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

Murat Üney, , Jérémie Houssineau, Emmanuel Delande, Simon J. Julier, Daniel
Clark This work was supported by the Engineering and Physical Sciences
Research Council (EPSRC) Grant number EP/K014277/1 and the MOD University
Defence Research Collaboration (UDRC) in Signal Processing.Murat Üney is with
the Institute for Digital Communications, School of Engineering, University of
Edinburgh, EH9 3FB, Edinburgh, UK (e-mail: m.uney@ed.ac.uk).Jérémie Houssineau
is with the National University of Singapore, Department of Statistics and
Applied Probability, National University of Singapore, Singapore
119077.Emmanuel Delande is with the Institute for Computational Engineering
and Sciences, University of Texas at Austin (edelande@ices.utexas.edu).Simon
J. Julier is with the Computer Science Department, University College London,
London (e-mail: s.julier@ucl.ac.uk).Daniel E. Clark is with Départment CITI,
Telecom-SudParis, 9, rue Charles Fourier 91011, EVRY Cedex, France (e-mail:
daniel.clark@telecom-sudparis.eu).

###### Abstract

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 that extend 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.

###### Index Terms:

random finite sets, multi-sensor fusion, exponential mixture density,
covariance intersection, target tracking

##  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) distributions [2].
A well-known algorithm that utilises EMDs of Gaussian densities is covariance
intersection [3]. In covariance intersection (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 divergences [5] of the fused density that is
sought 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 (unweighted) 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.

In [24, 25], it has been proved that EMDs have a probability density that at
no point in the state space overlooks the density of their components. This
property is proposed as a working definition of consistency in the context of
fusion [25]. Finite set density EMDs also satisfy this consistency condition
pointwise, at every finite collection of points.

Finite set distributions, on the other hand, factorise into a cardinality
distribution on the number of objects and a localisation density conditioned
on the cardinality [26]. 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., [27]). 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 and/or object existence
probabilities.

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 [26] 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 (PDFs) $f_{i}$ and $f_{j}$ on a state
space $\cal X$, let us consider finding another density $f$ in the space of
PDFs $\cal P$ such that $f$ 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 [28], and, has an established relevance to
estimation when ${\cal X}$ is a finite alphabet (which is often referred to as
hypothesis testing) [5].

The KLD of two distributions with densities $f$ and $g$ is computed as

$D(f||g)=\int_{\cal X}f(X)\log\frac{f(X)}{g(X)}\mathrm{d}X,$ (1)

where $D$ is always nonnegative and vanishes for $f=g$.

Let us denote the centroid of $f_{i}$ and $f_{j}$ with respect to a weighted
sum of KLD by $f_{\omega}$. This distribution is a solution to the associated
variational problem given by

$\displaystyle\text{(P)}\,\,\,\min_{f\in{\cal P}}J_{\omega}[f]$ $\displaystyle
J_{\omega}[f]$ $\displaystyle\triangleq$
$\displaystyle(1-\omega)D(f||f_{i})+\omega D(f||f_{j})$ (2)

where $\omega\in[0,1]$ is a design parameter selecting the weight of the
divergence of each point $f_{i}$ and $f_{j}$ in the space of probability
distributions $\cal P$ over $\cal X$, with respect to $f$.

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

$\displaystyle f_{\omega}(X)$ $\displaystyle=$
$\displaystyle\frac{1}{Z_{\omega}}f_{i}^{(1-\omega)}(X)f_{j}^{\omega}(X)$ (3)
$\displaystyle Z_{\omega}$ $\displaystyle=$ $\displaystyle\int_{\cal
X}f_{i}^{(1-\omega)}(X^{\prime})f_{j}^{\omega}(X^{\prime})\,\,\mathrm{d}X^{\prime},$
(4)

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

$J_{\omega}[f]=D(f||f_{\omega})-\log\int_{\cal
X}f_{i}^{(1-\omega)}(X)f_{j}^{\omega}(X)\,\,\mathrm{d}X,$ (5)

(see, for example, [29, Eq.(3)]), and, realising that the second term on the
right hand side does not depend on $f$ (see Appendix -A for a direct proof).
In fact, this term is the scaled Rényi divergence [30] of order $\omega$ from
$f_{j}$ to $f_{i}$, i.e.,

$\displaystyle J_{\omega}[f]$ $\displaystyle=$ $\displaystyle
D(f||f_{\omega})-(\omega-1)R_{\omega}(f_{j},f_{i}),$ (6) $\displaystyle
R_{\omega}(f_{j},f_{i})$ $\displaystyle\triangleq$
$\displaystyle\frac{1}{\omega-1}\log\int_{\cal
X}f_{j}^{\omega}(X)f_{i}^{(1-\omega)}(X)\mathrm{d}X,$ $\displaystyle=$
$\displaystyle\frac{1}{\omega-1}\log Z_{\omega}.$

Let us consider the weight parameter $\omega$ as a free variable, and find the
stationary point of $J_{\omega}$ in (2) with respect to $\omega$ for
$f=f_{\omega}$. For the case, the KLD term in (6) vanishes and (2) reduces to
a cost function for finding the Chernoff information of $f_{i}$ and $f_{j}$
[5] which is concave in $\omega$ 111To be specific, in [31], Chernoff
introduces $C(f_{j},f_{i})\triangleq-\log\min Z_{\omega}$ as a “measure of
divergence” between two distributions. This quantity can equivalently be found
by $\max-\log Z_{\omega}$ in which the argument of maximisation is nothing but
$J_{\omega}[f=f_{\omega}]$.. In [32], it is explained that there is a unique
stationary point $\omega^{*}$ which satisfies

$D(f_{\omega}||f_{i})=D(f_{\omega}||f_{j})\bigg{\rvert}_{\omega=\omega^{*}}.$
(7)

The density $f_{\omega}$ in (4) is obtained by normalising the weighted
geometric mean of $f_{i}$ and $f_{j}$, 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
$(\omega^{*},f_{\omega^{*}})$ using (7) and (3) for fusing $f_{i}$ and
$f_{j}$. This can be rephrased as a $\max\min$ mathematical programme:

$(\text{P2})\,\,\,\,(\omega^{*},f_{\omega^{*}})\triangleq\arg\max_{\omega\in[0,1]}\min_{f\in{\cal
P}}J_{\omega}[f].$ (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 $i$
and $j$, respectively. When $\cal X$ is $\mathbb{R}^{d}$, i.e., the
$d$-dimensional space of real vectors, 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., [33, 34]). An information geometric characterisation of $f_{\omega^{*}}$
for multivariate Gaussians and other exponential family distributions is
provided in [35] where it is proved that $f_{\omega^{*}}$ is the unique
intersection point of the exponential geodesic curve joining $f_{i}$ and
$f_{j}$ (obtained by varying $\omega$ from $0$ to $1$ in (3)) and its dual
hyperplane on the induced statistical manifold.

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 $f_{i}$ and
$f_{j}$ [32]. Perhaps for this reason, some authors refer to this fusion rule
as Chernoff fusion (see, for example [36] and the references therein).

###  II-C Other fusion rules utilising EMDs

Other fusion methods that use EMDs include consensus based approaches as
overviewed in Section I. These methods compute $f_{\omega}$ 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 tends 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 $\omega$ in (2) that would maximise the
“peakiness” of $f_{\omega}$ [7], or, to minimise the uncertainty captured by
$f_{\omega}$ quantified by its Shannon differential entropy [4].

###  II-D A notion of consistency in Fusion

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., $Z_{\omega}\leq 1$, and, consequently
EMDs (3) satisfy the following consistency condition:

$f_{\omega}(X)\geq\min\\{f_{i}(X),f_{j}(X)\\}$ (9)

for all points $X\in{\cal X}$ and $\omega\in[0,1]$. In other words, the fused
distribution does not overlook the probability mass assigned by $f_{i}$ and
$f_{j}$ onto the vicinity of any point in the state space. In this sense, this
condition corresponds to a notion of consistency [25], in the context of
distributed fusion333Note that the use of the term “consistency” here differs
from its use in classical statistics..

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 [37] and MB mixtures [38])
irrespective of their weight selection mechanism. In the next section, we
utilise (9) for analysing finite set EMDs and examine the fused global
cardinality distributions for inconsistencies and their consequences in
estimating object existence probabilities and/or the number of objects.

##  III Finite set EMDs and cardinality distributions

In the case of finite set valued random variables, $\cal X$ is the space of
finite subsets of $\mathbb{R}^{d}$ and the density $f$ is a set function
characterised by i) a cardinality distribution with probability mass function
(pmf) $p(n)$ over natural numbers $n=0,1,\ldots$, and, ii) localisation
densities $\rho_{n}(x_{1},...,x_{n})$ for ${n=1,2,\ldots}$ which are symmetric
in their arguments [26]. The corresponding density has a set valued argument
$X=\\{x_{1},\ldots,x_{n}\\}$ and is given by

$\displaystyle f(X)$ $\displaystyle=$ $\displaystyle
p\left(n\right)\sum_{\sigma\in\Sigma_{n}}\rho_{n}(x_{\sigma(1)},...,x_{\sigma(n)})$
(10) $\displaystyle=$ $\displaystyle
p\left(n\right)n!\rho_{n}(x_{\sigma^{\prime}(1)},...,x_{\sigma^{\prime}(n)})$

where $n=|X|$ and $|.|$ denotes set cardinality. Here, $\Sigma_{n}$ is the set
of all permutations of $(1,\ldots,n)$, and, $\sigma^{\prime}\in\Sigma_{n}$ in
the last line is an arbitrary permutation which is selected as the identity
permutation in the rest of this article.

Note that $p$ in (10) sums to one and $\rho_{n}$s integrate to unity. The
finite set density $f$ also integrates to one over $\cal X$, i.e.,

$\int_{\cal X}f(X)\mu(\mathrm{d}X)=1$

where $\mu$ is an appropriate measure. Let us select $\mu$ as

$\mu(\mathrm{d}X)=\sum_{n=0}^{\infty}\frac{\lambda_{n}(\mathrm{d}X\cap{\cal
X}_{n})}{n!}$

where ${\cal X}_{n}$ is the space of $n$-tuple of points in $\mathbb{R}^{d}$,
and, $\lambda_{n}$ is the Lebesgue (volume) measure on ${\cal X}_{n}$
444Further details on the topic can be found in Section II.B and Appendix B in
[11], and, the references therein.. An alternative form of this integral is
referred to as the set integral [8], i.e.,

$\int_{\cal X}f(X)\mu(\mathrm{d}X)=\int_{\mathbb{R}^{d}}f(X)\delta X,$

where the right hand side is the set integral of $f$ defined as 555Note that
the set integral in (LABEL:eqn:setint) is defined for an arbitrary
(measurable) function $f$, but, when $f$ is a finite point process density,
(LABEL:eqn:setint) is nothing but the total probability theorem applied on
(10) [39].

$\displaystyle\int_{\mathbb{R}^{d}}f(X)\delta X$ $\displaystyle\triangleq$
$\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}\int\limits_{{\mathbb{R}^{d}}}\ldots\int\limits_{{\mathbb{R}^{d}}}f(\\{x_{1},...,x_{n}\\})\mathrm{d}x_{1}\dots\mathrm{d}x_{n}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\int\limits_{{\mathbb{R}^{d}}}\ldots\int\limits_{{\mathbb{R}^{d}}}p(n)\rho_{n}(x_{1},...,x_{n})\mathrm{d}x_{1}\dots\mathrm{d}x_{n}.$

Let us consider the EMD of finite set distributions $f_{i}$ and $f_{j}$. For
the case (3) is valid with the scale factor in (4) found using the set
integral in (LABEL:eqn:setint), i.e.,

$Z_{\omega}=\int_{\mathbb{R}^{d}}f_{i}^{(1-\omega)}(X^{\prime})f_{j}^{\omega}(X^{\prime})\delta
X^{\prime}.$ (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
$X\subset\mathbb{R}^{d}$.

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

$f_{\omega}(X)=p_{\omega}(n)n!\rho_{\omega,n}(x_{1},\ldots,x_{n})$

where the localisation density for cardinality $n$ is

$\rho_{\omega,n}(x_{1},\ldots,x_{n})\triangleq\frac{1}{z_{\omega}(n)}\rho_{i,n}^{(1-\omega)}(x_{1},\ldots,x_{n})\\\
\times\rho_{j,n}^{\omega}(x_{1},\ldots,x_{n}),$ (13) $\displaystyle
z_{\omega}(n)$ $\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}^{d}}\cdots\int\limits_{\mathbb{R}^{d}}\rho_{i,n}^{(1-\omega)}(x^{\prime}_{1},\ldots,x^{\prime}_{n})$
(14)
$\displaystyle\mspace{10.0mu}\times\rho_{j,n}^{\omega}(x^{\prime}_{1},\ldots,x^{\prime}_{n})\mathrm{d}x^{\prime}_{1},\ldots,\mathrm{d}x^{\prime}_{n},$

and, the cardinality pmf is

$\displaystyle p_{\omega}(n)$ $\displaystyle=$
$\displaystyle\frac{1}{N_{\omega}}p_{i}^{(1-\omega)}(n)p_{j}^{\omega}(n)z_{\omega}(n)$
(15) $\displaystyle N_{\omega}$ $\displaystyle=$
$\displaystyle\sum_{n^{\prime}=0}p_{i}^{(1-\omega)}(n^{\prime})p_{j}^{\omega}(n^{\prime})z_{\omega}(n^{\prime}).$
(16)

where $z_{\omega}(0)=1$ by convention, and for $n\neq 0$, $z_{\omega}(n)<1$
unless $\rho_{i,n}$ and $\rho_{j,n}$ are identical. The latter is a direct
application of Hölder’s inequality (see, e.g., Theorem 188 in [40]). It can be
shown similarly that $N_{\omega}<1$.

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 pmf is the scaled product of the
cardinality EMD with the localisation density scale factors $z_{\omega}(n)$ in
(14). As a result, the consistency property of EMDs does not apply to the
fused cardinality distribution. Below, we first relate the consistency of the
fused cardinality pmf to the sequence of scale factors and give a condition
under which the fused cardinality distribution is inconsistent. Then, in the
rest of this section, we demonstrate that inconsistent cardinality
distributions occur under some typical operating conditions.

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

Consider the fused cardinality pmf $p_{\omega}$ in (15), (16). Consider the
following inconsistency condition for $p_{\omega}(n)$ obtained by negating the
consistency condition:

$p_{\omega}(n)<\min\\{\,\,p_{i}(n),\,p_{j}(n)\,\,\\}.$ (17)

This condition holds true if

$z_{\omega}(n)<\frac{\sum_{n^{\prime}\neq
n}p_{i}^{(1-\omega)}(n^{\prime})p_{j}^{\omega}(n^{\prime})z_{\omega}(n^{\prime})}{\frac{p_{i}^{(1-\omega)}(n)p_{j}^{\omega}(n)}{\min\\{\,\,p_{i}(n),\,p_{j}(n)\,\,\\}}-p_{i}^{(1-\omega)}(n)p_{j}^{\omega}(n)},$
(18)

where $z_{\omega}(n)$ 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 $n$th localisation scale
$z_{\omega}(n)$ is comparably small. This is in stark contrast with the fused
localisation densities in (13) which always satisfy the consistency condition

$\rho_{\omega,n}(x_{1},\ldots,x_{n})\geq\min\\{\rho_{i,n}(x_{1},\ldots,x_{n}),\rho_{j,n}(x_{1},\ldots,x_{n})\\}$

for all $x_{1},\ldots,x_{n}\in\mathbb{R}^{d}$ and $n$, as they are EMDs.

The scale factors modulating the cardinality pmf, i.e., $z_{\omega}(n)$, 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 $z_{\omega}(n)$ equaling unity only when $\rho_{i,n}$ and $\rho_{j,n}$
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
$\rho_{i,n}$ and $\rho_{j,n}$ will have a comparably small overlap in their
confidence regions. As a result, much smaller $z_{\omega}(n)$ 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

$p(n)=\begin{cases}1-\alpha,&\,n=0,\\\ \alpha,&\,n=1,\\\
0,&\,\text{otherwise}\end{cases}$ (19)

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

There is also a single localisation density $\rho_{n}$ for $n=1$ which we will
denote by $\rho$. Therefore, given two Bernoullis
$f_{i}=(\alpha_{i},\rho_{i})$ and $f_{j}=(\alpha_{j},\rho_{j})$, the sequence
$z_{\omega}(n)$ reduces to

$z_{\omega}(n)=\begin{cases}1,&\,n=0,\\\
z_{\omega}\triangleq\int_{\mathbb{R}^{d}}\rho_{i}^{(1-\omega)}(x)\rho_{j}^{\omega}(x)\mathrm{d}x,&\,n=1,\\\
0,&\,\text{otherwise}.\end{cases}\\\ $ (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]

$\alpha_{\omega}=\frac{\alpha_{i}^{(1-\omega)}\alpha_{j}^{\omega}z_{\omega}}{(1-\alpha_{i})^{(1-\omega)}(1-\alpha_{j})^{\omega}+\alpha_{i}^{(1-\omega)}\alpha_{j}^{\omega}z_{\omega}}$
(21)

is smaller than either of $\alpha_{i}$ or $\alpha_{j}$ if

$z_{\omega}<\frac{(1-\alpha_{i})^{(1-\omega)}(1-\alpha_{j})^{\omega}}{\alpha_{i}^{(1-\omega)}\alpha_{j}^{\omega}/\min\\{\,\alpha_{i}\,,\,\alpha_{j}\,\\}-\alpha_{i}^{(1-\omega)}\alpha_{j}^{\omega}}.$

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 $\alpha_{i}$ and
$\alpha_{j}$ are equal, then this condition reduces to $z_{\omega}<1$ which
always holds for all practical purposes as $\rho_{i}$ and $\rho_{j}$ should
not be expected to be identical. For $\alpha_{i}\neq\alpha_{j}$, this
inconsistency still occurs with overwhelming probability in Bernoulli fusion
which is demonstrated in the following example.

Fig. 1: Localisation densities of the Gauss-Bernoulli finite sets, i.e.,
$\rho_{i}$ (solid line) and $\rho_{j}$ (dash-dotted line), in Example III.3
for increasing sensing diversity as the covariance condition number is
increased as $\kappa=1,10,20,30,40$ (left to right).

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

Fig. 2: The scale factor (left) and the fused existence probabilities in the
Gauss-Bernoulli Example III.3. These quantities are calculated for varying
sensing diversity $\kappa$ (equivalently, the covariance condition number of
the Gaussians) and (exponential) mixture weight $\omega$ values.

Let us consider Bernoulli distributions with Gaussian localisation densities
given by

$\rho_{i}(x)={\cal
N}(x;\mathbf{m}_{i},\mathbf{C}_{i}),\,\,\,\,\rho_{j}(x)={\cal
N}(x;\mathbf{m}_{j},\mathbf{C}_{j}),$ (22)

where $\mathbf{m}$ is the mean vector and $\mathbf{C}$ is the covariance
matrix. The fused localisation density $\rho_{\omega}$ for the case is a
Gaussian with mean and covariance given by

$\displaystyle\mathbf{m}_{\omega}$ $\displaystyle=$
$\displaystyle\mathbf{C}_{\omega}\big{(}(1-\omega)\mathbf{C}_{i}^{-1}\mathbf{m}_{i}+\omega\mathbf{C}_{j}^{-1}\mathbf{m}_{j}\big{)}$
(23) $\displaystyle\mathbf{C}_{\omega}$ $\displaystyle=$
$\displaystyle\left((1-\omega)\mathbf{C}_{i}^{-1}+\omega\mathbf{C}_{j}^{-1}\right)^{-1}.$
(24)

The scale factor $z_{\omega}$ is found using integration rules for Gaussians
as

$\mspace{-20.0mu}z_{\omega}=\frac{\left|\mathbf{\mathbf{C}}^{-1}_{i}\right|^{(1-\omega)/2}\left|\mathbf{C}^{-1}_{j}\right|^{\omega/2}}{\left|\mathbf{C}_{\omega}\right|^{1/2}}\exp\left\\{-\frac{1}{2}\bigg{(}(1-\omega)\mathbf{m}_{i}^{T}\mathbf{\mathbf{C}}_{i}^{-1}\mathbf{m}_{i}\bigg{.}\right.\\\
\left.\bigg{.}+\omega\mathbf{m}_{j}^{T}\mathbf{\mathbf{C}}_{j}^{-1}\mathbf{m}_{j}-\mathbf{m}_{\omega}^{T}\mathbf{C}_{\omega}^{-1}\mathbf{m}_{\omega}\bigg{)}\right\\}.$
(25)

Let us consider two Bernoullis with existence probabilities
$\alpha_{i}=\alpha_{j}=0.8$ with localisation densities of mean vectors
$\mathbf{m}_{i}=[0.25,0.25]^{T}$ and $\mathbf{m}_{j}=[-0.75,-0.25]^{T}$,
respectively, where $(.)^{T}$ denotes vector transpose. We select the
covariance matrices as rotated versions of a diagonal covariance given by

$\displaystyle\mathbf{C}_{i}$ $\displaystyle=$
$\displaystyle\mathbf{R}(\pi/4)\mathbf{\Sigma}\mathbf{R}^{T}(\pi/4),$
$\displaystyle\mathbf{C}_{j}$ $\displaystyle=$
$\displaystyle\mathbf{R}(-\pi/4)\mathbf{\Sigma}\mathbf{R}^{T}(-\pi/4),$
$\displaystyle\mathbf{\Sigma}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\sigma_{1}^{2},&0\\\
0,&\sigma_{2}^{2}\end{array}\right],$ (28) $\displaystyle\mathbf{R}(\phi)$
$\displaystyle=$ $\displaystyle\left[\begin{array}[]{rr}\cos\phi,&-\sin\phi\\\
\sin\phi,&\cos\phi\end{array}\right].$ (31)

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 $\Sigma$ – equivalently,
$\mathbf{C}_{i}$ and $\mathbf{C}_{j}$ – is given by
$\kappa=\sigma_{1}^{2}/\sigma_{2}^{2}$ and has higher values for sensors with
range/cross-range ambiguity such as cameras/radars. We vary this quantity from
$\kappa=1$ to $40$. 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 $z_{\omega}$ and
$\alpha_{\omega}$ values obtained by varying the condition number $\kappa$
with small steps from $1$ to $40$ hence increasing the sensing diversity. The
exponential mixture weights $\omega$ take values from a dense grid over
$[0,1]$. 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 $\kappa$ which controls the sensing diversity. It
is convex with respect to the mixture weight $\omega$, 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 $\kappa$ and $\omega$. Moreover, the fused existence probability
drops below $0.5$ for large values of the sensing diversity parameter
$\kappa$. 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
$\alpha_{i}=\alpha_{j}=0.8$, detection might be missed if based on the fused
result instead, thereby 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. $\blacksquare$

###  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.,

$p(n)=\frac{\mathrm{e}^{\lambda}\lambda^{n}}{n!}$ (32)

where $\lambda$ is the expected number of objects. The localisation densities
factorise over the density for $n=1$ as

$\rho_{n}(x_{1},\ldots,x_{n})=\prod_{i=1}^{n}\rho_{1}(x_{i}),$ (33)

making it possible to parameterise the entire finite set distribution with a
scalar and a single density666We drop the subscript in $\rho_{1}$ for the rest
of this subsection and denote it by $\rho$..

For two Poissons $f_{i}=(\lambda_{i},\rho_{i})$ and
$f_{j}=(\lambda_{j},\rho_{j})$, the sequence $z_{\omega}(n)$ is a geometric
sequence found by subsituting from (33) for both $i$ and $j$ into (14). This
sequence is found as

$\displaystyle z_{\omega}(n)$ $\displaystyle=$ $\displaystyle z_{\omega}^{n}$
(34) $\displaystyle z_{\omega}$ $\displaystyle\triangleq$ $\displaystyle
z_{\omega}(1)=\int_{{\mathbb{R}^{d}}}\rho_{i}^{(1-\omega)}(x^{\prime})\rho_{j}^{\omega}(x^{\prime})\mathrm{d}x,$
(35)

where $z_{\omega}<1$ unless $\rho_{i}$ and $\rho_{j}$ are identical, as
aforementioned.

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

$\lambda_{\omega}=\lambda_{i}^{(1-\omega)}\lambda_{j}^{\omega}z_{\omega}.$
(36)

######  Proposition III.4 (Poisson inconsistency in expectation)

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

$\lambda_{\omega}<\min\\{\lambda_{i},\lambda_{j}\\}.$ (37)

This condition holds whenever

$z_{\omega}<\frac{\min\\{\lambda_{i},\lambda_{j}\\}}{\max\\{\lambda_{i},\lambda_{j}\\}}.$
(38)

The proof follows easily from substituting (38) in (36) and (37). 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 $n$. The above result is on the expected value of
$n$ in Poisson finite set densities. The condition in (38) is satisfied with
overwhelming probability in practice leading to inconsistencies as observed,
for example, in [27]. For example, for $\lambda_{i}=\lambda_{j}=\lambda$, this
reduces to the common ratio $z_{\omega}$ 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
$\lambda$ as the estimated number of objects777Maximum a posteriori (MAP)
estimation is not used with Poisson cardinality distributions as (32) is not
guaranteed to have a unique maximum. Notice that, for example, (32) evaluates
at the same value for both $n=0$ and $n=1$ for $\lambda=1$.. 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
$z_{\omega}$.

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

IID cluster finite set distributions relax the Poisson cardinality pmf in (32)
and take arbitrary cardinality pmfs underpinning the C-PHD filter [41]. The
localisation densities still take the factorised form in (33) leading to the
identical geometric series $z_{\omega}(n)$ in (34). For the case, Proposition
III.1 specialises as follows:

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Fig. 3: Illustrations of the results in Example III.6: LABEL:sub@fig:BB Two
cardinality distributions peaking at $n=5$. LABEL:sub@fig:BBfusion Fused
cardinalities for some intermediate values of $\omega$ and $0.1\leq
z_{\omega}\leq 0.9$. LABEL:sub@fig:calI The inconsistency upper bound in (39).
LABEL:sub@fig:eta The inconsistency threshold in (40). LABEL:sub@fig:mapest1
MAP estimates using the fused cardinalities for varying $\omega$ and
$z_{\omega}$. LABEL:sub@fig:cardHigh Cardinality distributions peaking at
$n=35$. LABEL:sub@fig:BBfusionHigh Fused cardinalities for some intermediate
values of $\omega$ and $0.1\leq z_{\omega}\leq 0.9$. LABEL:sub@fig:mapBBHigh
MAP estimates using the fused cardinalities for varying $\omega$ and
$z_{\omega}$.

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

Given two IID cluster finite set distributions $f_{i}=(p_{i}(n),\rho_{i})$ and
$f_{j}=(p_{j}(n),\rho_{j})$, the fused cardinality distribution $p_{\omega}$
satisfies the inconsistency condition in (17) in Proposition III.1 for the
number of objects $n$ and non-zero $p_{i}(n),p_{j}(n)$ if

$z_{\omega}<{\cal I}_{\omega,n}\\\ $

holds, where the term on the right hand side is

${\cal
I}_{\omega,n}\triangleq\left(N_{\omega}\frac{\min\\{p_{i}(n),p_{j}(n)\\}}{p^{(1-\omega)}_{i}(n)p^{\omega}_{j}(n)}\right)^{1/n},$
(39)

$z_{\omega}$ is given in (35), and, $N_{\omega}$ is obtained by substituting
(34) in (16).

The proof follows from substituting (34) in (18) and using (16) after
rearrangement of the terms. The inconsistency condition in (39) depends both
on $n$ and $\omega$, 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 (39) is smaller than one and hence $\cal I$
approaches to one as $n$ grows. Therefore, for some threshold $\eta$, the
fused object number probabilities will be lower than the input cardinality
reports for all $n>\eta$. Such a threshold can easily be found from (39) as

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{\log\left(N_{\omega}\gamma_{\omega}\right)}{\log
z_{\omega}},$ (40) $\displaystyle\gamma_{\omega}$ $\displaystyle\triangleq$
$\displaystyle\min_{n^{\prime}}\frac{\min\\{p_{i}(n^{\prime}),p_{j}(n^{\prime})\\}}{p^{(1-\omega)}_{i}(n^{\prime})p^{\omega}_{j}(n^{\prime})}$

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 C-PHD 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 $p_{i}(n)=B(n;k=5,P=0.95)$ and
$p_{j}(n)=B(n;k=5,P=0.92)$ where these distributions give the probability that
$n$ objects exist simultaneously among $k=5$ possibilities each with an
existence probability of $P$ (Fig. 3LABEL:sub@fig:BB). Of particular interest
is the characteristics of $p_{\omega}$ as $z_{\omega}$ and $\omega$ vary in
$0.1\leq z_{\omega}\leq 0.9$ and $0\leq\omega\leq 1$, respectively. Fig.
3LABEL:sub@fig:BBfusion presents fused distributions obtained by varying
$z_{\omega}$ and some intermediate values of $\omega$. Note that the
cardinality $n$ at which the fused distributions peaks varies with
$z_{\omega}$ as suggested in Corollary III.5. In particular, the inconsistency
bound in (3c) is illustrated in Fig. 3LABEL:sub@fig:calI which monotonically
increases with $n$ as discussed. The inconsistency threshold for $n$ as given
in (40) is given in Fig. 3LABEL:sub@fig:eta. Note that for a large ratio of
$z_{\omega}$ and $\omega$ 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 $\hat{n}_{i}=\hat{n}_{j}=5$. 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 $n$ value. Specifically, $p_{i}(n)=B(n;k=35,P=0.98)$ and
$p_{j}(n)=B(n;k=35,P=0.975)$ are used (see Fig. 3LABEL:sub@fig:cardHigh) which
have individual map estimates of $\hat{n}_{i}=\hat{n}_{j}=35$. The fused
cardinalities in Fig. 3LABEL:sub@fig:BBfusionHigh illustrate that for a larger
subset of $(z_{\omega},\omega)$ 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 (39) approaches to $1$ with increasing $n$, as the peak cardinality
increases, the IID inconsistency detoriates decision making more.

###  III-D Summary of results

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 $z_{\omega}(n)$. Proposition III.1 provides a
general condition on the fused distribution to be inconsistent with the input
distributions at a cardinality value $n$. 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 $\lambda$ that specifies the distribution for all $n$. Proposition
III.4 provides a condition of inconsistency in this parameter, similarly as an
upper bound on $z_{\omega}(n)$. It is pointed out that because $z_{\omega}(n)$
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
settings888The authors at this point would like to conjecture that
$z_{\omega}(n)<1$ 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., $f_{i}(X)=f(X|Z^{i}_{1:t})$ and
$f_{j}(X)=f(X|Z^{j}_{1:t})$ for realisations $Z^{i}_{1:t}$ and $Z^{j}_{1:t}$
of (independent) measurement processes associated with sensors $i$ and $j$,
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 pmf on the
scaling factor series $z_{\omega}(n)$. 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 $f_{i}$ and $f_{j}$:

$\displaystyle\text{(P3)}\,\,\,\,\,{\text{For}}\,\,\,n$ $\displaystyle=$
$\displaystyle 1,2,\ldots$
$\displaystyle(\omega_{n}^{*},\rho_{\omega_{n}^{*},n})$
$\displaystyle\triangleq$
$\displaystyle\arg\max_{\omega\in[0,1]}\min_{\rho_{n}\in{\cal
P}_{n}}J_{\omega,n}[\rho_{n}]$ (41) $\displaystyle J_{\omega,n}[\rho_{n}]$
$\displaystyle\triangleq$
$\displaystyle(1-\omega)D(\rho_{n}||\rho_{i,n})+\omega
D(\rho_{n}||\rho_{j,n}).$

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

Following our discussion in Section II, solutions of these uncoupled problems
have an EMD form given by (13) and (14)999It is easy to show that because
$\rho_{i,n}$ and $\rho_{j,n}$ are symmetric in their arguments,
$\rho_{\omega,n}$ also exhibits this symmetricity.. One difference here
compared to the solution of problem P2 is that for each $n$, a different
optimal weight $\omega^{*}_{n}$ 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
$(\omega^{*}_{n},\rho_{\omega^{*}_{n},n})$, the fused cardinality distribution
becomes an additional 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

$\displaystyle\mspace{-30.0mu}\text{(P4)}\,\,\,\,\,\,(\omega_{c}^{*},\tilde{p}_{\omega_{c}^{*}})$
$\displaystyle\triangleq$
$\displaystyle\arg\max_{\omega\in[0,1]}\min_{p\in{\cal P}_{c}}J_{\omega,c}[p]$
(42) $\displaystyle J_{\omega,c}[p]$ $\displaystyle\triangleq$
$\displaystyle(1-\omega)D(p||p_{i})+\omega D(p||p_{j}).$

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

$\displaystyle\tilde{p}_{\omega}(n)$ $\displaystyle=$
$\displaystyle\frac{1}{\tilde{N}_{\omega}}p_{i}^{(1-\omega)}(n)p_{j}^{\omega}(n)$
(43) $\displaystyle\tilde{N}_{\omega}$ $\displaystyle=$
$\displaystyle\sum_{n^{\prime}=0}p_{i}^{(1-\omega)}(n^{\prime})p_{j}^{\omega}(n^{\prime}).$
(44)

evaluated at $\omega=\omega^{*}_{c}$.

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

$\tilde{p}_{\omega}(n)\geq\min\\{p_{i}(n),p_{j}(n)\\}$

is satisfied for all $n$ and for all $\omega$ regardless of $z_{\omega}(n)$.
Thus, $\tilde{p}_{\omega}$ prevents 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

$\displaystyle\tilde{f}^{*}(X)$ $\displaystyle=$
$\displaystyle\tilde{f}_{\Omega}(X)\bigg{\rvert}_{\Omega=(\omega_{c}=\omega^{*}_{c},\omega_{1}=\omega^{*}_{1},\omega_{2}=\omega^{*}_{2},\ldots)}$
(45) $\displaystyle\tilde{f}_{\Omega}(X)$ $\displaystyle\triangleq$
$\displaystyle
p_{\omega_{c}}\left(n\right)n!\rho_{\omega_{n},n}(x_{1},...,x_{n})$ (46)

where $\omega^{*}_{c}$ is found by solving the maximisation in (42) with a
cardinality distribution given by (43) and (44). Here, $\omega^{*}_{n}$ solves
the maximisation in (41) 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 $\tilde{f}$ 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

$\tilde{f}_{\omega}(X)\triangleq\tilde{f}_{\Omega}(X)\bigg{\rvert}_{\Omega=(\omega_{c}=\omega,\omega_{1}=\omega,\omega_{2}=\omega,\ldots)}$
(47)

for some $\omega$. $\tilde{f}_{\omega}$ is pointwise inconsistent, i.e.,

$\tilde{f}_{\omega}(X)<\min\\{f_{i}(X),f_{j}(X)\\}$ (48)

if

$\frac{E_{\tilde{p}_{\omega}}\\{z_{\omega}(n)\\}}{z_{\omega}(n)}<\frac{\min\\{f_{i}(X),f_{j}(X)\\}}{f_{\omega}(X)}<=1$
(49)

where $f_{\omega}$ is the finite set EMD given in (13)–(16).

###### Proof:

By comparing (13)–(16) and (43)–(46), it can be seen that the two finite set
densities of concern are related by

$\tilde{f}_{\omega}(X)=\frac{E_{\tilde{p}_{\omega}}\\{z_{\omega}(n)\\}}{z_{\omega}(n)}f_{\omega}(X).$
(50)

where the expectation is with respect to (43).

Substitution of (50) in (48) yields the first inequality in (49). Note that,
$f_{\omega}$ 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 $\tilde{f}_{\omega}$
is guaranteed only for those $X$ with cardinality $n$ for which the scaling
factor of the localisation density $z_{\omega}(n)\leq 1$ equals to the
expectation. If $z_{\omega}(n)$ is greater than the expectation to the extent
that (49) is satisfied, then $\tilde{f}_{\omega}$ 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

Algorithm 1 Newton iterations for solving the $n$th problem in P3.

1:Input: $\rho_{i,n}$,$\rho_{j,n}$ $\triangleright$ Localisation densities

2:Input: $\omega^{(0)}\in[0,1]$, $\epsilon$ $\triangleright$ Initial value,
termination threshold

3:$k\leftarrow 1$, $\omega^{(1)}\leftarrow\infty$

4:while $|\omega^{(k)}-\omega^{(k-1)}|>\epsilon$ do $\triangleright$
termination condition

5: $z_{k}\leftarrow z_{\omega}(n)\rvert_{\omega={\omega^{(k)}}}$ using (14)

6: $z^{\prime}_{k}\leftarrow
z^{\prime}_{\omega}(n)\rvert_{\omega={\omega^{(k)}}}$ using (53)

7: $z^{\prime\prime}_{k}\leftarrow
z^{\prime\prime}_{\omega}(n)\rvert_{\omega={\omega^{(k)}}}$ using
(LABEL:eqn:zprimeprime)

8:
$\omega^{(k+1)}\leftarrow\omega^{(k)}-z^{\prime}_{k}z_{k}/\left(z^{\prime\prime}_{k}z_{k}-\left(z^{\prime}_{k}\right)^{2}\right)$

9: $k\leftarrow k+1$

10:end while

11:Return $\omega^{*}_{n}\leftarrow\omega^{(k)}$

12:Return $\rho_{\omega^{*}_{n},n}$ using (13)

The variational problems P3 and P4 are $\max\min$ optimisation problems
similar to (8). Hence, the minimisations given $\omega$ 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))

$\displaystyle G_{n}(\omega_{n})$ $\displaystyle\triangleq$ $\displaystyle
J_{\omega,n}[\rho_{n}]\bigg{\rvert}_{\rho_{n}=\rho_{\omega,n}}$
$\displaystyle=$ $\displaystyle-(\omega-1){\cal
R}_{\omega}(\rho_{n,i},\rho_{n,j})$ $\displaystyle=$ $\displaystyle-\log
z_{\omega}(n)$

which is a concave function of its one dimensional argument that takes values
from a bounded interval. Newton iterations converge to a solution and have an
excellent convergence rate [42]. Starting from an initial value
$\omega^{(0)}\in[0,1]$, recursive increments are made by the ratio of the
first and second order derivatives, i.e.,

$\displaystyle\mspace{-20.0mu}\frac{G^{\prime}_{n}(\omega_{n})}{G^{\prime\prime}_{n}(\omega_{n})}\bigg{\rvert}_{{\omega_{n}=\omega^{(k)}}}\mspace{-10.0mu}$
$\displaystyle=$
$\displaystyle\frac{z^{\prime}_{\omega}(n)z_{\omega}(n)}{z^{\prime\prime}_{\omega}(n)z_{\omega}(n)-\left(z^{\prime}_{\omega}(n)\right)^{2}}\bigg{\rvert}_{{\omega=\omega^{(k)}}}$
(52) $\displaystyle z^{\prime}_{\omega}(n)$ $\displaystyle\triangleq$
$\displaystyle\int\rho_{i,n}^{1-\omega}(x)\rho_{j,n}^{\omega}(x)\log\frac{\rho_{j,n}(x)}{\rho_{i,n}(x)}\mathrm{d}x$
(53) $\displaystyle z^{\prime\prime}_{\omega}(n)$ $\displaystyle\triangleq$
$\displaystyle\int\rho_{i,n}^{1-\omega}(x)\rho_{j,n}^{\omega}(x)\left(\log\frac{\rho_{j,n}(x)}{\rho_{i,n}(x)}\right)^{2}\mathrm{d}x$

where $\omega^{(k)}$ is the value found in the $k$th iteration. Here,
$z_{\omega}(n)$ is given in (14) and its derivatives in (53) 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 $\omega_{n}^{*}=\omega_{n}^{(k)}$ is found for
which the corresponding EMD $\rho_{\omega_{n}^{(k)},n}$ is the fused density.

An analogous iterative algorithm for finding the consistently fused
cardinality distribution 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 2 Newton iterations solving P4 for consistent cardinality fusion.

1:Input: $p_{i}(n)$,$p_{j}(n)$ $\triangleright$ Cardinality pmfs

2:Input: $\omega^{(0)}\in[0,1]$, $\epsilon$ $\triangleright$ Initial value,
termination threshold

3:$k\leftarrow 1$, $\omega^{(1)}\leftarrow\infty$

4:while $|\omega^{(k)}-\omega^{(k-1)}|>\epsilon$ do $\triangleright$
termination condition

5: $N_{k}\leftarrow N_{\omega}\rvert_{\omega={\omega^{(k)}}}$ using (16)

6: $N^{\prime}_{k}\leftarrow\sum
p_{i}^{1-\omega}(n)p_{j}^{\omega}(n)\log\big{(}p_{j}(n)/p_{i}(n)\big{)}$

7: $N^{\prime\prime}_{k}\leftarrow\sum
p_{i}^{1-\omega}(n)p_{j}^{\omega}(n)\bigg{(}\log\big{(}p_{j}(n)/p_{i}(n)\big{)}\bigg{)}^{2}$

8:
$\omega^{(k+1)}\leftarrow\omega^{(k)}-N^{\prime}_{k}z_{k}/\left(N^{\prime\prime}_{k}N_{k}-\left(N^{\prime}_{k}\right)^{2}\right)^{2}$

9: $k\leftarrow k+1$

10:end while

11:Return $\omega^{*}_{c}\leftarrow\omega^{(k)}$

12:Return $p_{\omega^{*}_{c}}$ using (15)

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 [43]. 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 (33)– 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,
$D(\rho_{n}||\rho_{n,i})=nD(\rho||\rho_{i})$ where $\rho$ parameterises
$\rho_{n}$, and, the family of problems in P3 satisfy

$J_{\omega,n}[\rho_{n}]=nJ_{\omega,1}[\rho].$ (55)

Consequently, the optimal solution to P3 for cardinality $n$ is parameterised
by the optimal solution to $n=1$ thereby restricting it to the case for only
$n=1$.

Bernoulli finite sets have nonzero cardinality pmf only for $n\leq 1$
naturally restricting P3 to $n=1$. If the parameterising densities $\rho_{i}$
and $\rho_{j}$ are Gaussians, then P3 specifies a covariance intersection
procedure [4]. For this case, P4 has a closed form solution given by [44]

$\displaystyle\omega^{*}_{c}$ $\displaystyle=$
$\displaystyle\frac{\log\bigg{(}\frac{\log(1-\alpha_{i})/(1-\alpha_{j})}{\log(\alpha_{j}/\alpha_{i})}\bigg{)}-\log\bigg{(}\frac{\alpha_{i}}{1-\alpha_{i}}\bigg{)}}{\log\frac{1-\alpha_{i}}{1-\alpha_{j}}+\log\frac{\alpha_{j}}{\alpha_{i}}}$
(56) $\displaystyle\alpha^{*}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{i}^{1-\omega^{*}_{c}}\alpha_{j}^{\omega^{*}_{c}}}{\alpha_{i}^{1-\omega^{*}_{c}}\alpha_{j}^{\omega^{*}_{c}}+(1-\alpha_{i})^{1-\omega^{*}_{c}}(1-\alpha_{j})^{\omega^{*}_{c}}}$
(57)

For Poisson cardinality pmfs, similarly a closed form solution exists for P4
which is given by [44]

$\displaystyle\omega^{*}_{c}$ $\displaystyle=$
$\displaystyle\frac{-\log\big{(}\log(\lambda_{j}/\lambda_{i})\big{)}+\log(\lambda_{j}/\lambda_{i}-1)}{\log(\lambda_{j}/\lambda_{i})}$
(58) $\displaystyle\lambda^{*}$ $\displaystyle=$
$\displaystyle\lambda_{i}^{1-\omega^{*}_{c}}\lambda_{j}^{\omega^{*}_{c}}$ (59)

It is worthwhile to notice that both Bernoulli and Poisson distributions are
exponential family distributions and the above solutions bear the geometric
properties aforementioned in Section II-B and proved in [35].

###  IV-C Demonstration of cardinality consistent fusion

In this section, we revisit the examples in Section III involving cardinality
inconsistencies and demonstrate the efficacy of the solutions of Problems P3
and P4 in these fusion scenarios.

######  Example IV.2 (Gauss-Bernoulli case revisited)

Let us consider the Gauss-Bernoulli case in Example III.3 in the light of the
discussion above. Fusion of localisation distributions in P3 involve the
fusion of only a single pair for $n=1$, for the case. These distributions
$\rho_{i}$ and $\rho_{j}$ given by (22) are Gaussians, therefore, Algorithm 1
is equivalently an iterative covariance intersection algorithm that optimises
$\omega$ to achieve the KLD equality criteria in (7). The Newton update for
the parameter $\omega$ in Step 8 of Algorithm 1 is carried out as follows:
Evaluation of $z_{\omega}$ in Step 5 at $\omega=\omega^{(k)}$ is made using
its closed form expression in (25). Given $z_{\omega}$, the derivative in Step
6 is also found in closed form using the following identity

$z^{\prime}_{\omega}=z_{\omega}\big{(}D(\rho_{\omega}||\rho_{i})-D(\rho_{\omega}||\rho_{j})\big{)},$
(60)

which can easily be verified by dividing both sides of (53) to
$z_{\omega}(n)$. This quantity is computed by evaluating the KLD of multi-
variate Gaussian densities given by (see, for example, [45, A.23])

$D(\rho_{\omega}||\rho_{i})={1/2}\log|\mathbf{C}_{i}\mathbf{C}_{\omega}^{-1}|\\\
+\mathbf{tr}\\{\mathbf{C}_{i}^{-1}\big{(}(\mathbf{m}_{\omega}-\mathbf{m}_{i})(\mathbf{m}_{\omega}-\mathbf{m}_{i})^{T}+\mathbf{C}_{\omega}-\mathbf{C}_{i}\big{)}\\},$
(61)

where $\mathbf{tr}\\{.\\}$ is the trace of its matrix argument. The second
derivative in Step 7 on the other hand, is found approximately using the Monte
Carlo method [46, Chp.3] targeting the integration in (LABEL:eqn:zprimeprime)
divided by $z_{\omega}(n)$. Therefore, $L$ samples are generated from
$\rho_{\omega}$ for this step, i.e., $x^{(l)}\sim\rho_{\omega}$ for
$l=1,\ldots,L$. Using these samples, the approximation seeked is given by

$z^{\prime\prime}_{\omega}\approx{z_{\omega}}\times\frac{1}{L}\sum_{l=1}^{L}\Big{(}\log\frac{\rho_{j}(x^{(l)})}{\rho_{i}(x^{(l)})}\Big{)}^{2}$
(62)

where the approximation error decreases with ${\cal O}(1/\sqrt{L})$.

We input the Gaussian pairs in Example III.3 that are obtained by varying the
covariance condition number $\kappa$ – equivalently the sensing diversity– to
Algorithm 1 and use the computational procedures above. Fig. 4 depicts
optimally weighted EMDs for $\kappa=1,10,20$. The optimal weight $\omega^{*}$
output by Algorithm 1 as a function of the condition number $\kappa$ is given
in Fig. 5. Here, the termination threshold is set to
$\epsilon=1\mathrm{e}{-4}$ and the number of samples used for the Monte Carlo
estimate in Step 7 is $L=1000$. Convergence is declared after an average of
$3.4$ and a maximum of $5$ iterations. Note that $\omega^{*}$ yields a fused
result that bears more influence from $\rho_{i}$ as $\kappa$ increases.

Fig. 4: Optimally weighted EMDs of Gaussian localisation distributions
obtained by using Algorithm 1: Here, $\rho_{i}$ (solid line) and $\rho_{j}$
(dash-dotted line) are inputs for $\kappa=1,10,20$ (left to right – see
Example III.3 for details) and $\rho_{\omega^{*}}$ (magenta dashed line) are
fused outputs with optimal weights found as $\omega^{*}=0.500,0.397$ and
$0.387$ (left to right).

Fig. 5: Optimal weight parameters found using Algorithm 1 as a function of the
sensing diversity $\kappa$ as explained in detail in Example III.3.

Fig. 6: Fused existence probabilities: Cardinality consistent fusion via
Problem P4 (green line) in comparison with fused existence probabilities
obtained using KL averaging (blue line–see, e.g.,[17]–[23]) depicted as a
function of sensing diversity parameter $\kappa$ as detailed in Example III.3.
The red-dashed line is the canonical Bayesian decision threshold for deciding
the existence of an object.

Cardinality fusion problem P4 has an analytical solution for the case. Using
(56) for $\alpha_{i}=\alpha_{j}=0.8$ (see Example III.3), we find that
$\omega_{C}^{*}=0.5$ and $\alpha^{*}=0.8$ regardless of the solution of
Problem P3 above, i.e., the optimal weight parameters $\omega^{*}$ of the
localisation distributions or the corresponding normalisation constants
$z_{\omega^{*}}$s.

Let us compare this result with conventional EMD fusion with weights selected
using “Kullback-Leibler averaging” (KL averaging) as used in, for example,
[17]–[23]. In this approach, the fused finite set density is the Bernoulli EMD
with weight parameter $\omega=1/2$. In other words, the fused result solves
Problem P for $\omega=1/2$. The coupling of the fused existence probability
with $z_{\omega}$ and the lack of a weight selection mechanism results with
the fused existence probabilities given in Fig. 6 which illustrates a
monotonically increasing disagreement with the input beliefs on the existence
of an object as the sensing diversity increases. This trend results with the
fused existence probability falling below the canonical decision threshold of
$0.5$ 101010Note that this graph is nothing but the cross-section of the
existence probability graph in Fig. 2 along $\omega=0.5$.. The proposed
cardinality consistent fusion, on the other hand, preserves the confidence of
input distributions on the existence of an object irrespective of the sensing
diversity. The KL averaging fusion outputs a localisation density that is
similarly the EMD of the localisation distributions with weight $\omega=1/2$.
This margin between this value and the optimum point found by the proposed
algorithm grows significantly with $\kappa$, in this example (see Fig. 5).

Note that these results are also relevant for the work in literature on fusion
of multi-Bernoulli [20], and, labelled random finite set families as their
fusion is often reduced to performing Bernoulli-Bernoulli fusion for multiple
pairs using, for example, KL averaging [23].

######  Example IV.3 (Example III.6 revisited)

Let us demonstrate Algorithm 2 in solving the cardinality fusion problem P4.
First, we consider the binomial cardinality distribution pair illustrated in
Fig. 3LABEL:sub@fig:BB. Note that, the algorithm allows for exact computations
in all steps. The termination threshold is selected as $\epsilon=1.0e-4$. In
$2$ iterations Algorithm 2 declares convergence to the optimal weight
parameter $\omega_{C}^{*}=0.5182$. The corresponding fused cardinality pmf is
the EMD with this weight and depicted in Fig. 7LABEL:sub@fig:cfA. We repeat
the same procedure for the cardinality pair in Fig.
3LABEL:sub@fig:BBfusionHigh. The proposed algorithm convergences in $2$ steps
to $\omega_{C}^{*}=0.5090$. The resulting cardinality distribution is depicted
in Fig. 7LABEL:sub@fig:cfB. Note that the MAP estimates of the number of
objects is in agreement with the inputs. The consistency here is underpinned
by that the cardinality fusion here is independent of localisation densities
and $z_{\omega}$.

(a)

(b)

Fig. 7: Consistent cardinality fusion using Algorithm 2: Inputs (blue and
green dashed lines) are the binomial pairs introduced in Example III.3. The
optimal weights converged are $\omega_{C}^{*}=0.5182$ and $0.5090$ in
LABEL:sub@fig:cfA and LABEL:sub@fig:cfB, respectively.

In order to contrast this result with that obtained by KL averaging (see,
e.g.,[17]), let us consider the fused cardinalities depicted for $\omega=1/2$
in the top right panes in Fig. 3LABEL:sub@fig:BBfusion and
LABEL:sub@fig:BBfusionHigh in which the coupling of cardinality fusion with
$z_{\omega}$ is demonstrated. Let us remind also that the MAP object number
estimate depends on $z_{\omega}$ (Fig.s 3LABEL:sub@fig:mapest1 and
LABEL:sub@fig:mapBBHigh) with an increasing bias towards underestimation as
the input distribution peaks shift towards right indicating higher number of
objects (Fig.s 3LABEL:sub@fig:mapBBHigh). The proposed algorithm, on the other
hand, finds a consistent common ground of the input cardinality distributions
which is also optimal with respect to Problem P4.

##  V Conclusion

This work considered the recently growing literature on the use of EMDs –or,
weighted geometric means- of finite set densities in multi-sensor fusion for
multi-object tracking. EMDs of distributions are pointwise consistent,
however, we have proved in this article that they are prone to inconsistency
in their cardinality distributions which can lead to serious decision errors
related to the number of objects sensed. We have demonstrated that pointwise
consistency does not imply consistency in cardinality and vice versa. We
remedy this problem by redefining the variational optimisation problem that
underlies EMD fusion. Then, we specify iterative solutions and establish a
conceptual framework for cardinality consistent fusion of finite set densities
which also accommodates EMDs.

Following these results, possible future directions include investigation of
numerical computational schemes in order to use within this variational
framework. The extension of this variational perspective to accommodate $N$
sources is also anticipated to be a worthwhile direction to pursue.

###  -A Solution of the Problem P in (2)

In this appendix, we provide a direct proof for the assertion that the
solution of Problem P in (2) for any $\omega$ is the EMD given in (3),(4) when
$f$ is constrained to be a density, i.e., to integrate to unity. This
constraint on the feasible set of solutions together with the cost functional
of the problem are captured in the Lagrangian given by [47]

$\displaystyle{\cal L}[f,\lambda]$ $\displaystyle\triangleq$ $\displaystyle
J_{\omega}[f]+\lambda G[f],$ (63) $\displaystyle G[f]$
$\displaystyle\triangleq$ $\displaystyle\left(1-\int f(X)\mathrm{d}X\right).$

Here, $\lambda$ is a free variable referred to as a Lagrange multiplier which
–at its stationary point– imposes the constraint of integration to unity on
those $f$ which are also stationary. This point together with the convexity of
$J_{\omega}$ in $f$ results with the stationary point of (63) $f^{*}$ being
the solution to P in (2) [47].

The necessary (and sufficient) condition of stationarity that $f^{*}$ should
satisfy is given by i) the functional derivative of the Lagrangian, i.e.,

$\bigg{.}\delta\left(J_{\omega}+\lambda
G\right)[f;\delta_{X}]\bigg{\rvert}_{f=f^{*}}=0$ (64)

for all $X$, and, ii) the partial differential with respect to the multiplier
$\lambda$, i.e.,

$\bigg{.}\frac{\partial\left(J_{\omega}[f]+\lambda
G[f]\right)}{\partial\lambda}\bigg{\rvert}_{\lambda=\lambda^{*}}=0.$ (65)

The functional derivative in (64) can be expressed in terms of the partial
derivative with respect to $f(X)$, i.e.,

$\delta\left(J_{\omega}+\lambda
G\right)[f;\delta_{X}]=\frac{\partial\left(J_{\omega}[f]+\lambda
G[f]\right)}{\partial f(X)}$

for all $X$. By using the definition of KLD in (1), and rules of
differentiation, this expression leads to

$\frac{\partial\left(J_{\omega}[f]+\lambda G[f]\right)}{\partial f(X)}=\log
f(X)+\frac{f(X)}{f(X)}\\\ -(1-\omega)\log f_{i}(X)-\omega\log
f_{j}(X)-\lambda.$ (66)

The equation above is zero when the multiplier $\lambda$ takes the value

$\lambda=1+\log\frac{f(X)}{f^{(1-\omega)}_{i}(X)f^{\omega}_{j}(X)},$ (67)

for which the corresponding $f(X)$ is found as

$f(X)=\exp(\lambda-1)f^{(1-\omega)}_{i}(X)f^{\omega}_{j}(X).$ (68)

We substitute from the equality above into the partial differentiation in (65)
and obtain

$\displaystyle\mspace{-50.0mu}\frac{\partial\left(J_{\omega}[f]+\lambda
G[f]\right)}{\partial\lambda}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\lambda}\left((\lambda-1)\exp(\lambda-1)\int
f^{(1-\omega)}_{i}(X)f^{\omega}_{j}(X)\mathrm{d}X\right)$
$\displaystyle\,\,\,\,+\frac{\partial}{\partial\lambda}\left(\lambda-\lambda\exp(\lambda-1)\int
f^{(1-\omega)}_{i}(X)f^{\omega}_{j}(X)\mathrm{d}X\right)$

$\displaystyle=$
$\displaystyle\frac{\partial}{\partial\lambda}\left(\lambda-\exp(\lambda-1)\int
f^{(1-\omega)}_{i}(X)f^{\omega}_{j}(X)\mathrm{d}X\right)$ (69)
$\displaystyle=$ $\displaystyle 1-\exp(\lambda-1)\int
f^{(1-\omega)}_{i}(X)f^{\omega}_{j}(X)\mathrm{d}X.$

After (69) is set to zero, $\lambda^{*}$ is found as

$\lambda^{*}=-\log\int f^{(1-\omega)}_{i}(X)f^{\omega}_{j}(X)\mathrm{d}X+1$
(70)

and, the solution $f^{*}$ is found by subsituting from (70) into (68) as

$f^{*}(X)=\exp\bigg{(}-\log{\int
f^{(1-\omega)}_{i}(X)f^{\omega}_{j}(X)\mathrm{d}X}\bigg{)}f^{(1-\omega)}_{i}(X)f^{\omega}_{j}(X).$

Following a rearrangement of the terms, the expression above takes the form
given by

$f^{*}(X)=\frac{1}{\int
f^{(1-\omega)}_{i}(X)f^{\omega}_{j}(X)\mathrm{d}X}f^{(1-\omega)}_{i}(X)f^{\omega}_{j}(X),$
(71)

which can be identified as the EMD in (3),(4).

###  -B Proof of Proposition III.1

The proof follows from decomposing $p_{\omega}(n)$ in (15) and (16) as follows

$p_{\omega}(n)=\\\
\frac{p_{i}^{(1-\omega)}(n)p_{j}^{\omega}(n)z_{\omega}(n)}{p_{i}^{(1-\omega)}(n)p_{j}^{\omega}(n)z_{\omega}(n)+\sum_{n^{\prime}\neq
n}p_{i}^{(1-\omega)}(n^{\prime})p_{j}^{\omega}(n^{\prime})z_{\omega}(n^{\prime})},$
(72)

and substituting on the left hand side of the inequality in (17). The
inequality can easily be solved for $z_{\omega}(n)$ leading to (18).

###  -C Derivation of Algorithm 2

We start by finding the first and second order derivatives of $G_{n}$ in
(IV-B):

$\displaystyle G^{\prime}_{n}(\omega_{n})$ $\displaystyle\triangleq$
$\displaystyle\frac{dG_{n}(\omega)}{d\omega}\bigg{\rvert}_{\omega=\omega_{n}}$
(73) $\displaystyle=$
$\displaystyle-\frac{dz_{\omega}(n)/d\omega}{z_{\omega}(n)}\bigg{\rvert}_{\omega=\omega_{n}},$
$\displaystyle G^{\prime\prime}_{n}(\omega_{n})$ $\displaystyle\triangleq$
$\displaystyle\frac{d^{2}G_{n}(\omega)}{d\omega^{2}}\bigg{\rvert}_{\omega=\omega_{n}}$
(74) $\displaystyle=$
$\displaystyle-\frac{d^{2}z_{\omega}(n)/d\omega^{2}\times
z_{\omega}(n)-\left(dz_{\omega}(n)/d\omega\right)^{2}}{\left(z_{\omega}(n)\right)^{2}}\bigg{\rvert}_{\omega=\omega_{n}}.$

where $z_{\omega}(n)$ is given by (14). Newton iterations [42] use recursive
increments to the scalar argument of maximisation. These increments are found
by evaluating the ratio of (73) and (74) which is found as

$\displaystyle\frac{G^{\prime}_{n}(\omega_{n})}{G^{\prime\prime}_{n}(\omega_{n})}$
$\displaystyle=$
$\displaystyle\frac{z^{\prime}_{\omega}(n)z_{\omega}(n)}{z^{\prime\prime}_{\omega}(n)z_{\omega}(n)-\left(z^{\prime}_{\omega}(n)\right)^{2}}\bigg{\rvert}_{\omega=\omega_{n}},$
(75) $\displaystyle z^{\prime}_{\omega}(n)$ $\displaystyle\triangleq$
$\displaystyle\frac{dz_{\omega}(n)}{d\omega}$ (76) $\displaystyle
z^{\prime\prime}_{\omega}(n)$ $\displaystyle\triangleq$
$\displaystyle\frac{dz^{\prime}_{\omega}(n)}{d\omega}$ (77)

in terms of $z_{\omega}(n)$ and its first and second order derivatives.

Next, let us find the derivatives of $z_{\omega}(n)$. The first order
derivative follows after substituting (14) in (76) as

$\displaystyle\frac{dz_{\omega}(n)}{d\omega}$ $\displaystyle=$
$\displaystyle\int\rho_{i,n}(x)\frac{d}{d\omega}\left(\frac{\rho_{j,n}(x)}{\rho_{i,n}(x)}\right)^{\omega}\mathrm{d}x$
$\displaystyle=$
$\displaystyle\int\rho^{1-\omega}_{i,n}(x)\rho^{\omega}_{j,n}(x)\log\frac{\rho_{j,n}(x)}{\rho_{i,n}(x)}\mathrm{d}x$

which is equivalent to (53). The second order derivative seeked is found by
substituting from the above equality into (77) as

$\displaystyle\frac{dz^{\prime}_{\omega}(n)}{d\omega}$ $\displaystyle=$
$\displaystyle\int\rho_{i,n}(x)\log\frac{\rho_{j,n}(x)}{\rho_{i,n}(x)}\frac{d}{d\omega}\left(\frac{\rho_{j,n}(x)}{\rho_{i,n}(x)}\right)^{\omega}\mathrm{d}x$
$\displaystyle=$
$\displaystyle\int\rho^{1-\omega}_{i,n}(x)\rho^{\omega}_{j,n}(x)\bigg{(}\log\frac{\rho_{j,n}(x)}{\rho_{i,n}(x)}\bigg{)}^{2}\mathrm{d}x$

which is equivalently given in (LABEL:eqn:zprimeprime).

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