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Measurement-to-Track Association and Finite-Set Statistics

Multi-hypothesis trackers (MHT's), which are based on the measurement-to-track association (MTA) concept, have long been asserted to be"Bayes-optimal."Recently, rather bolder claims have come to the fore:"The right model of the multitarget state is that used in the multi-hypothesis tracker (MHT) paradigm, not the RSF [random finite set] paradigm."Or, the RFS approach is essentially a mathematically obfuscated reinvention of MHT. In this paper it is shown that: (a) although MTA's can be given a Bayesian formulation, this formulation is not fully consistent with Bayesian statistics; (b) phenomenologically, an MTA is a heuristic extrapolation of an intuitive special case to general multitarget scenarios; (c) MTA's are, therefore, not physically real entities and thus cannot (as with MHT's) be employed as state representations of a multitarget system; (d) MHT's are, consequently, heuristic approximations of the actual Bayes-optimal approach, the multitarget Bayes filter; (d)the theoretically correct measurement modeling approach is the RSF multitarget likelihood function L_Z(X) = f(Z|X); (f) although MTA's do occur in f(Z|X), they are the consequence of a mere change of notation during the RFS derivation of f(Z|X); and (g) the generalized labeled multi-Bernoulli (GLMB) filter of Vo and Vo is currently the only provably Bayes-optimal and computationally tractable approach for true multitarget tracking using MTA's.


I. INTRODUCTION
By the early 1990's at least, Bayesian statistics had become the overwhelmingly dominant foundation for target tracking. Such was its caché, in fact, that it was not uncommon for authors to claim that a proposed approach was "Bayesoptimal" merely because Bayes' rule had been utilized in some fashion. In particular, it was-and still is-claimed that multi-hypothesis trackers (MHT's) are not only theoretically rigorous, but theoretically rigorous within the Bayesian framework ("Bayes-optimal"). For example: "...MHT algorithms themselves can be, and indeed were, derived through rigorous mathematics ["in the theory of Bayesian filtering"]." Such claims have tended to be made while overlooking the following points: 1) The term "Bayes optimal" refers to one thing only: state estimation. 2) In target tracking, the term "state variable" also has a specific meaning: it must be a faithful mathematical model of some unknown but physically real property of whatever targets are of interest. 3) "Rigorous mathematics" that proceeds from faulty mathematical and/or phenomenological assumptions is spuriously rigorous.
This paper expands upon an argument originally advanced in 2007 in Section 10.7.2 of [8]. Its purpose is fourfold: 1) Systematically assess the "Bayes-optimal" and "theoretically rigorous" claims made for MHT. 2) In particular, assess the mathematical and phenomenological underpinnings of the concept that underlies MHT's and related algorithms: the measurement-totrack association (MTA). 3) Clarify the relationship between classical MTA-based approaches, and the random finite set (RFS) approach of finite-set statistics (FISST) [9]. 4) Refute certain erroneous criticisms of the RFS approach.
For example, that "The right model of the multitarget state is that used in the multi-hypothesis tracker (MHT) paradigm, not the RFS paradigm." Or more expansively, that the RFS approach is essentially a mathematically obfuscated reinvention of MHT theory. See [15] for an overview of multitarget tracking that covers the MHT, RFS, and other approaches.
The paper is organized as follows. Following brief summaries of Bayesian statistics and MTA theory in Sections II and III, we examine the concept of an association likelihood in Section IV. Then, in Section V, we turn to a Bayesian assessment of MTA's. This will lead us to argue that, even though MTA's can be given a Bayesian formulation, this formulation is more consistent with classical than with Bayesian statistics.
A phenomenological assessment of MTA's in Section VI will lead us to further conclude that: • MTA's are not physically real entities and thus cannot be employed as state representations of a multitarget system. Which is to say, it is not phenomenologically reasonable to claim that certain measurements originated with certain tracks. Rather, the most that can legitimately be asserted is the following: If targets with state-set X are present, then there is a probability (density) f (Z|X) that they will generate a measurement-set Z.
This will lead us to consider, in Sections VII and VIII, the RFS notion of a multitarget likelihood function L Z (X) = f (Z|X). There we will note that, even though the formula for f (Z|X) involves MTA's, they do not-as in MTA theory-arise from questionable heuristic intuition. Rather, they arise from a mathematically rigorous RFS derivation based on a statistically and phenomenologically rigorous RFS measurement model. Specifically, they are the consequence of a mere change of mathematical notation. This will then naturally lead us, in Sections VII and VIII, to a summary of RFS multitarget measurement modeling and, in Section IX, to a discussion of the rigorous meaning of the term "Bayesoptimal" in a multitarget context.
We will then turn to the following question: Since the MTA and RFS theories both involve MTA's, how are they related? In Section X, we will describe a mathematical connection between RFS and MTA likelihoods. This will lead us to finally deduce that: • MHT's are heuristic approximations of the actual Bayesoptimal approach, namely the multitarget Bayes filter. We will conclude with two final questions: Is there an MTA-oriented multitarget tracker that is provably Bayesoptimal? If so, is it computationally tractable? The affirmative answer to these questions-the generalized labeled multi-Bernoulli (GLMB) filter of Vo and Vo-is the subject of Section XI. This discussion will produce a theoretically grounded tracking interpretation of MTA's: • An MTA is a purely mathematical entity-namely, the index of one possible weighted hypothesis about which track labels exist in the scene and which track distributions correspond to those labels. Even so, this paper should not be construed as a denigration of MHT's. When the MHT was introduced by Reid in 1979 [12], computer processing was primitive by today's standards. Reid addressed this difficulty by using MTA's to decompose the multitarget tracking problem into a coordinated system of extended Kalman filters. Since then, theoretical and practical advances by his successors have made MHT's the workhorses of multitarget tracking. Quite understandably, however, limited computing power, combined with a lack of suitable mathematical theory, also made it difficult to adhere to proper levels of theoretical rigor. Now that computional and mathematical tools are sufficiently mature, it is important to move from conventional wisdom to scientific clarity.
II. BAYESIAN ANALYSIS Let X be a space whose elements ξ are the states of the physical entities of interest. Here, ξ ∈ X should uniquely and exhaustively model and correspond to the actual physical states of the system.
The physical entities are observed by a sensor with measurements ζ ∈ Z. The goal is to estimate ξ based on ζ. In a Bayesian analysis, the two are related by a likelihood function (measurement distribution) which gives the probability (or probability density) that measurement ζ will be collected if an entity with state ξ is present. In particular the normalization condition f (ζ|ξ)dζ = 1 must be true for every ξ ∈ X.
In classical statistics, the unknown state ξ is assumed to be a nonrandom constant. In Bayesian statistics, however, ξ is assumed to a random variable, the statistical behavior of which is characterized by some prior probability distribution f 0 (ξ). If a measurement ζ is collected, then the posterior probability distribution of ξ, conditioned on ζ, is where f (ζ) = f (ζ|ξ) · f 0 (ξ)dξ and where ' ·dξ' denotes the integration concept for X. In Bayes filtering theory, if ζ = ζ k+1 was collected at time t k+1 , then the prior f 0 (ξ) is the predicted distribution where f k+1|k (ξ|ξ ′ ) is the Markov state-transition density, and in which case the posterior distribution is f (ξ|ζ) = f k+1|k+1 (ξ|ζ 1:k+1 ). One can determine the "best" value of ξ using some Bayesoptimal state estimator, such as the maximum a posteriori (MAP) estimate (if it exists): An estimator is Bayes-optimal if it minimizes the Bayes risk for some cost function C(ξ, ξ ′ ) defined on states ξ, ξ ′ ( [14], pp. 54-63). This is the only meaning of "Bayes-optimal."

III. MEASUREMENT-TO-TRACK ASSOCIATIONS (MTA'S)
The MTA approach presumes the "small target" sensor model. A detection process (such as thresholding) is applied to a sensor signature, resulting in a set Z of point detections. Every target is assumed to be distant enough that it generates at most a single detection, but close enough that different targets produce distinct detections.
An MTA is a hypothesis about which tracks in X generated which measurements in Z. That is, assume that the measurements in Z ′ ⊆ Z were generated by the tracks in X ′ ⊆ X. Then the excess tracks in X − X ′ are "missed detections" and the excess measurements in Z −Z ′ are "false detections" or "clutter." In addition, there is a bijection (oneto-one and onto) function γ : X ′ ↔ Z ′ , which specifies that x ∈ X ′ generates γ(x) ∈ Z ′ . If X ′ = ∅ then Z ′ = ∅ and all of the measurements in Z are false detections.
Stated with greater mathematical specificity, an MTA is a 4-tupleα = (ν, X ′ , Z ′ , γ) such that: (a) ν is an integer According to Section II,α = (ν, X ′ , Z ′ , γ) cannot be a valid state representation since it depends on the measurementspace parameters Z ′ and γ. In particular,α cannot be specified without knowing, ahead of time and for any time, what measurement-set Z will be collected.
To address this conundrum, choose orderings x 1 , ..., x n and z 1 , ..., z m of the elements of X and Z, respectively. Then redefine an MTA to be a function α : In this case X ′ = {x i ∈ X| α(i) > 0} and γ( This strategem does not completely resolve the conundrum, since the cardinality |Z| of Z still must be known ahead of time and for any time. One can sidestep this difficulty by again redefining an MTA, this time as a pair (m,α (m) ) where m ≥ 0 is an integer andα (m) is a functionα (m) : {1, ..., |X|} → {0, 1, ..., m} such thatα (m) (i) =α (m) (i ′ ) > 0 implies i = i ′ . The unknown state variable m is thereby conceptually disengaged from the known measurement cardinality |Z| .

IV. THE LIKELIHOOD OF AN MTA
In what follows, let (a) L z (x) = f (z|x) be the single-target likelihood function (measurement distribution); (b) p D (x) be the (state-dependent) probability of detection; and (c) κ(z) be the intensity function of a Poisson clutter process, with λ = κ(z)dz the clutter rate (expected number of clutter measurements in each frame) and c(z) = κ(z)/λ the clutter spatial distribution. Then in Eq. (7.32) of [3] it was shown that the "global association likelihood" of an MTA is the probability (density) that the i'th track generates z, given the degree to which the track can be detected; and where the probability that it is undetected is: As an example, suppose that there is no clutter and no missed detections: λ = 0 and p D = 1. Then MTA's α reduce to permutations π : {1, ..., n} ↔ {1, ..., n} and Eq. (6) reduces to ( [3], Eq. (7.29)) ℓ Z|X (π) = ℓ(z π(1) |1) · · · ℓ(z π(n) |n).
The likelihood function ℓ Z|X (α) is not normalized. Its normalization is easily seen to be:

V. MTA'S AND BAYESIAN ANALYSIS
MTA's can be applied to multitarget tracking in numerous ways: single-hypothesis trackers, hypothesis-based MHT's, track-based MHT's, etc. [15]. Regardless of the approach, sooner or later the following question must be answered: How many targets are present, and what are their states?
To answer this question in a Bayesian fashion, we construct the posterior distribution on MTA's where L Z (m,α (m) ) =f (Z|m,α (m) ) is the likelihood function and p 0 (m,α (m) ) is a prior on the MTA's. The most probable MTA is then: Given this, the estimated number of targets is the number of i's such thatα (m) (i) > 0. Also, for any i such that α (m) (i) > 0, we get the corresponding estimated target state by updating f (x|i) using the measurement zα (m) (i) and the single- The likelihood function in Eq. (11) is where f (Z|α) is as in Eq. (10). Note that f (Z|m,α (m) )δZ = 1 for all (m,α (m) ).
Eq. (13) conceptually disengages m (unknown state parameter) from |Z| (known measurement parameter). The MTA approach can thereby be endowed with a Bayesian formulation. However, Eq. (13) seems somewhat peculiarcontrived, even-because it implies that m is always a nonrandom constant. This is at variance with the usual Bayesian presumption that unknown state variables are random variables. It is in perfect agreement, however, with the classical-statistics presumption that they are nonrandom constants. We therefore conclude that: • The MTA approach is not entirely consistent with Bayesian statistics.

VI. MTA'S AND PHENOMENOLOGY
Beyond this purely mathematical incongruity, one must address a more serious physical one: • However formulated, is an MTA actually a physical entity? That is, is it phenomenologically sensical to claim that certain measurements originated with certain tracks? This seems doubtful. An MTA is a heuristic extrapolation of the following special case to general multitarget scenarios. Suppose that we have a sensor with no missed or false detections. Further suppose that we have n targets that are well-separated with respect to the noise resolution of this sensor, as specified by L z (x) = f (z|x). In this case it seems self-evident that there is a permutation π 0 of the measurements z 1 , ..., z n such that z π0(i) originated with x i for all i = 1, ..., n-because there is only a small probability that z π0(i) could have originated with any target other than x i . Expressed in terms of association likelihoods, When all targets are very close together, however, it becomes statistically impossible to maintain that any particular measurement was generated by any particular target. Such a claim becomes even more difficult to maintain if the sensor has missed and false detections. To further insist that there is a "Bayes-optimal" MTA is to impose a phenomenologically spurious stucture upon the modeling of the physical system. Worse, by imposing physically extraneous information we potentially insert a hidden statistical bias into our analysis.
There is an additional issue: • Is an MTA a valid state representation of a multitarget system to begin with? This seems dubious. First, the MTA concept is specific to a particular sensor measurement model-one in which some detection process is applied to a sensor signature, resulting in a set of point detections. If the sensor has some other model, however-for example, a signature such as a pixelized image or a rotating-radar range-bin amplitude trace-then the MTA concept is completely meaningless.
Second, to apply MTA theory we must choose a priori orderings of the elements of both Z and X. Generally speaking, such orderings have no phenomenological basis. By imposing them, we potentially introduce additional unknown biases into our analysis.
Third and most importantly, a multitarget system is by definition an ensemble consisting of an unknown number of targets with unknown states. Thus its state representation must be based on the single-target states x 1 , ..., x n of those targets for n ≥ 0-and on nothing else (since, otherwise, phenomenologically extraneous information is potentially introduced). How do we proceed?
One frequently proposed approach is to employ concatenated vectors x = (x 1 , ..., x n ) (along with the null vector φ for n = 0). Such an approach is conceptually questionable [18], [15]. Estimation error is an important aspect of multitarget tracking. It must be possible to compute the distance between the "ground truth" multitarget state and a multitarget tracker's estimate of it. As an example, let x 1 = x 2 be Euclidean states. Then (x 1 , x 2 ) and (x 2 , x 1 ) are two possible vector representations of a twotarget system with states is not 0. Likewise, what is the distance between the two-target state (x 1 , x 2 ) and single-target state (y)? Or between (y) and the no-target state φ?
A multitarget state is more correctly modeled as a finite set {x 1 , ..., x n } with n ≥ 0; and there are well-defined and computationally tractable metrics for finite sets, such as the optimal sub-pattern assignment (OSPA) metric (see Section 6.2 of [3]).
At the same time, the fact that finite sets are unordered does not mean-as is often asserted-that they cannot be used to model temporally-connected tracks. This is because, in general, each x i has a unique identifying label-see Sections II and XI.
Consequently, the following is the only phenomenologically legitimate claim that can be ventured about the relationship between measurements and tracks: • The finite set Z of measurements was generated by the finite set X of tracks, with probability (density) f (Z|X) that this is the case. This insight is useless unless we can also answer the following question: • What is the concrete formula for f (Z|X)? These issues immediately lead us to finite-set statistics, in which L Z (X) = f (Z|X) is known as the multitarget likelihood function or multitarget measurement distribution.

VII. THE MULTITARGET LIKELIHOOD FUNCTION
Let Z = {z 1 , ..., z n } with |Z| = m and X = {x 1 , ..., x n } with |X| = n. Then f (Z|X) was derived in Eq. (12.139) of [8], and reiterated in Eq. (7.21) of [3]: where the summation is taken over all MTA's α : Note that Eq. (14) does not functionally depend on the particular orderings chosen for the elements of Z and X.
Eq. (14) might seem to require MTA theory since it involves MTA's. The important point to understand, however, is that the MTA's occurring in Eq. (14)) do not, as in MTA theory, arise from heuristic intuition. Rather they arise from a change of notation in a mathematically rigorous RFS derivation based on a statistically and phenomenologically rigorous RFS model. The purpose of the next section is to demonstrate this claim.

A. The RFS Measurement Model
were C is the Poisson clutter RFS; Υ(x) with |Υ(x)| ≤ 1 is the random measurement-set generated by a target with state x; and Σ is the total measurement-RFS for the entire scene.
B. The Belief Measure of Σ If Z has the Fell-Matheron topology then the statistics of Σ are completely characterized by its belief measure for all closed subsets T ⊆ Z. If target and clutter measurements are generated independently, then C, Υ(x 1 ), ..., Υ(x n ) are independent and where, if 1 T (x) is the indicator function of subset T , C. The p.g.fl. of Σ Substituting test functions 0 ≤ g(z) ≤ 1 for 1 T (x), we get the p.g.fl. of Σ ([3], Eq. (7.19)): where −1] (24)

D. The Set-Theoretic Formula for f (Z|X)
The multitarget likelihood function f (Z|X) is [9], [8]: where "δ/δZ" is the functional derivative with respect to Z. When applied to Eq. (22), the general product rule for functional derivatives (see [3], Eq. (3.68)) yields: where the summation is taken over all mutually disjoint (and possibly empty) subsets W 0 , W 1 , ..., W n of Z whose union is Z. Setting g = 0, from Eq. (28) we get

E. The MTA Formula for f (Z|X)
Because of Eq. (30), the only surviving terms in the summation in Eq. (29) are those for which W 1 , ..., W n are either empty or singleton; and W i contributes a factor to the product in Eq. (28) only if it is a singleton. Thus for a given choice of W 1 , ..., W n , define α : {1, ..., n} → {0, 1, ..., m} implicitly by {z α(i) } = W i if W i = ∅ and α(i) = 0 otherwise. Note that α is an MTA in the sense of Section III. Conversely if we are given α, Thus there is a one-to-one correspondence between MTA's α and lists W 1 , ..., W n of mutually disjoint empty or singleton subsets of Z. Furthermore, only those i's with α(i) > 0 contribute a factor to the product. Consequently, Eq. (29) can be rewritten as: from which Eq. (14) immediately follows.

F. The RFS Interpretation of MTA's
This leads us to the following inferences: 1) The MTA's α in Eq. (14) do not arise from heuristic intuition. Rather, they are the consequence of a change of mathematical notation-i.e., as a mathematically equivalent way of rewriting the purely set-theoretic formula of Eq. (29). 2) Eq. (14) involves all possible MTA's, with no MTA having a greater impact on the value of f (Z|X) than any other. Thus Eq. (14) does not assign any phenomenological reality to MTA's as isolated entities. 3) Also, f (Z|X) does not functionally depend on particular orderings of the elements of Z or X. Thus the potential statistical biases associated with the MTA approach, as identified in Section VI, cannot occur.

IX. MULTITARGET BAYES OPTIMALITY
This material reiterates the discussion in Section 5.3 of [3]. Suppose that f 0 (X) is the multitarget prior distribution and that we have collected a measurement-set Z. Then as per Section II, the multitarget posterior distribution is where now the integral is the set integral ([3], Section 3.3) 1 n! f ({x 1 , ..., x n })dx 1 · · · dx n .
(34) Also as per Section II, a multitarget state estimator is a functionX(Z) of the measurements Z whose values are finite state-sets. It is Bayes-optimal if it maximizes the multitarget Bayes risk with respect to some cost function C(X, Y ) defined on multitarget state-sets X, Y . This is the only meaning of the term "Bayes-optimal" in a multitarget context. The joint multitarget (JoM) and marginal multitarget (MaM) estimators (see Section 14.5 of [8]) have been shown to be Bayes-optimal; and the former has been shown to be statistically consistent.

X. RELATIONSHIP BETWEEN MTA AND FISST
Assume a priori that n targets are known to exist and are statistically independent. Then the multitarget distribution that describes them is where X = {x 1 , ..., x n } with |X| = n and where the summation is taken over all permutations π on 1, ..., n.
If f (Z|X) is defined as in Eq. (14), then the following equation, Eq. (7.48) of [3], establishes the basic relationship between RFS theory and MTA theory: That is, the probability (density) f (Z) that the measurementset Z will be collected from n independent tracks is the same thing as the total (unnormalized) likelihood of association between Z and X. This demonstrates that (Remark 15, p. 173 of [3]): • The MHT approach is a heuristic approximation. For, the optimal approach to multitarget tracking is the multitarget Bayes filter:

XI. THE GLMB FILTER
Suppose, instead, that f 0 (X) is chosen non-heuristicallyspecifically, that it is a generalized labeled multi-Bernoulli (GLMB) distribution (to be defined shortly). Let a GLMB distribution be denoted as f (X|p) where p is a parametervector (also to be defined shortly).
The three most important properties of GLMB distributions are as follows: 1) If the previous posterior distribution f k|k (X|Z 1:k ) is a GLMB distribution f (X|p k|k ) then so is the predicted distribution: f k+1|k (X|Z 1:k ) = f (X|p k+1|k ) for some p k+1|k . 2) If the predicted distribution is f k+1|k (X|Z 1:k ) is a GLMB distribution f (X|p k+1|k ) then so is the new posterior distribution: f k+1|k+1 (X|Z 1:k+1 ) = f (X|p k+1|k+1 ) for some p k+1|k+1 . 3) An arbitrary labeled multitarget distribution f (X) can be approximated by a GLMB distribution that has the same PHD and cardinality distribution as f (X) [11]. 2 Thus if we choose f 0 (X) = f (X|p k+1|k ) then f 0 (X) is not a heuristic choice-it is the actual predicted distribution: f (X|p k+1|k ) = f k+1|k (X|Z 1:k ). A similar claim can be made for the time-update step. If f k|k (X|Z 1:k ) is approximated heuristically as some a priori distribution f − (X), then the corresponding predicted distribution f k+1|k (X|Z 1:k ) is almost never correct because f − (X) is not an actual posterior distribution. But if we instead choose f − (X) = f (X|p k|k ) then f − (X) is not a heuristic choice, since in this case it is the actual posterior distribution: f (X|p k|k ) = f k|k (X|Z 1:k )which in turn means that f k+1|k (X|Z 1:k ) is the actual predicted distribution.
Properties 1 and 2 state that the family of GLMB distributions is an exact closed-form solution of the multitarget Bayes filter. It follows that, when restricted to GLMB distributions, the labeled multitarget Bayes filter can be equivalently replaced by a filter on the parameters alone-i.e., by the GLMB filter (invented by Vo and Vo in 2011 [17], [16]): The GLMB filter is not only provably Bayes-optimal but is a true multitarget tracker. That is, because it explicitly accounts for target labeling (see Section XI-A), it inherently incorporates a provably Bayes-optimal track-management scheme. A chapter-length discussion of the GLMB filter can be found in [3]. See also [4] for a formulation based solely on p.g.fl.'s.
What follows is a brief overview of the GLMB filter. It is organized as follows: (a) labeled RFS's; (b) GLMB distributions; (c) the GLMB filter; (d) a comparison of MHT's and the GLMB filter; and (e) a discussion of "unlabeled" exact closed-form filters.

A. Labeled RFS's
Track labeling in an RFS context was first addressed in 1997 in [1], pp.135,196-197. However, the first implementations of RFS filters did not take track labels into account. Later implementations, such as Gaussian mixture CPHD filters ( [3], pp. 244-250), addressed labeling heuristically. The subject was not addressed systematically until the 2011 and 2013 "labeled RFS" (LRFS) papers [17], [16] by Vo and Vo. We address labeling via the following change of notation. Single-target states are now assumed to have the formx = 2 The cardinality distribution p Ξ (n) and PHD D Ξ (x) of RFS Ξ are: ∈X where x is a conventional target state (e.g., kinematic variables and, if appropriate, target type) and ℓ is a track label. 3 The integral onX is defined by where f (x, ℓ)dx = 0 for all but a finite number of ℓ. The corresponding set integral is ℓ 1 ), ..., (x n , ℓ n )})dx 1 · · · dx n .

Now letX
be a finite subset ofX. The set of labels of the targets inX is denoted asX Given this,X is a labeled multitarget state-set if |X L | = |X|-i.e., if its elements have distinct labels. An RFSΞ ⊆ X is a labeled RFS (LRFS) if for all realizationsΞ =X ofΞ. The distribution of an LRFSΞ must have the following property: fΞ(X) = 0 if |X L | = |X|. Thus, for example, a Poisson RFSΞ ofX is not an LRFS.

B. GLMB Distributions
A multitarget probability distributionf (X) onX is a generalized labeled multi-Bernoulli (GLMB) distribution if it has the following distribution and p.g.fl [17], [16]:  3 The symbol "ℓ" has previously been used to denote association likelihoods, ℓ Z|X (α), but is repurposed here to denote labels. The meaning will always be clear from context.

C. The GLMB Filter
The following intuitive overview of the GLMB filter is adapted from Section 15.4.2 of [3]. Suppose that: 1) Every label ℓ in L has the form ℓ = (k, i) where t k with k ≥ 0 is the time that the track was created; and where integer i ≥ 1 distinguishes the track from any other track created at time t k . Let ℓi,α 1:k−1 (x) will be of two types: i) the distribution of a track that persisted from the previous time t k−1 , and which thus arose from the previous time-history α 1:k−1 ; or ii) the distribution of a newly-appearing track, and which therefore does not depend on α 1:k−1 . 2) Let Z 1:k : Z 1 , ..., Z k be the time-sequence of measurement-sets at time t k . Then the measurementupdated distribution at time t k is GLMB of the form: where, ifX L = {ℓ 1 , ..., ℓ n } with |X L | = n, then ω k|k α 1:k ({ℓ 1 , ..., ℓ n }) is the weight of the hypothesis that: a) there are n tracks with distinct labels ℓ 1 , ..., ℓ n ; and b) their respective track distributions are given by s k|k ℓ1,α 1:k (x) =s k|k α 1:k (x, ℓ 1 ), ..., s k|k ℓn,α 1:k (x) = s k|k α 1:k (x, ℓ n ); and c) these distributions arose as a consequence of the time-history α 1:k of MTA's; and d) for each i = 1, ..., n, the track distribution s k|k ℓi,α 1:k (x) will be of two types: i) the distribution of a track that was not detected and which therefore arose from the previous time-history α 1:k−1 ; or ii) the distribution of a track that was detected and which therefore arises from the current timehistory α 1:k .

D. The GLMB Filter and MHT
Like many MHT-type algorithms, the GLMB filter propagates time-histories of MTA's. A major conceptual difference, however, is that in the GLMB filter an MTA α j : L 0:j → {0, 1, ..., |Z j |} in an MTA time-sequence α 1:k = (α 1 , ..., α k ) is not a representation of the multitarget state. Rather, it is: • an index of a weighted hypothesis about (a) which labels exist in the scene; and, (b) which track distributions correspond to those labels. In particular, no attempt is made to estimate the best MTA at any given time-step. Rather, the GLMB filter estimates the best state-set using an approximation of a Bayes-optimal multitarget state estimator.
The baseline computational complexity of the GLMB filter is roughly the same as that of track-oriented MHT: it is combinatorial in both m (the current number of measurements) and n (the current number of tracks). However, one can greatly decrease complexity using statistical sampling methods. The Gibbs sampler is a computationally efficient special case of the Metropolis-Hasting MCMC algorithm which, in this application, has an exponential convergence rate. Vo and Vo have used it-together with a merging of the time-update p k|k → p k+1|k and measurement-update p k+1|k → p k+1|k+1 into a joint update p k|k → p k+1|k+1 -to devise an implementation of the GLMB filter with computational order O(mn 2 ) [19]. This results in an at least two orders of magnitude computational improvement, as compared to the original GLMB filter implementation described in [20].

E. "Unlabeled"Exact Closed-Form Filters
After the GLMB filter was introduced in 2011 [17], a few authors began investigating unlabeled exact closed-form filters. First and most notably, [22,Thms. 1,2] employed "hybrid Poisson and multi-Bernoulli" RFS's Ξ of X with p.g.fl.'s of the form Unlabeled exact closed-form filters are both theoretically and practically redundant. They are inherently inferior to the GLMB filter because they are not true multitarget trackers. Furthermore, labeling permits a big decrease in computational complexity in the prediction step of the GLMB filter [16]. This decrease is unavailable for unlabeled distributions, and in particular for those as in Eq. (49). Indeed, severe and purely ad hoc approximations [22,Eqs. (61,73)], [21] are necessary to address this difficulty-thereby inviting skepticism about the "exactness" part of any "exact closed-form" claim.
The "unknown targets" model in Eq. (49) is theoretically questionable. By implication, the "known targets" must be modeled by the summation in Eq. (49). Given this, Eq. (49) implies that the unknown-target RFS and known-target RFS are statistically independent-an impossibility, since the two are inherently correlated but the latter is non-Poisson.
Finally, the following claim must be addressed: • "...[the 2013 Vo-Vo GLMB filter paper [16] ] shows that the labelled case can be handled within the unlabelled framework by incorporating a label element in to the underlying state space" [22, p. 1675]. This assertion is manifestly untrue. An RFS filter on X cannot be converted to an LRFS filter onX simply by substituting (x, ℓ) whenever x occurs in the filter equations. This is because such substitutions do not forbid state-sets with non-distinct labels-i.e., |X| > |X L | becomes possible. For example, Poisson RFS's onX are not LRFS's-with the consequence that the factor e D[h −1] in Eq. (49) is inherently non-LRFS. Thus it is not possible to use Eq. (49) as the basis of a theoretically rigorous LRFS filter-and thereby of a theoretically rigorous true multitarget tracker.

XII. CONCLUSIONS
This paper has addressed the following claims made about multi-hypothesis trackers (MHT's), as well as the fundamental concept upon which they are based, the measurement-to-track association (MTA): 1) Claim 1: MHT's are not only theoretically rigorous, but theoretically rigorous within the Bayesian framework ("Bayes-optimal"). 2) Claim 2: "The RFS model of the multiple target state is an approximation, because the Bayes posterior RFS is not exact, but is an approximation based on the earlier invocations of the PHD approximation used to close the Bayesian recursion. The Bayes posterior RFS is an approximation even before the PHD approximation is invoked. The right model of the multitarget state is that used in the multi-hypothesis tracker (MHT) paradigm, not the RFS paradigm." 3) Claim 3: The RFS approach-and specifically the generalized labeled multi-Bernoulli (GLMB) filter of Vo and Vo [17], [16], [20]-is essentially a mathematically obfuscated reinvention of MHT. Claim 1 can be ascribed to uncritical acceptance of unexamined conventional wisdom. Specifically, in this paper it has been demonstrated that: 1) MTA's are not phenomenologically real. Rather, they are purely mathematical entities arising from a change of notation in the RFS derivation of the multitarget likelihood function for the "standard" multitarget measurement model. 2) The MHT/MTA approach is neither theoreticially rigorous nor strictly Bayesian. It is, rather, an intuitiveheuristic approximation of the Bayes-optimal approach to multitarget tracking, the multitarget Bayes filter.
Unlike them, however, it is a provably Bayes-optimal exact closed-form solution of the labeled multitarget Bayes filter, which can be considerably faster than conventional combinatorial algorithms. As for Claims 2 and 3, they appear attributable to a superficial understanding of the finite-set statistics literature. For example, the first sentence of Claim 2 repeats a common misconception: that the "RFS model of the multitarget state" is the same thing as the "PHD approximation" of that state.
To the contrary, the actual "RFS model of the multitarget state" is the evolving random finite multitarget state-set Ξ k|k . The PHD filter results when we assume that Ξ k+1|k is, approximately and for every k ≥ 0, a Poisson RFS Ξ Poiss k+1|k . This Ξ Poiss k+1|k is an approximation, not a model. Furthermore, it is only the simplest of a series of increasingly more accurate RFS approximations: i.i.d.c., multi-Bernoulli, labeled multi-Bernoulli, and generalized labeled multi-Bernoulli [9], [2], [3].