Exact Closed-Form Multitarget Bayes Filters

The finite-set statistics (FISST) foundational approach to multitarget tracking and information fusion has inspired work by dozens of research groups in at least 20 nations; and FISST publications have been cited tens of thousands of times. This review paper addresses a recent and cutting-edge aspect of this research: exact closed-form—and, therefore, provably Bayes-optimal—approximations of the multitarget Bayes filter. The five proposed such filters—generalized labeled multi-Bernoulli (GLMB), labeled multi-Bernoulli mixture (LMBM), and three Poisson multi-Bernoulli mixture (PMBM) filter variants—are assessed in depth. This assessment includes a theoretically rigorous, but intuitive, statistical theory of “undetected targets”, and concrete formulas for the posterior undetected-target densities for the “standard” multitarget measurement model.


Important RFSs
Various RFSs of importance to this paper are most easily described using their PGFLs:
In this paper we will be concerned with f k (Z|X) for only the "standard" multitarget measurement model, which has PGFL: Here, at time t k , p k D (x) is the sensor probability of detection, f k (z|x) = L k z (x) is the sensor measurement density, k k (z) is the intensity function of a Poisson clutter process, and L k g (x) = g(z) · f k (z|x)dz. For notational simplicity we will usually suppress the time-index k-e.g., p k D (x) = p D (x),L k z (x) = L z (x), etc. Likewise, we will be concerned with f k|k−1 (X|X ) for only the "standard" multitarget motion model, which has PGFL: Here For notational simplicity we will usually suppress the time-index k-e.g., p k|k−1 S = p S , M k|k−1 x = M x , etc.

Labeled Random Finite Sets (LRFSs)
Track labeling (or, more generally, target identity) in an RFS context was first addressed in 1997 in [25] (pp. 135, 196-197) and in 2007 in [14] (pp. 505-508). However, the first implementations of RFS filters did not take track labels into account because of computational concerns. Later implementations, such as the Gaussian mixture cardinalized probability hypothesis density (GM-CPHD) filter, addressed labeling heuristically [13] (pp. 244-250). The labeling issue was not addressed in a theoretically rigorous and systematic fashion until 2011 in the labeled RFS (LRFS) papers of Vo and Vo [7,8].
In LRFS theory, single-target states are assumed to have the form x = (u, ) ∈ × L 0 where u ∈ is a kinematic target state-vector and is an element of a countable set L 0 of target labels. The integral on × L 0 is defined by: (23) where, by assumption, f (u, )du = 0 for all but a finite number of . The corresponding set integral is: Let X = {(u 1 , 1 ), . . . , (u n , n )} ⊆ × L 0 . Then the set of labels of the targets in X is denoted as X L = { 1 , . . . , n }. Given this, X is a labeled multitarget state-set if |X L | = |X|-i.e., if its elements have distinct labels, in which case targets are uniquely identified. An RFS Ξ ⊆ × L 0 is a labeled RFS (LRFS) if |Ξ L | = |Ξ| for all realizations Ξ = X of Ξ. Consequently, the distribution of an LRFS Ξ has the following property: In LRFS theory, labels are unknown random state variables, which must be Bayes-optimally estimated along with the unknown random kinematic states u 1 , . . . , u n . By way of contrast, in conventional track-management approaches labels are deterministic, heuristic bookkeeping devices.
The LRFS approach requires appropriate definitions of p k|k−1 The primary distinction is that f k|k−1 (u, |u , ) = δ , ·f k|k−1 (u|u , )-i.e., targets do not change labels. For purposes of multitarget tracking and classification (see Remark 4), these quantities will usually depend on the labels. However, for general tracking it can usually be assumed that p k|k−1

Important Labeled RFSs
These are most simply defined in terms of their PGFLs, where 0 ≤ h(u, ) ≤ 1 are labeled test functions:

2.
Labeled Multi-Bernoulli Mixture (LMBM) LRFS: Any labeled multitarget distribution can be approximated by a GLMB distribution that has the same PHD and cardinality distribution [26].

The GLMB Filter
This filter was introduced in 2011 in [8] and elaborated in [6,7]. Suppose that L 0 = {0,1, . . . ,} × {1, . . . } and if = (k,i) then t k is the time that track was initiated and i ≥ 1 distinguishes it from any other track created at time t k . At time t k , a finite number of labels in L k = {k} × {1, . . . } are assigned to hypothesized newly-appearing tracks. Thus, at time t k , the set L [0:k] of all currently assigned track labels is a finite subset of L 0:k = {0,1, . . . , k} × {1, . . . }; and each such label is an unknown discrete random variable ∈ L [0:k] which must be estimated.
Given this, the family of GLMB distributions solves the labeled multitarget Bayes filter in exact closed form. In particular, at time t k : where the summation is taken over all (α 1 , . . . ,α k ) ∈ A 1 × . . . × A k ; where each α i :L [0:i] → {0,1, . . . , |Z i |} is a label-to-measurement association-i.e., if α i ( ) = α i ( ) > 0 then = ; where s k|k α 1 ,...,α k (u, ) is a target spatial distribution; and where A i denotes the set of all such associations α i at time t k .
The GLMB filter is a true Bayesian multitarget tracker because it is guaranteed to propagate target tracks with unique track labels (a "true" tracker), which in turn are realizations of unknown random identity-variables (a "Bayesian" tracker).
Moreover, because it is an exact closed-form solution of the labeled multitarget Bayes filter, the GLMB filter has provably Bayes-optimal track-management. At time t k−1 , an (approximate) Bayes-optimal multitarget state estimate X k−1|k−1 is extracted from f k−1|k−1 (X|Z 1:k−1 ). At time t k , a similar estimate X k|k is extracted from f k|k (X|Z 1:k ). If (u, ) ∈ X k−1|k−1 and (u , ) ∈ X k|k then (u, ) and (u , ) both belong to the track with label . If (u , ) ∈ X k|k for any u then track has been dropped. If (u, ) ∈ X k|k but (u , ) ∈ X k−1|k−1 for any u then a track with label has been initiated or reacquired.
Due to the number of association-vectors (α 1 , . . . , α k ) increases without bound, the summation in Equation (25) must be pruned at every time-step. The information loss due to pruning can be characterized exactly-i.e., the L 1 norm between the pruned and unpruned distributions is the sum of the weights of the pruned terms [6] (Proposition 5).
Using Gibbs stochastic sampling techniques, the GLMB filter can be implemented with computational order O(n 2 m) where m is the current number of measurements and n the current number of tracks [5]. This is particularly advantageous when clutter is dense. The most recent such implementations can simultaneously track over a million 2D targets in significant clutter using off-the-shelf computing equipment [27].

Remark 4.
Every target has a unique identity state variable [25] (pp. 135, 196-197). A track label is a provisional identity assigned to a target in lieu of its actual identity. The GLMB filter can be generalized from joint multitarget detection and tracking to joint multitarget detection, tracking, and identification. This is accomplished by incorporating identity information into target labels [9].

The LMBM Filter
In [10] it was shown that the family of LMBM distributions solves the labeled multitarget Bayes filter in exact closed form. The corresponding LMBM filter is, therefore, a true Bayesian multitarget tracker with provably Bayes-optimal track management. It is somewhat less computationally expensive than the GLMB filter, but also less accurate since LMBM distributions are less accurate approximations of labeled multitarget distributions than GLMB distributions.
All target-birth RFSs are assumed to be Poisson-in our notation, G k|k−1 −1] . Given this, the PMBM filter propagates PMBM distributions in exact closed form. Specifically,  [11,12] is somewhat sketchy. The following PGFL-based verification of it will be useful in the sequel.
U-PMBM Filter Time-Update. According to Equation (20) and substituting G k|k−1 , the PGFL prediction formula is: where, by assumption, G k−1|k−1 [h] is PMBM: Thus, predicted PGFL is easily seen to be PMBM: where: U-PMBM Filter Measurement-Update. Let Z = {z 1 , . . . , z m } with |Z| = m be collected at time t k . According to Equation (21) the measurement-updated PGFL is: where, by Equation (22) ] and, by assumption, G k|k−1 [h] is PMBM: , where: and the measurement-updated PGFL is: where: Since each F l [g,h] has the same Poisson factor e κ[g−1]+D[h(1+p D L g−1 )−1] , it is sufficient to show that the measurement-update of a PMB PGFL is a PMBM PGFL Accordingly, in what follows we neglect the index l in F l [g,h].

Undetected-Target Interpretation of the U-PMBM Filter
The PMBM filter therefore, solves the unlabeled multitarget Bayes filter in exact closed form. However, the PMBM approach goes beyond this to adopt a specific physical interpretation of PMBM RFSs. Let: be the PMBM PGFL at time t k . It is clear from Equations (29) and (41) that the timeand measurement-updates for the Poisson factors are, respectively, as a model of the "undetected targets" at time t k [4,11,12]. According to [11] (p. 1103), these are " . . . targets that have never been detected"-i.e., not detected at times t 1 , . . . , t k . It was subsequently stipulated that " . . . detected targets cannot become undetected targets" [4] (p. 246).
The primary justification for the PMBM approach is the following: "One significant benefit of the inclusion of a Poisson component is in initialization of the tracker . . . The Poisson distribution provides a convenient mechanism for specifying a prior distribution on the number and position of targets when little information is available" [12] (p. 1670).
However, this potential advantage is negated by a major theoretical obstacle: Poisson RFSs require non-unique labels and so are not LRFSs (see Remark 3). Due to this, they cannot be used in any theoretically rigorous, true multitarget tracker.
A more subtle obstacle is this: it is theoretically impossible to prune PMBM distributions in a practically useful manner. When a GLMB distribution (Equation (25)) is pruned, the pruned distribution is a GLMB distribution. When a PMBM distribution is pruned, however, it is usually not even a multitarget density function. First consider an LMB distribution f Ξ ({x 1 , . . . , x n }) [14] (Equation (11.133)). Any term in it has the form f i 1 ,...,i n (x 1 , . . . , x n -and therefore, not a multitarget density-neither is any other pruning of f Ξ . Now, let f Ξ be a PMB distribution: where α: {1, . . . , ν} → {0, 1, . . . , n} is an association. Its terms have the same form as before, except that the f i can be equal to D but those f i that are not D are distinct. Since f i 1 ,...,i n is symmetric only when f j = D for all j = 1, . . . , n, no pruning of f Ξ other than this case is a multitarget density. What is theoretically permissible is to prune an MBM (resp. PMBM) PGFL by eliminating one or more of its MB (resp. PMB) PGFL terms. However, pruning the individual terms of the corresponding MB (resp. PMB) distributions is not permissible-which is exactly what is required to eliminate specific small-weight hypotheses.

"Label-Augmented" PMBM (LA-PMBM) Filter
As was noted at the beginning of Section 3.1, unlabeled RFS-based filters, such as the GM-CPHD filter, can heuristically propagate tracks even though they are not true multitarget trackers. The U-PMBM filter can propagate tracks using similar heuristics, but it-like the GM-CPHD filter-is not a true multitarget tracker since it is unlabeled. Accordingly, in 2015 it was modified as follows: " . . . [the Vo-Vo paper [7]] shows that the labelled case can be handled within the unlabeled framework by incorporating a label element in to the underlying state space" [12] (p. 1675). That is, it was claimed that the PMBM filter can be extended to the labeled case by replacing the unlabeled single-target state space with the labeled state space × L 0 . This modified PMBM filter will be referred to as the "label-augmented" PMBM (LA-PMBM) filter. It must propagate PMBM RFSs of × L 0 with PGFLs: Now, however, the PHD D k|k and spatial distributions s k|k l,i must have the respective forms D k|k (u, ) and s k|k l,i (u, ), where D k|k (u, )du = 0 and s k|k l,i (u, )du = 0 for all but a finite number of and where s k|k l,i [1] = s k|k l,i (u, )du = 1 for all l,i. There is a serious theoretical difficulty, however: the s k|k l,i are not track distributions. For if otherwise, s k|k l,i (u, )du = 1 would imply that s k|k l,i [1] > 1, a contradiction. Therefore, s k|k l,i appears to be physically meaningless.
Beyond this, the above claim-that "the labelled case can be handled within the unlabeled framework"-is untrue. As was noted in Remark 3, a Poisson RFS on × L 0 is not an LRFS since it requires nondistinct target labels. Consequently, it is not possible for the LA-PMBM filter to be a true multitarget tracker. Instead, it " . . . is able to maintain track continuity implicitly based on the information provided by metadata" [4] (p. 245)-that is, only heuristically.

"Hybrid Labeled-Unlabeled" RFSs
Like the U-PMBM filter, the LA-PMBM filter is not a true multitarget tracker-a fact that was pointed out in 2017 in [23] (Section XI-E). Apparently to address this issue, it was modified in 2018 as follows [4] (p. 246): A single common label-*, say-is assigned to all "undetected targets" at all times, whereas "detected targets" are uniquely labeled as in LRFS theory. Additionally, the "undetected-target" RFS at any time is assumed to be a Poisson RFS on × { *} (a slightly later paper, [3], also appears to employ the H-PMBM approach, except that * is implicit rather than explicit.) No careful theoretical foundation for the hybrid approach was provided in [4]. It is, therefore, necessary to construct one here. The label space is L 0 = L \ *∪{ *} where L \ * = L 0 − { *}. Given a finite subset X = {(u 1 , 1 ), . . . , (u n , n )} ⊆ × L 0 , as usual let X L = { 1 , . . . , n } denote the set of labels in X. Additionally, let X* = {(u, ) ∈ X| = *} be the subset of X of targets that are "undetected"; and let X \ * = X − X* be the targets in X that are "detected." Then it is assumed that the only legitimate state-sets X are those such that |X \ *| = |X L − { *}|-i.e., those for which the detected targets have distinct labels other than *. Let us refer to these as "hybrid state-sets." Let Ξ be a "hybrid RFS"-i.e., an RFS of × L 0 whose instantiations are hybrid. Then it must be the case that f Ξ (X) = 0 if X is not hybrid. Thus, every distribution defined for hybrid X must include the factor δ |X \ * |,|X L −{ * }| .

"Hybrid Labeled-Unlabeled" PMBM (H-PMBM) Filter
Now let: be the PGFL of a PMBM RFS Ξ on × L 0 with * ∈ L 0 , where the MBM factor in Equation (45) has been replaced by an LMBM factor since "detected targets" are now assumed to be uniquely labeled; and where L k|k l are finite subsets of L 0 and s k|k l, (u)du = s k|k l (u, )du = 1 for all ∈ L k|k l . Since the Poisson factor F u k|k [h] = e D k|k [h−1] applies only to "undetected targets" with common label *, it must be the case that * L k|k l for every l = 1,..,N k|k and that D k|k (u, ) = δ , * · D * k|k (u) for some D * k|k (u). We will refer to Equation (48) as an "H-PMBM PGFL".
Given this, f d k|k (X d |Z 1:k ) and f u k|k (X u |Z 1:k ) " . . . .can be propagated in parallel, in both cases by carrying out a prediction step and an update step . . . " [4] (p. 248); where the undetected-target Moreover, the following claim is made about these two filters: "In the following development of the prediction and update steps, we use the fact that the posterior pdf of the overall multitarget state RFS . . . factorizes as . . . " (in current notation): However, Equation (49) is untrue. For by Bayes' rule:

Theoretical Issues With the H-PMBM Filter
From Section 4.1 we know that the PMBM filter on × L 0 is guaranteed to propagate PMBM distributions on × L 0 in exact closed form. However, does it propagate hybrid PMBM distributions in exact closed form? This does not appear to be the case.
Consider, for example, Equation (36) with k = 1 and with the single-target state space being × L 0 (with *∈ L 0 ) rather than : Here, D 1|0 since p 1 D and L 1 z are independent of labels, and: Given that D 1|0 B (u, ) = δ , * · D 1|0 B * (u), D 1|1 (u, ) has the correct form for an H-PMBM PGFL However, the product is not an LMB PGFL This is because the individual Bernoulli factors are indexed by the elements of Z 1 , not by labels in L 0 .
Thus, as a heuristic workaround, a distinct label z * is assigned to each z ∈ Z 1 : "For each measurement . . . a new Bernoulli component is created, to which [a] unique label . . . is assigned" [4] (following Equation (107)); and: " . . . [e]ach measurement at each time step gives rise to a new potentially detected target. That is, there is the possibility that a new measurement is the first detection of a target, but it can also correspond to another previously detected target or clutter, in which case there is no new target. As this target may exist or not, its resulting distribution is Bernoulli and we refer to it as [sic] 'potentially detected target'" [3] (p. 1885).
It follows that the labeled track distribution of the Bernoulli representation of the "potentially detected target" corresponding to z∈ Z 1 must be s 1|1 (u, z ) = s 1|1 z (u, z ). This workaround results in at least three theoretical difficulties:

1.
There is an inherent theoretical conflict between labeling using ∈ L 0 and labeling using z∈ Z 1 .

2.
In [4] (pp. 245-246) the following was stated: "In cases of limited prior birth information, one typically uses a heuristic to generate new Bernoulli components based on measurements from the previous time step (Reuter et al. [28]). Such heuristics can be avoided with the MB-Poisson model . . . " This is untrue on both counts. First, and as was noted following Equation (17), approaches that dynamically estimate the target-birth process "based on measurements from the previous time step"-i.e., based on Z k-1 -are theoretically permissible. Examples include [29] and [30]-and [28]. Second, note that the "MB-Poisson model" employs a "heuristic to generate new Bernoulli components based on measurements from": the current time-step! Thus, how is it conceptually different from the approach in [28]? 3.
More seriously, the dynamical transition of undetected targets to detected ones occurs during the measurement-update, as mediated by f k (Z k |X), rather than-as theoretically should be the case-during the time-update, as mediated by f k|k−1 (X|X ,Z 1:k−1 ). Thus, f k (Z k |X) has been implicitly assumed to have the form f k (Z k |X,X )-which is not the case (see Equation (54)).
The H-PMBM filter therefore, does not appear to have a theoretically rigorous, closed-form mechanism for assigning labels to newly-detected "undetected targets." And this fact is a direct consequence of the Poisson factor in Equation (48).
However, there is a far more fundamental theoretical and phenomenological difficulty: the hybrid approach has no basis in physical reality. Targets are physically real entities regardless of whether or not they are detected. They have distinct (but unknown) real-world identities and therefore, inherently have distinct (unknown) labels. As was noted in Remark 4, target labels in L 0 are provisional identities assigned in lieu of more precise identifying information. LRFS labels are, therefore, not "artificial variables that are added to the target states" [3] (p. 1884). Rather, they are standbys for the realizations of a physically real random state-variable: target identity. The H-PMBM approach, by way of contrast, requires targets with label * to have multiple kinematic states simultaneously-a physical impossibility.

A Statistical Theory of Undetected Targets
The meaning of the "undetected target" concept is extremely unclear. Thus, the purpose of this section is to devise a statistically rigorous-and yet intuitive-theory of undetected targets. As stated in the Introduction, our ultimate goal is to construct a concrete formula for the posterior "undetected targets" PGFL G u k|k [h|Z 1:k ]. The argument presented is as follows.
1. Section 5.1: We surmise that the "undetected target" concept is meaningful only at the instant that an observation process-in the form of the standard multitarget likelihood function f k {Z k |X)-is applied.

The "Undetected Target" Concept
At its most elemental level, the concept of "detected" vs. "undetected" target at time t k is independent of previous measurement history. The multitarget predicted distribution f k:k-1 (X|Z 1:k-1 ) determines how probable any given multitarget state-set X will be at time t k . However, only the current multitarget likelihood function f k (Z|X) determines which elements of X are detected vs. undetected at time t k .
The question then becomes: Given a finite subset X ⊆ , which elements of X generated measurements in Z k and which did not? The most that we can say is that, for each Y ⊆ X, there is some probability p d k (Y|X) that all elements of Y ⊆ X generated measurements in Z k . The detected-target set is, therefore, a discrete RFS Ξ d k ⊆ X. Likewise, there is some probability p u k (V|X) that no elements of V ⊆ X generated measurements in Z k . The undetected-target set is therefore, a discrete RFS Ξ u k ⊆ Xwith Ξ u k = X − Ξ d k . Given this, the following questions will be addressed:

Remark 5.
In general a multitarget state-set X will be labeled. However, the labeled version of Equation (54) is almost identical in form to Equation (54): f k (Z|X) = e −λ κ Z if X is not labeled (not just if X = ∅). Additionally, the "undetected target" concept was originally raised in the context of unlabeled RFSs. Thus, it is sufficient to use Equation (54).
Choose a particular α ∈ A. Then X d:α = {x i ∈ X|α(i) > 0} is the set of x i ∈ X that-according to the hypothesis α-generated measurements in Z. Likewise, X u:α = {x i ∈ X|α(i) = 0} is the set of those that did not. Now note the following: MTAs are in one-to-one correspondence with pairs (Y,τ) where Y ⊆ X with |Y| ≤ |Z k | and where τ:Y ⇒ Z k is a one-to-one function (i.e., τ(y) = τ(y ) implies y = y ).
For on the one hand, let us be given α.
On the other, let us be given a pair (Y,τ). Then for each i ∈ {1, . . . , n} define It is easily verified that the transformations α → (Y α ,τ α ) and (Y,τ) → α (Y,τ) are inverses of each other. Now define: where the unitless ratio: is a measure of how "target-like" vs. "clutter-like" the measurement τ(y) is (additionally, note that the " * " in "L * Z " and " f * k " no longer refers to the label " *" in Section 4.4.) Given this, note that the multitarget likelihood function can be rewritten as: It therefore, follows that f * k (Z|X)δZ = 1 for all X. Thus, f * k (Z|X) is the same thing as f k (Z|X), but under perfect-detection conditions.

The General Detected-Target Likelihood Function
Given these preliminaries, let X be a fixed finite subset of and define: where, note, (1 X ) Y = 1 if Y ⊆ X and (1 X ) Y = 0 otherwise. This is a continuous density in Z and a discrete distribution in Y: where the final equation follows from Equation (11). Equation (61) is the probability that all of the elements of the subset Y of X generated measurements; and is largest when p D (x) ≈ 1 for x ∈ Y and p D (x) ≈ 0 for x ∈ X − Y, where "≈" denotes approximate equality. It is the distribution of the detected-target RFS in X: Equation (57) has the following interpretation: L Z (X) is the unweighted average of hypotheses p d k (Z, Y|X) regarding the likelihood that subset Y of X generated measurements in Z. The factor (1 − p D ) X−Y p Y D quantifies the "raw detectability" of Y, whereas f * k (Z k |Y) measures the degree to which detectability is degraded by clutter under perfect-detectability conditions.
We need to transform p d k (Z, Y|X) so that it becomes a continuous density f d k (Z, Y|X) with respect to Y. This is accomplished as follows. For X = {x 1 , . . . , x n } with |X| = n and Y = {y 1 , . . . , y ν } with |Y| = ν, define: where the summation is taken over all one-to-one functions τ:{1, . . . , ν} ⇒ {1, . . . , n}. This is a multitarget density function with respect to Y. Note that Equation (64) can be rewritten in the same form as Equation (59): where the first summation is taken over all one-to-one functions τ:Y ⇒ X and where we define ρ τ (y) = δ τ(y) (y). Given this, in Sections 6.1-6.3 it is respectively shown that: where Equations (66)-(68) are true for all finite X,Y ⊆ and all multitarget densities f (X).
We are now in a position to define the general detected-target likelihood function: It is the likelihood that, given a target-set X, the following are simultaneously true: Z k is the measurement-set collected at time t k ; and Y ⊆ X is a subset of targets in X that generated measurements in Z k . In Section 6.4 the following is verified:

The General Detected-Target Density
Let us be given the prior distribution f k|k−1 (X|Z 1:k−1 ). Since f d k (Z k , Y|X) does not depend on Z 1:k−1 , then f d k (Z k , Y|X) = f d k (Z k , Y|X, Z 1:k−1 ) and so from Bayes' rule and the total probability theorem we obtain: However, if we instead use (1 X ) X-V , then since (1 X ) X-{x} = (1 X ) X = 1 we would get the incorrect result and: where the final equation results from Equation (11.251) of [14].

PGFLs of the Detected/Undetected Target Densities
The PGFL corresponding to f d k|k (Y|Z 1:k−1 ) (Equation (75)) is: The PGFL corresponding to f u k|k (Y|Z 1:k−1 )(Equation (87)) is simpler than f u k|k (Y|Z 1:k−1 ): where the final equation results from Equation (11.251) of [14]. Finally, in Section 6.5 it is shown that: Thus, posterior PGFL at time t k is an amalgam of the undetected-target and detected-target PGFLs. Note that if p D = 1 (all targets are perfectly detectable) then G u k|k [h|Z 1:k ] = 1 (there are no undetected targets) and G k|k [h|Z 1:k ] = G d k|k [h|Z 1:k ] (all targets are detected).
In what follows it will be demonstrated that the second claim leads to a contradiction, whereas the first one appears to be consistent with the formal theory of undetected targets.
Claim (2) Leads to a Contradiction. According to Equation (100), G u k|k [h|Z 1:k ] is the PGFL of targets that are undetected only at time t k . According to Claim (2), if a target is undetected at t k then it was also undetected at times t 1 , . . . , t k−1 . Given this, it must be the case that G u k|k [h|Z 1:k (20); and so G 1|1 [h|Z 1 ] is a PMB PGFL as in Equation (36): In Sections 6.6 and 6.7 it is respectively shown that: These equations are consistent with the "undetected targets" interpretation, Equation (45), since: From Equation (20), the next predicted PGFL is PMB: It can be shown that G u 2|2 [h|Z 1:2 ] is PMB, not Poisson. The claim that "detected targets" cannot become "undetected targets", therefore, leads to a contradiction. The proof of this fact for general G u 2|2 [h|Z 1:2 ] will not be proved here, since it suffices to address the following informative special case. , In Section 6.9 it is shown that the detected-target posterior PGFL is Bernoulli: In Section 6.8 it is shown that the undetected-target posterior PGFL is not Poisson: Here, q u is the probability that the posterior undetected-target RFS is nonempty-i.e., it is the target's composite probability of undetectability. Note that q u + q d = q + .
Additionally, note that q u parses the distinction between nonexistent vs. undetectable targets. If q> 0 (the target exists) and p D = 0 (it is undetectable), then no information can be collected about it and so its composite undetectability is q u = q. For example, if q = 1 then q u = 1-i.e., if a definitely-existing target is undetectable then it is compositely undetectable.
At the other extreme, if q = 0 (it does not exist) then q u = 0 (it is compositely detectable: 1 − q u = 1). This seems counter-intuitive since a nonexistent target would seem to be inherently undetectable. However, a nonexistent target is neither detectable nor undetectable. Whereas an existent target can generate either an actual measurement z or the null measurement ∅, a nonexistent target cannot generate any measurement. Thus, a nonexistent target has been "detected" if, as must be the case, it has not generated any measurement. In this sense, all nonexistent targets are compositely detectable. Now suppose that q = 1-i.e., that the target definitely exists. Then: That is, the target's composite undetectability q u is a composite of its "raw undetectability" s[1 − p D ] and the degree to which clutter density impairs its detectability. It varies between q u = 0 when κ = 0 and q u = 1 when κ = ∞. That is, the composite undetectability of a definitely-existing target is 0 if there is no clutter; and its composite detectability 1 − q u is 0 if the clutter density is infinite (and, thus, the signal-to-noise ratio is extremely small).

Proof of Equation (66)
Recall that n!C |X|,n is the number permutations of the elements of X taken n at a time; and that C |X|,n is the number of subsets of X of cardinality n. Then: where F ν (X) denotes the set of subsets of X of cardinality ν. Then:

Conclusions
This review paper has assessed and compared the following proposed exact closed-form solutions of the multitarget Bayes filter: