# Fusion of finite set distributions: Pointwise consistency and global cardinality

A recent trend in distributed multi-sensor fusion is to use random finite set filters at the sensor nodes and fuse the filtered distributions algorithmically using their exponential mixture densities (EMDs). Fusion algorithms which extend the celebrated covariance intersection and consensus based approaches are such examples. In this article, we analyse the variational principle underlying EMDs and show that the EMDs of finite set distributions do not necessarily lead to consistent fusion of cardinality distributions. Indeed, we demonstrate that these inconsistencies may occur with overwhelming probability in practice, through examples with Bernoulli, Poisson and independent identically distributed (IID) cluster processes. We prove that pointwise consistency of EMDs does not imply consistency in global cardinality and vice versa. Then, we redefine the variational problems underlying fusion and provide iterative solutions thereby establishing a framework that guarantees cardinality consistent fusion.

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## I Introduction

In networked sensing, nodes perform local filtering and exchange filtered distributions as opposed to communicating raw measurements [1]

. The problem of fusion is to find an estimate for the

a posteriori distribution over some state space conditioned on two or more (conditionally) independent sensor data streams, given local posteriors computed by local filtering of each data stream individually.

A large body of work utilises exponential mixtures of distributions (EMDs) for fusion. These mixtures are found by taking the weighted geometric mean of their components followed by scaling to ensure integration to unity. They have been widely used for fusion of single object (probability) distribution

[2]. A well-known algorithm that utilises EMDs of Gaussian densities is covariance intersection [3]. In CI, the weights of the components in the mixture are selected using various criteria [4]

. The underlying variational problem considers minimising a cost that equals to the weighted sum of Kullback-Leibler divergence

[5] of the fused density that is seeked, with respect to the mixture components. The stationary density and set of weights for this problem specifies an EMD which is deemed as a middle-ground of the components in a way analogous to logarithmic opinion pooling of experts [6].

The EMD form has been adopted for finite set densities in order to address fusion in the case of multiple objects [7]. Following the introduction of tractable recursive filters [8] such as the probability hypothesis density (PHD) filter [9]

, and, explicit filtering algorithms using Gaussian mixture model (GMM) representations

[10] and sequential Monte Carlo (SMC) techniques [11], numerical algorithms that extend CI fusion to Bernoulli, PHD, and cardinalised PHD (C-PHD) were proposed [12, 13, 14]. These methods have been proved useful in improving localisation accuracy in multi-sensor problems including those involving heterogenous sensors [15].

Another utilisation of EMDs for fusion of finite set distributions has been within the network consensus framework [16]. Briefly, iterative message passing algorithms which asymptotically compute the equally-weighted mixture, i.e., the geometric mean of the components, at all nodes of a sensor network are proposed for C-PHD [17], multi-Bernoulli [18], generalised MB [19, 20], Bernoulli [21], and, labelled [22, 23] finite set filters.

Finite set distributions factorise into a cardinality distribution on the number of objects and a localisation density conditioned on the cardinality [24]. The aforementioned variational principle when used with finite set distributions results with EMDs that satisfy point-wise consistency [25] with respect to the components. In this article, we show that the cardinality distributions of EMDs are not endowed with such consistency guarantees, in general. Such inconsistencies might result with smaller existence probabilities or estimates on the number of objects when the fused results are used instead of either of the inputs. This phenomena which might undermine the benefits of using diversity in sensing has been empirically observed by other researchers as well (see, e.g., [26]). Here, we provide explicit mathematical formulae specifying conditions under which the cardinality distributions of finite set EMDs are inconsistent. We demonstrate in examples that these inconsistencies are encountered sometimes with overwhelming probability under typical operating conditions and might lead to large discrepancies in, for example, the estimated number of objects.

Based on these results, we argue that the variational problem needs to be decoupled for the cardinality and the localisation distributions (i.e., scaled Janossy [24] distributions). Doing so separates the fusion of cardinality distributions and localisation terms. This approach results with the same localisation densities as the direct adoption of the variational problem, and, avoids any inconsistencies in the cardinality distribution. We show that pointwise consistency does not imply consistency in cardinality and vice versa. Then, we derive iterative algorithms for cardinality consistent fusion of finite set distributions.

The outline of the article is as follows: In Section II, we discuss fusion rules that accommodate EMDs in the light of the associated variational problems and pointwise consistency of EMDs. We provide our results regarding the cardinality inconsistencies of finite set EMDs in Section III, together with examples. Then, we redefine the variational problem underpinning fusion and derive solutions for cardinality consistent fusion in Section IV. Conclusions and future directions are provided in Section V.

## Ii Fusion as a variational problem

### Ii-a EMDs as weighted KLD centroids

Given two probability density functions

and on a state space , let us consider finding another density in the space of PDFs such that captures the information contained in both of the input distributions. An intuitive approach which is geometric in flavour would involve finding the centroid of the input distributions based on a distance/divergence metric. Kullback-Leibler divergence (KLD) is such a divergence metric which is used in information geometry in a way similar to the squared Euclidean distance [27], and, has an established relevance to estimation when is a finite alphabet (which is often referred to as hypothesis testing) [5].

The KLD of two distributions with densities and is computed as

 D(f||g)=∫Xf(X)logf(X)g(X)dX, (1)

where is always nonnegative and vanishes for .

Let us denote the centroid of and with respect to a weighted sum of KLD by . This distribution is a solution to the associated variational problem given by

 (P)minf∈PJω[f] Jω[f] ≜ (1−ω)D(f||fi)+ωD(f||fj) (2)

where is a design parameter selecting the weight of the divergence of each point and in the space of probability distributions over , with respect to .

The solution to problem (P) with the cost (2) is unique and found as

 fω(X) = 1Zωf(1−ω)i(X)fωj(X) (3) Zω = ∫Xf(1−ω)i(X′)fωj(X′)dX′, (4)

which can easily be seen after rearranging the cost in (2) as

 Jω[f]=D(f||fω)−log∫Xf(1−ω)i(X)fωj(X)dX, (5)

(see, for example,  [28, Eq.(3)]), and, realising that the second term on the right hand side does not depend on (see Appendix -A for a direct proof). In fact, this term is the scaled Rényi divergence [29] of order from to , i.e.,

 Jω[f] = D(f||fω)−(ω−1)Rω(fj,fi), (6) Rω(fj,fi) ≜ 1ω−1log∫Xfωj(X)f(1−ω)i(X)dX, = 1ω−1logZω.

Let us consider the weight parameter as a free variable, and find the stationary point of in (2) with respect to for . For the case, the KLD term in (6) vanishes and (2) reduces to a cost function for finding the Chernoff information of and  [5] which is concave in 111To be specific, in [30], Chernoff introduces as a “measure of divergence” between two distributions. This quantity can equivalently be found by in which the argument of maximisation is nothing but .. In [31], it is explained that there is a unique stationary point which satisfies

 D(fω||fi)=D(fω||fj)∣∣∣w=ω∗. (7)

The density in (4) is obtained by normalising the weighted geometric mean of and , and, thus referred to as their geometric mean density (GMD), or, exponential mixture density (EMD). In this article we adopt the latter.

### Ii-B Covariance intersection and generalisations

The above discussion outlines a fusion algorithm which outputs the pair using (7) and (3) for fusing and . This can be rephrased as a mathematical programme:

 (P2)(ω∗,fω∗)≜argmaxω∈[0,1]minf∈PJω[f]. (8)

The input densities here are a posteriori in nature as they are propagated by local filters, i.e., they are conditioned on the data-streams of sensors and , respectively. When is and the distributions involved are Gaussians, this approach reduces to a set of linear algebraic operations which are known as the “covariance intersection” algorithm [3]. In this setting, how well an approximation the EMD (3) is to the joint posterior222Here, we refer to the posterior distribution conditioned on the data streams of both sensors which is infeasible to compute given the limited communication and computational resources of the networked setting. is studied in terms of bounds over the uncertainty spread characterised by covariance matrices (see, e.g., [32, 33]). For exponential family distributions including multi-variate Gaussians, is the unique intersection point of the exponential geodesic curve joining  and  (obtained by varying from to in (3

)) and its dual hyperplane on the induced statistical manifold

[34].

For general distributions, (3), (4) and (7) are still valid as a solution to the variational fusion problem in (8). The optimal weight selected through (7) equates the cost in (2) to the Chernoff information [5] between and  [31]. Perhaps for this reason, some authors refer to this fusion rule as Chernoff fusion (see, for example [35] and the references therein).

The uncertainty spread in EMDs of arbitrary distributions is characterised in terms of pointwise bounds. In [25], the authors show that the scale factor in (4) is less than or equal to one, i.e., , and, consequently EMDs (3) satisfy the following consistency condition:

 fω(X)≥min{fi(X),fj(X)}. (9)

for all points and . In other words, the fused distribution does not overlook the probability mass assigned by and onto the vicinity of any point in the state space. In this sense, this condition corresponds to a notion of consistency [25].

Other fusion methods that use EMDs include consensus based approaches as overviewed in Section I. These methods compute by iterative message passings between nodes. However, instead of finding stationary weights of the variational problem in (2), this network averaging approach can compute only an equally weighted EMD, and, when the number of iterations tend to infinity. Some other methods differ from the generalised CI approach described above in their weight selection criteria: Some authors argue that it might be more beneficial to select the value of in (2) that would maximise the “peakiness” of  [7], or, to minimise the uncertainty captured by quantified by its Shannon differential entropy [4].

In this article, our concern is the consistency properties of EMDs of finite set distributions. These distributions have been commonly used to represent multi-object scenes [8]. The following discussion is valid for any fusion scheme that employs EMDs and random finite set (RFS) distributions in order to quantify uncertainty in, for example, “the number of objects,” (e.g., Poisson, i.i.d. cluster RFSs [8]), “existence probabilities” (e.g., Bernoulli, multi-Bernoulli, generalised labelled MB RFSs [36] and MB mixtures [37]) irrespective of their weight selection mechanism.

## Iii Finite set EMDs and cardinality distributions

In the case of finite set valued random variables,

is the space of finite subsets of and the density is a set function characterised by a cardinality distribution with probability mass function (pmf) over natural numbers and localisation densities for which are symmetric in their arguments [24]. The corresponding density has a set valued argument and is given by

 f(X) = p(n)∑σ∈Σnρn(xσ(1),...,xσ(n)) (10) = p(n)n!ρn(xσ′(1),...,xσ′(n))

where and denotes set cardinality. Here, is the set of all permutations of , and, in the last line is an arbitrary permutation which is selected as the identity permutation in the rest of this article.

Note that in (10) sums to one and s integrate to unity. The finite set density also integrates to one over , i.e.,

 ∫Xf(X)μ(dX)=1

where is an appropriate measure. Let us select as

 μ(dX)=∞∑n=0λn(dX∩Xn)n!

where is the space of -tuple of points in , and, is the Lebesgue (volume) measure on  333Further details on the topic can be found in Section II.B and Appendix B of [11], and, the references therein.. An alternative form of this integral is referred to as the set integral [8], i.e.,

 ∫Xf(X)μ(dX)=∫Rdf(X)δX,

where the right hand side is the set integral of defined as 444Note that the set integral in (III) is defined for an arbitrary (measurable) function , but, when is a finite point process density, (III) is nothing but the total probability theorem applied on (10[38].

 ∫Rdf(X)δX ≜ ∞∑n=01n!∫Rd…∫Rdf({x1,...,xn})dx1…dxn = ∞∑n=0∫Rd…∫Rdp(n)ρn(x1,...,xn)dx1…dxn.

Let us consider the EMD of finite set distributions and . For the case (3) is valid with the scale factor in (4) found using the set integral in (III), i.e.,

 Zω=∫Rdf(1−ω)i(X′)fωj(X′)δX′. (12)

This scale factor is also less than one and consequently the finite set EMD satisfies the pointwise consistency condition in (9) for every finite subset .

In order to investigate the cardinality distribution of the EMD, let us substitute and in the form given in (10) into (12) and (3), and, obtain the finite set EMD as

 fω(X)=pω(n)n!ρω,n(x1,…,xn)

where the localisation density for cardinality is

 ρω,n(x1,…,xn)≜1zω(n)ρ(1−ω)i,n(x1,…,xn)×ρωj,n(x1,…,xn), (13)
 zω(n) = ∫Rd⋯∫Rdρ(1−ω)i,n(x′1,…,x′n) (14) ×ρωj,n(x′1,…,x′n)dx′1,…,dx′n,

and, the cardinality pmf is

 pω(n) = 1Nωp(1−ω)i(n)pωj(n)zω(n) (15) Nω = ∑n′=0p(1−ω)i(n′)pωj(n′)zω(n′). (16)

where by convention, and for , unless and are identical. The latter is a direct application of Hölder’s inequality (see, e.g., Theorem 188 in [39]). It can be shown similarly that .

Let us focus on the fused cardinality pmf in (15). This distribution is not an EMD of the cardinality distributions of the components unlike the fused localisation distributions in (13) that are EMDs of the input localisation densities. In fact, the fused cardinality is the scaled product of the cardinality EMD with the localisation density scale factors in (14). As a result, the consistency property of EMDs does not apply to the fused cardinality. Below, we first relate the consistency of the fused cardinality distribution to the sequence of scale factors and give a condition under which the fused cardinality is inconsistent. Then, in the rest of this section, we demonstrate that inconsistent cardinalities occur under some typical operating conditions.

###### Proposition III.1 (Inconsistency in cardinality distribution)

Consider the fused cardinality pmf in (15), (16). Consider the following inconsistency condition for obtained by negating the consistency condition:

 pω(n)

This condition holds true if

 zω(n)<∑n′≠np(1−ω)i(n′)pωj(n′)zω(n′)p(1−ω)i(n)pωj(n)min{pi(n),pj(n)}−p(1−ω)i(n)pωj(n), (18)

where is given in (14).

The proof is given in Appendix -B. The above proposition points out that the fused cardinality distribution opts to disagree with local results on the probability of number of objects when the th localisation scale is comparably small. This is in stark contrast with the fused localisation densities in (13) which always satisfy the consistency condition

 ρω,n(x1,…,xn)≥min{ρi,n(x1,…,xn),ρj,n(x1,…,xn)}

for all and , as they are EMDs.

The scale factors modulating the cardinality pmf, i.e., , are found by taking the inner products of the input localisation densities raised to fractional powers. As explained above, these terms are upper bounded by one with equaling unity only when and are equal (see [25] for an alternative proof). In fusion networks, however, one of the main goals is to benefit from sensing diversity which means and will have a comparably small overlap in their confidence regions. As a result, much smaller values should be expected in typical operating conditions.

Now, let us consider some particular RFS families and demonstrate the consequences of Proposition III.1.

### Iii-a Bernoulli finite set EMDs and fused existence probabilities

Bernoulli finite set distributions select at most one object from a population. Collections, and mixtures thereof are used to represent multi-object models the fusion of which reduces to EMD fusion of Bernoulli pairs (see, e.g., [19, 20]). For a Bernoulli finite set, the cardinality pmf in (10) is given by

 p(n)=⎧⎨⎩1−α,n=0,α,n=1,0,otherwise (19)

where the parameter is referred to as the existence probability of the object modelled.

There is also a single localisation density for which we will denote by . Therefore, given two Bernoullis and , the sequence reduces to

 zω(n)=⎧⎪⎨⎪⎩1,n=0,zω≜∫Rdρ(1−ω)i(x)ρωj(x)dx,n=1,0,otherwise. (20)
###### Corollary III.2

The inconsistency condition given by Proposition III.1 for Bernoulli finite set distributions reduces to that the existence probability of the EMD given by [14]

 αω=α(1−ω)iαωjzω(1−αi)(1−ω)(1−αj)ω+α(1−ω)iαωjzω (21)

is smaller than either of or if

 zω<(1−αi)(1−ω)(1−αj)ωα(1−ω)iαωj/min{αi,αj}−α(1−ω)iαωj.

The proof follows from substituting the sequence (20) in Proposition III.1, and, in particular in (17) and (18). This condition is very often satisfied in sensing applications as explained before. For example, if and are equal, then this condition reduces to which always holds for all practical purposes as and should not be expected to be identical. For , this inconsistency still occurs with overwhelming probability in Bernoulli fusion which is demonstrated in the following example.

###### Example III.3 (Gauss-Bernoulli EMDs)

Let us consider Bernoulli distributions with Gaussian localisation densities given by

 ρi(x)=N(x;mi,Ci),ρj(x)=N(x;mj,Cj), (22)

where

is the mean vector and

is the covariance matrix. The fused localisation density for the case is a Gaussian with mean and covariance given by

 mω = (1−ω)C−1imi+ωC−1jmj (23) Cω = ((1−ω)C−1i+ωC−1j)−1. (24)

The scale factor is found using integration rules for Gaussians as

 zω=∣∣C−1i∣∣(1−ω)/2∣∣C−1j∣∣ω/2|Cω|1/2exp{−12((1−ω)mTiC−1imi+ωmTjC−1jmj−mTωC−1ωmω)}. (25)

Let us consider two Bernoullis with existence probabilities with localisation densities of mean vectors and , respectively, where denotes vector transpose. We select the covariance matrices as rotated versions of a diagonal covariance given by

 Ci = R(π/4)ΣRT(π/4), Cj = R(−π/4)ΣRT(−π/4), Σ = [σ21,00,σ22], R(ϕ) = [cosϕ,−sinϕsinϕ,cosϕ].

This covariance structure is typical with sensors placed at different positions and taking their measurements from different aspect angles of the surveillance zone. The condition number of (i.e., ) has higher values for sensors with range/cross-range ambiguity such as cameras/radars which we vary from to . Fig. 1

depicts the uncertainty ellipses of sample Gaussians by using three times the standard deviation along the eigen vector directions.

The behaviours of the fused existence probability in (21) and the scale factor in (25) are our concern. Fig. 2 presents both the and values obtained by varying the condition number with small steps from to hence increasing the sensing diversity. The exponential mixture weights take values from a dense grid over . As pointed out in this section, the scale factor values are always smaller than unity, and, can often take very small values. The scale factor monotonically decreases with which controls the sensing diversity. It is convex with respect to the mixture weight , as pointed out in Section II.

The fused existence probabilities given in Fig. 2 demonstrate the inconsistency in cardinality. In this example, this quantity is always smaller than the input existence probabilities admitting inconsistency for all selections of  and . Moreover, the fused existence probability drops below for large values of the sensing diversity parameter . This threshold is often used as the Bayesian decision boundary for detection and despite that the input sources are fairly confident on the existence of an object with existence probabilities of , this decision might be rejected based on the fused result undermining the benefits of sensing diversity. As a result, the inconsistency in cardinality may lead to inconsistency in decision making when EMDs of finite set distributions are used.

### Iii-B EMDs of Poisson finite set distributions

Poisson finite set densities are capable of representing many objects and underpin popular multi-object filters such as the PHD filter [9]. Their cardinality pmf in (10

) is given by a Poisson distribution, i.e.,

 p(n)=eλλnn! (26)

where is the expected number of objects. The localisation densities factorise over the density for  as

 ρn(x1,…,xn)=n∏i=1ρ1(xi), (27)

making it possible to parameterise the entire finite set distribution with a scalar and a single density555We drop the subscript in for the rest of this subsection and denote it by ..

For two Poissons and , the sequence is a geometric sequence found by subsituting from (27) for both  and  into (14). This sequence is found as

 zω(n) = znω (28) zω ≜ zω(1)=∫Rdρ(1−ω)i(x′)ρωj(x′)dx, (29)

where unless and are identical, as aforementioned.

The expected number of objects with respect to an EMD with weight parameter is given by [14]

 λω=λ(1−ω)iλωjzω. (30)
###### Proposition III.4 (Poisson inconsistency in expectation)

Let us consider an inconsistency condition for Poisson cardinality distributions in terms of their expectations:

 λω

This condition holds whenever

 zω

The proof follows easily from substituting (32) in (30) and (31). It is instructive to contrast this result with Proposition III.1. The latter holds for any class of finite set densities and considers their cardinality distributions for different . The above result is on the expected value of in Poisson finite set densities. The condition in (32) is satisfied with overwhelming probability in practice leading to inconsistencies as observed, for example, in [26]. For example, for , this reduces to the common ratio being less than one which should –as previously discussed– always be expected to be the case in practice.

The inconsistency in decision making for the case is related to the estimation of the number of objects. In Poisson finite set models, the minimum mean squared error (MMSE) estimation principle is used which leads to the use of as the estimated number of objects666Maximum a posteriori (MAP) estimation is not used with Poisson cardinality distributions as (26) is not guaranteed to have a unique maximum. Notice that, for example, (26) evaluates at the same value for both and for .. As a result, the EMD density always underestimates the number of objects despite that the source densities might be consistently suggesting otherwise, in practice. The magnitude of the error stemming from this bias depends on the value of .

### Iii-C EMDs of IID cluster finite set densities

IID cluster finite set distributions relax the Poisson cardinality pmf in (26) and take arbitrary cardinality pmfs underpinning the CPHD filter [40]. The localisation densities still take the factorised form in (27) leading to the identical geometric series in (28). For the case, Proposition III.1 specialises as follows:

###### Corollary III.5 (IID cluster inconsistency)

Given two IID cluster finite set distributions and , the fused cardinality satisfies the inconsistency condition in (17) in Proposition III.1 for the number of objects and non-zero if

 zω

holds, where the term on the right hand side is

 Iω,n≜⎛⎜⎝Nωmin{pi(n),pj(n)}p(1−ω)i(n)pωj(n)⎞⎟⎠1/n, (33)

is given in (29), and, is obtained by substituting (28) in (16).

The proof follows from substituting (28) in (18) and using (16) after rearrangement of the terms. The inconsistency condition in (33) depends both on and , and, it is not straightforward to relate the inconsistent bins to object number estimation either in the MMSE or MAP rules. On the other hand, the base of the exponent in (33) is smaller than one and hence approaches to one as grows. Therefore, for some threshold , the fused object number probabilities will be lower than the input cardinality reports for all . Such a threshold can easily be found from (33) as

 η = log(Nωγω)logzω, (34) γω ≜ minn′min{pi(n′),pj(n′)}p(1−ω)i(n′)pωj(n′)

As a result, one should expect estimation biases to become more severe for higher object numbers. For a small number of objects, these effects do not necessarily yield biases in MAP estimations, which also explains the accurate estimates obtained using EMD fusion of CPHD filters in simulated scenarios, e.g., in [14]. Next, we demonstrate this point in an example involving fusion of two binomial cardinality distributions.

###### Example III.6

Let us consider the EMD fusion of two finite set distributions with binomial cardinalities given by and where these distributions give the probability that objects exist simultaneously among possibilities each with an existence probability of (Fig. 3LABEL:sub@fig:BB). Of particular interest is the characteristics of as and vary in and , respectively. Fig. 3LABEL:sub@fig:BBfusion presents fused distributions obtained by varying and some intermediate values of . Note that the cardinality at which the fused distributions peaks varies with as suggested in Corollary III.5. In particular, the inconsistency bound in ((c)c) is illustrated in Fig. 3LABEL:sub@fig:calI which monotonically increases with as discussed. The inconsistency threshold for as given in (34) is given in Fig. 3LABEL:sub@fig:eta. Note that for a large ratio of and values, this threshold is larger than five and the MAP estimate for the cardinality given in Fig. 3LABEL:sub@fig:mapest1 agrees with the individual MAP estimates of . However, there are also MAP estimates that indicate less than five objects caused by the IID inconsistency.

These computations are repeated for cardinality distributions peaking at a higher value. Specifically, and are used (see Fig. 3LABEL:sub@fig:cardHigh) which have individual map estimates of . The fused cardinalities in Fig. 3LABEL:sub@fig:BBfusionHigh illustrate that for a larger subset of pairs the IID inconsistency occurs, now, as discussed above. The resulting errors in estimating the number of objects is given in Fig. 3LABEL:sub@fig:mapBBHigh which verifies our expectation: Based on that (33) approaches to with increasing , as the peak cardinality increases, the IID inconsistency detoriates decision making more.

As a summary, this section has shown that when EMDs of finite set densities are used for their fusion, the resulting cardinality distribution will bear inconsistencies depending on . Proposition III.1 provides a general condition on the fused distribution to be inconsistent with the input distributions at a cardinality value . This condition is specialised for Bernoulli finite set densities in Corollary III.2. Example III.3 has demonstrated that this condition holds with overwhelming probability for Bernoulli EMDs. In Poisson cardinality distributions, there is a single parameter that specifies the distribution for all . Proposition III.4 provides a condition of inconsistency in this parameter, similarly as an upper bound on . It is pointed out that because is determined by the sensing diversity as well as sensor measurement histories in a sensor network, its value should be expected to be less than one in these settings777The authors at this point would like to conjecture that with probability one in a multi-sensor setting in which the finite set densities to be fused are posteriors obtained from recursive Bayesian filtering of local sensor data, i.e., and for realisations and of (independent) measurement processes associated with sensors and , respectively. which in turn shows that Poisson EMDs are very prone to inconsistencies, as well. IID cluster processes have more general cardinality distributions. For the case, Proposition III.1 specialises to Corollary III.5 which reveals that inconsistencies should be expected in MAP estimates of the cardinality, when the input densities indicate a high number of objects. These points are demonstrated in Example III.6.

## Iv Cardinality consistent fusion of finite set distributions

In this section, we propose a new approach that accommodates EMD fusion while avoiding the cardinality inconsistencies detailed in Section III. These inconsistencies result from the dependency of the fused cardinality on the scaling factor series . One way to remove this dependency is to decouple the fusion problem for different cardinalities by asserting a separate variational problem for each cardinality as opposed to using P2 in (8) with finite set distributions as a single entity.

### Iv-a Variational problem definitions

Let us first consider finite set distributions as parameterised in (10) and remind that problem P2 is solved with distributions in the form given in (13)–(16). Now, let us consider the following family of variational problems given and :

 (P3)Forn = 1,2,… (ω∗n,ρω∗n,n) ≜ argmaxω∈[0,1]minρn∈PnJω,n[ρn] (35) Jω,n[ρn] ≜ (1−ω)D(ρn||ρi,n)+ωD(ρn||ρj,n).

Here, is the space of localisation densities with arguments which are symmetric in their arguments. Note that P3 is a set of P2 that has the localisation distributions for each cardinality as the entries, separately. Equivalently, P3 asserts the variational problem of fusion be treated as a conditional problem to be solved given .

Following our discussion in Section II, solutions of these uncoupled problems have an EMD form given by (13) and (14)888It is easy to show that because and are symmetric in their arguments, also exhibits this symmetricity.. One difference here compared to the solution of problem P2 is that for each , a different optimal weight will be output, in general, as opposed to a single one. In addition –and, more importantly– problem P2 decouples fusion of the cardinality distributions thus given

, the fused cardinality distribution becomes an extra degree of freedom in the fused finite set distribution. In other words, the fusion of cardinality distributions can now be carried out in an isolated fashion in addition to problem

P2 as a solution to

 (P4)(ω∗c,~pω∗c) ≜ argmaxω∈[0,1]minp∈PcJω,c[p] (36) Jω,c[p] ≜ (1−ω)D(p||pi)+ωD(p||pj).

Following the discussion in Section II, the solution to problem P4 is the EMD of the cardinality pmfs

 ~pω(n) = 1~Nωp(1−ω)i(n)pωj(n) (37) ~Nω = ∑n′=0p(1−ω)i(n′)pωj(n′). (38)

evaluated at .

This distribution differs from the cardinality of the solution to P2 (given in (15) and (16)) in that it does not involve , and, is an EMD of the input finite set cardinalities. Therefore, the consistency condition (see (9))

 ~pω(n)≥min{pi(n),pj(n)}

is satisfied for all and for all regardless of . This avoids the decision errors stemming from the cardinality inconsistencies of the solutions to P2 as detailed in the previous section.

As a result, P3 and P4 yield a fused finite set density featuring cardinality consistency given by

 ~f∗(X) = ~fΩ(X)∣∣∣Ω=(ωc=ω∗c,ω1=ω∗1,ω2=ω∗2,…) (39) ~fΩ(X) ≜ pωc(n)n!ρωn,n(x1,...,xn) (40)

where is found by solving the maximisation in (36) with a cardinality distribution given by (37) and (38). Here, solves the maximisation in (35) with a localisation distribution in (13) and (14). These localisation distributions – similar to the cardinality distribution– are consistent individually, as they are EMDs of the inputs.

The pointwise consistency of over the space of finite sets, however, is not guaranteed. In order to clarify this point, we provide the following proposition:

###### Proposition IV.1 (Pointwise inconsistency)

Let us consider

 ~fω(X)≜~fΩ(X)∣∣∣Ω=(ωc=ω,ω1=ω,ω2=ω,…) (41)

for some . is pointwise inconsistent, i.e.,

 ~fω(X)

if

 E~pω{zω(n)}zω(n)

where is the finite set EMD given in (13)–(16).

Proof. By comparing (13)–(16) and (37)–(40), it can be seen that the two finite set densities of concern are related by

 ~fω(X)=E~pω{zω(n)}zω(n)fω(X). (44)

where the expectation is with respect to (37).

Substitution of (44) in (42) yields the first inequality in (43). Note that, satistifies the pointwise consistency condition in (9), hence, the right hand side of the inequality is smaller than or equal to one.

This proposition points out that pointwise consistency of is guaranteed only for those with cardinality for which the scaling factor of the localisation density equals to the expectation. If the latter is greater than the expectation to the extent that (43) is satisfied, then exhibits pointwise inconsistency despite being consistent in the global cardinality and localisation distributions. As a conclusion, pointwise consistency does not imply consistency in global cardinality in fusion of finite set densities and vice versa.

### Iv-B Solving the cardinality consistent fusion problems

The variational problems P3 and P4 are optimisation problems similar to (8). Hence, the minimisations given are solved by the EMDs of their argument distributions (see the discussion in Section II and Appendix -A). The objective of the outer maximisation in P3 is therefore (see also (6))

 Gn(ωn) ≜ Jω,n[ρn]∣∣∣ρn=ρω,n = −(ω−1)Rω(ρn,i,ρn,j) = −logzω(n)

which is a concave function of its one dimensional argument that takes values from a bounded interval. Newton iterations provide a solution with optimal convergence properties [41]. Starting from an initial value , recursive increments are made by the ratio of the first and second order derivatives, i.e.,

 = z′ω(n)zω(n)z′′ω(n)zω(n)−(z′ω(n))2∣∣∣ω=ω(k) (46) z′ω(n) ≜ ∫ρ1−ωi,n(x)ρωj,n(x)logρj,n(x)ρi,n(x)dx (47) z′′ω(n) ≜ ∫ρ1−ωi,n(x)ρωj,n(x)(logρj,n(x)ρi,n(x))2dx

where is the value found in the th iteration. Here, is given in (14) and its derivatives in (47) and (LABEL:eqn:zprimeprime) are found in Appendix -C. Algorithm 1 explicitly specifies this iterative solution which takes the localisation densities as inputs together with an initial value and termination condition. Upon convergence, the optimal value is found for which the corresponding EMD is the fused density.

An analogous iterative algorithm for finding the consistently fused cardinality as a solution to P4 is given in Algorithm 2. Note that the computations involved here can be carried out exactly for distributions with finite support, in practice.

Algorithm 1, on the other hand, should accommodate adequate computational schemes for exactly or approximately evaluating the integrals involved. In the latter case, it admits the interpretation of being a stochastic gradient approach [42]. Specification of such procedures is beyond the scope of this work.

There are, nevertheless, structural simplifications in both P3 and P4 for different families of finite set families. For Poisson and IID cluster finite set distributions, the localisation densities are parameterised by a single density over a single state variable –as given in (27)– which is the same for different cardinalities. In addition, the solution to the inner minimisation in P3 (equivalently P in (2)) is also a Poisson and IID cluster, respectively, in the Poisson and IID cluster cases [14]. Thus, where parameterises