling corresponds to a situation where there is ambiguity in labelling, i.e. in the assignment of labels to locations (where a "location" here means simply an unlabelled single-target state. The phenomenon is well-known in literature, and known to occur in the situation where targets move in close proximity to each other and afterwards separate. The occurrence of mixed labelling has been empirically observed using particle filter implementations of the Bayesian MTTL recursion. In this memorandum, we will instead demonstrate the occurrence of mixed labelling (in the situation of closely spaced targets) using only the Bayesian recursion itself, for a scenario containing two targets and no target births or deaths. We will also show how mixed labelling generally persists after the targets become well-separated, and how mixed labelling might not happen when the unlabelled single-target state contains non-kinematic quantities.