Accurate Likelihood Evaluation for Multiple Model PMHT Algorithms

A variety of authors have incorporated multiple target motion models into the probabilistic multi-hypothesis tracking (PMHT) algorithm using a discrete Markov chain to model the motion model switching process. However, in these papers the observed data likelihood function is not written down for this model, nor is it evaluated because all possible model assignment sequences must be considered over the PMHT batch. These two issues are addressed in this paper under the assumption that the Markov chain switching model affects the target state process but not the target measurement process: the observed data likelihood function for the PMHT algorithm is given along with a method for evaluating it. A closely related method of including multiple target motion models in the PMHT algorithm that results in a finite mixture distribution of motion models is described as well. In addition, it is shown that using multiple-model smoothing algorithms such as an IMM smoother to estimate the target states in a multiple model PMHT algorithm will not maximize the observed data likelihood function. Finally, it is shown that the MAP target state estimates for linear Gaussian targets with multiple motion models can be computed using a bank of Kalman smoothers. This result fills a gap in the existing literature.