ACCEPTED FOR PUBLICATION IN IEEE-TAES. ©IEEE. 1 Relationship between Finite Set Statistics and the Multiple Hypothesis Tracker Edmund Brekke, Member, IEEE, and Mandar Chitre, Senior Member, IEEE Abstract—The multiple hypothesis tracker (MHT) and finite set statistics (FISST) are two approaches to multitarget tracking which both have been heralded as optimal. In this paper we show that the multitarget Bayes filter with basis in FISST can be expressed in terms the MHT formalism, consisting of association hypotheses with corresponding probabilities and hypothesis-conditional densities of the targets. Furthermore, we show that the resulting MHT-like method under appropriate assumptions (Poisson clutter and birth models, no target-death, linear-Gaussian Markov target kinematics) only differs from Reid’s MHT with regard to the birth process. Index Terms—Multitarget tracking, Data association, MHT, FISST, Random Finite Sets I. I NTRODUCTION T He standard multitarget tracking problem can be decom- posed into the two subproblems of filtering and data association. Filtering concerns the estimation of a target’s kinematic state from a time series of measurements. Data association concerns how one determines which measurements originate from which targets, and which measurements are to be considered useless clutter. Many of the tasks carried out in a surveillance system are most appropriately understood in the context of data association. For example, establishing tracks on new targets is fundamentally a data association problem, since a decision to establish a new track amounts to a decision regarding the origin of a sequence of measurements. A Bayesian formalism is natural in target tracking since prior knowledge about target kinematics is easily quantified by a prior distribution. For filtering without measurement origin uncertainty one can evaluate the posterior probability density function (pdf) of the state given the sequence of measurements so far observed. If the filtering problem is Gaussian as well as linear, then the posterior is a Gaussian, whose sufficient statistics are found by the Kalman filter (KF) [1]. For non-linear and non-Gaussian problems, no closed- form representation of the posterior can be found in general, but the posterior can still be approximated by techniques such as sequential Monte-Carlo methods (SMC) [2]. When data association is involved things get more com- plicated, and it is not immediately clear that a well-defined posterior exists. Consequently, the concept of optimality be- comes problematic for data association problems. Truly Bayes- optimal solutions, i.e., solutions which minimize a posteriori E. Brekke is with the Department of Engineering Cybernetics, Norwegian University of Science and Technology, O. S. Bragstads plass 2D, 7034 Trondheim, Norway. e-mail: edmund.brekke@ntnu.no. M. Chitre is with the Acoustic Research Lab, Tropical Marine Science Institute, National University of Singapore, 14 Kent Ridge Road, Singapore 119222, e-mail: mandar@arl.nus.edu.sg. Manuscript received October 21, 2016; revised December 15, 2017. expected loss, can only be achieved if the posterior is available. From an intuitive perspective, it seems fairly obvious that Bayes-optimal solutions to the standard multitarget tracking problem must, in one way or another, consider all feasible data association hypotheses. A milestone was reached in 1979 when Donald Reid proposed the multiple hypothesis tracker (MHT) [3], which evaluates posterior probabilities of all feasible data association hypotheses. Several variants of the MHT exist. In this paper we are solely concerned with Reid’s original algorithm. In contrast to many of the later developments, it has two attributes which are very important here. First, Reid’s MHT is a recursive method, which updates hypothesis probabilities according to a recursive formula. Second, although one typically would want to find the hypothesis with the highest posterior probability, maximum a posteriori (MAP) estimation is never mentioned in [3]. The output of Reid’s MHT as described in [3] is simply a collection of possible hypotheses, with corresponding probabilities. In practice, implementations of MHT-methods rely on pruning, simplifications and sliding window techniques to mitigate its exponential complexity. While a practical MHT therefore will be suboptimal, it is commonly believed that the ideal MHT without approximations is optimal in some unspecified sense. A different approach known as finite set statistics (FISST) was developed by Ronald Mahler [4, 5, 6] in the 2000’s. FISST is a reformulation of point process theory tailored to multi- target tracking [7]. In FISST, both targets and measurements are generally treated as random finite sets, i.e., as set-valued random variables. This allows one to express a Bayes-optimal solution to the full tracking problem using a single prediction equation and a single update equation. This optimal recursion, commonly known as the multitarget Bayes filter, is just as intractable as the MHT, but Mahler and coworkers have devel- oped several approximative solutions such as the probability hypothesis density (PHD) filter, the cardinalized probability hypothesis density (CPHD) filter and the multitarget multi- Bernoulli (MeMBer) filter [6, 8]. The relationship between FISST and previously established tracking methods is a controversial topic which has been explored in several papers, although never fully resolved. The integrated probabilistic data association (IPDA) and the joint IPDA (JIPDA) have been linked to the FISST formalism in [9] and [10]. The Set JPDA employs concepts based on FISST to improve the JPDAF [11]. It was shown in [12] that the CPHD filter is equivalent to MHT when maximally one target is present. Several classical tracking methods were discussed from the perspective of point process theory in [13]. The relationship between MHT and FISST has been explored in