# Exact and Approximate Heterogeneous
Bayesian Decentralized Data Fusion

Ofer Dagan, Nisar R. Ahmed Manuscript received Month day, 2021; revised Month
day, 2021.The authors are with the Smead Aerospace Engineering Sciences
Department, University of Colorado Boulder, Boulder, CO 80309 USA (e-mail:
<EMAIL_ADDRESS>Nisar.Ahmed@colorado.edu).

###### Abstract

In Bayesian peer-to-peer decentralized data fusion, the underlying
distributions held locally by autonomous agents are frequently assumed to be
over the same set of variables (homogeneous). This requires each agent to
process and communicate the full global joint distribution, and thus leads to
high computation and communication costs irrespective of relevancy to specific
local objectives. This work studies a family of heterogeneous decentralized
fusion problems, where the set of problems in which either the communicated or
the processed distributions describe different, but overlapping, states of
interest that are subsets of a larger full global joint state is considered.
We exploit the conditional independence structure of such problems and provide
a rigorous derivation for a family of exact and approximate heterogeneous
conditionally factorized channel filter methods. We further extend existing
methods for approximate conservative filtering and decentralized fusion in
heterogeneous dynamic problems. Numerical examples show more than 99.5%
potential communication reduction for heterogeneous channel filter fusion, and
a multi-target tracking simulation shows that these methods provide consistent
estimates.

###### Index Terms:

Bayesian decentralized data fusion (DDF), distributed robot systems, multi-
robot systems, sensor fusion.

##  I Introduction

Bayesian decentralized data fusion (DDF) has a wide range of applications,
such as cooperative localization [1], multi-target tracking [2], multi-robot
localization and mapping [3], and more. Decentralized data fusion, while
generally less accurate compared to centralized data fusion, offers advantages
in terms of scalability, flexibility and robustness. One of the challenges of
decentralized data fusion stems from the difficulty of accounting for common
data and dependencies between communicating agents and avoiding ‘rumor
propagation’, where dependencies between data sources are incorrectly ignored.

In decentralized multi-agent systems aiming at some joint mission, such as
autonomous cooperative robot localization [1],[4], [5], or tracking [2], the
requirement for each agent to recursively update and communicate a global full
joint posterior probability distribution function (pdf) over an identical
(homogeneous) set of states leads to large overhead in local processing and
communication bandwidth. It is therefore desirable to enable processing,
communication and fusion of a posterior pdf over a subset of different but
overlapping states; we name such a process _heterogeneous fusion_. Consider
for example the 30 robot cooperative localization scenario given in [1]. If
each agent has 4 unknown position states, the full joint distribution has 120
variables, and requires processing a $120\times 120$ covariance matrix
(assuming Gaussian distribution). This includes states of agents ‘far away’
from each other in the network, which has negligible effect on the local
position estimate and might be considered ‘irrelevant’. But, if each agent
includes in its estimate only immediate ‘relevant’ neighbors’ states which
form a subset of the global joint distribution, then the local heterogeneous
joint distribution shrinks, e.g. to 16 states for a 3-neighbor topology. This
has a clear computation and communication gain over the homogeneous
alternative. However it might lead to indirect correlations between variables
not mutually monitored by both agents and result in an overconfident estimate.
To the best of our knowledge, existing methods have yet to solve this problem
and accurately account for the indirect correlations.

DDF algorithms can be considered exact or approximate, depending on how they
account for dependencies in the data shared between agents, to guarantee that
every piece of data is counted not more than once. In exact methods, these
dependencies are explicitly accounted for, either by pedigree-tracking, which
can be cumbersome and impractical in large ad hoc networks [6], or by adding a
_channel filter_ (CF), which requires enforcing a tree-structured
communication topology [7, 8]. Approximate methods assume different levels of
correlation between the communicating agents and fuse them in such a way that
the common data is promised not to be double counted, thus ensuring
conservativeness of the fused posterior. The most commonly used approximate
method is covariance intersection (CI) [9], when agents share only the first
two moments (mean and covariance) of their distributions, or the geometric
mean density (GMD) for general pdfs fusion [10]. In CI, the fused result is a
weighted average of the information vector and matrix describing the first and
second moments of the underlying distributions, respectively, where the weight
is optimized based on a predetermined cost function, e.g., determinant of the
posterior fused covariance matrix. Both exact and approximate fusion methods
usually assume that the distributions to be fused as well as the posterior
distributions are homogeneous, i.e. describe the same full joint state random
vector. However, these methods cannot be directly applied to heterogeneous
fusion, where pdfs over overlapping parts of the full joint pdfs are fused.

This paper builds upon the work presented in [11] and further develops a
rigorous Bayesian probabilistic approach for fusion of heterogeneous pdfs in
tree-structured networks. The key idea is to utilize conditional independence
properties between subsets of variables (states) in order to lower the
dimension of the local joint distributions to be fused. In [11] the
applicability of the conditionally factorized channel filter (CF2) algorithms
for dynamic systems is limited since the full time history has to be tracked
for conditional independence. This paper improves and extends the theory and
applicability of the work presented in [11] by making the following
contributions:

  1. 1.

The applicability of the CF2 heterogeneous fusion algorithms are extended to
dynamic problems by (i) enabling conservative filtering (ii) developing the
information augmented state (iAS) version of the augmented state algorithm
[12].

  2. 2.

The definition and desiderata for a heterogeneous fusion rule are more
formally and clearly defined, along with practical measures for evaluation.

  3. 3.

The theory developed in this paper is brought to practice with a detailed
explanation of two of the CF2 algorithms and pseudo-code for the case of
Gaussian distributions is presented.

  4. 4.

The scalability of the heterogeneous fusion methods is demonstrated by an
extended example, calculating the communication and computation savings, and
covering different problem sizes.

  5. 5.

New simulations of dynamic decentralized multi-target tracking scenarios are
provided, to demonstrate conservative filtering and compare it to the full
time history smoothing approach.

The rest of this paper consists of three main parts. The first (Sec. II-V)
discusses heterogeneous fusion in terms of general probability distributions
(pdfs); Sec. II defines the heterogeneous decentralized fusion problem and
reviews the state of the art; Sec. III derives a family of exact and
approximate conditionally factorized CF methods; Conditional independence in
dynamic systems, including augmented state (AS) density and conservative
filtering for general pdfs, is discussed in Sec. IV, and Sec. V details the
proposed fusion algorithm. In the second part (Sec. VI) Gaussian distributions
are treated as special case to develop: a closed-form fusion CF2 rule; the
information augmented state (iAS) method; and a conservative Kalman filtering
method. In the third part (Sec. VI-F-VII), numerical examples demonstrate the
potential communication and computation gains of the described methods (Sec.
VI-F) via static and dynamic multi-agent multi-target tracking simulations
(Sec. VII). Sec. VIII then draws conclusions and suggests avenues for
additional work.

##  II Problem Statement and Related Work

To motivate the problem, consider a simple static target tracking problem with
one target and two tracking agents as a running example (Fig. 1a). Both agents
$i$ and $j$ track the position of the target, described by the random state
vector $x$, and are assumed to have perfect knowledge of their own position,
but unknown constant biases in the agent-target relative measurement vectors
$y_{i,k}$ and $y_{j,k}$, described by the random state vectors $s_{i}$ and
$s_{j}$, respectively, where $k$ indicates the time step. The agents can also
take relative observations $m_{i,k}$ and $m_{j,k}$ to different landmarks to
locally collect data on their measurement biases.

As shown in Fig. 1a, the agents’ local biases $s_{i}$ and $s_{j}$ become
coupled due to measurements $y_{i,k}$ and $y_{j,k}$ of the common target $x$.
In the case of homogeneous information fusion, the two agents preform
inference over and communicate the full joint pdfs describing all the random
variables, including each other’s local biases. But in the heterogeneous
fusion case, agents might hold a pdf over only a subset of the random
variables, for example over the target ($x$) and the agent’s local bias (e.g.,
$s_{i}$), making the dependencies between the local biases hidden. Therefore
an agent might not be aware of the existence of another local bias random
state vector (e.g., $s_{j}$) held by another agent. These dependencies are key
to the problem, and the main challenge in heterogeneous fusion compared to
homogeneous fusion is to account for them during fusion, where they are not
explicitly represented in the local posterior pdfs.

(a)$s_{i}$$s_{j}$$x$$y_{i,k}$$y_{j,k}$$m_{i,k}$$m_{j,k}$$k=1:N$$k=1:N$(b)$s_{i}$$s_{j}$$x_{1}$$x_{2}$$x_{k}$$y_{i,1}$$y_{j,1}$$y_{i,2}$$y_{j,2}$$y_{i,k}$$y_{j,k}$$m_{i,k}$$m_{j,k}$$k=1:N$

Figure 1: (a) Static and (b) Partially dynamic Bayesian networks for two local
random vectors $s_{i},s_{j}$ (local measurement biases) and one common random
vector $x$ (target state). In (a), $s_{i}$ and $s_{j}$ are conditionally
independent given the static state $x$; in (b) the full time history $x_{1:k}$
is required for conditional independence.

###  II-A Problem Statement

Assume a network of $n_{a}$ autonomous agents, performing recursive
decentralized Bayesian updates to their prior pdf, with the goal of monitoring
some global random state vector $\chi_{k}$ at time $k$. Each agent
$i\in\\{1,...,n_{a}\\}$ is tasked with a local subset of states
$\chi_{i,k}\subseteq\chi_{k}$, an $n_{i}$-dimensional vector of random
variables at time $k$. An agent can update its local prior pdf for
$\chi_{i,k}$, by (i) using Bayes’ rule to recursively update a posterior pdf
for $\chi_{i,k}$ with independent sensor data $Y_{i,k}$ described by the
conditional likelihood $p(Y_{i,k}|\chi_{i,k})$, and (ii) performing peer-to-
peer fusion of external data relevant to $\chi_{i,k}$ from any neighboring
agent $j\in N_{a}^{i}$, where $N_{a}^{i}$ is the set of agents communicating
with $i$.

The heterogeneous fusion question is now: what is a peer-to-peer fusion rule
$\mathbb{F}$ which given the local prior distribution
$p_{i}(\chi_{i,k}|Z^{-}_{i,k})$ and a distribution over ‘relevant’ random
states from a neighboring agent $j$, $p_{j}(\chi_{j,k}^{r}|Z^{-}_{i,k})$,
returns a local fused posterior combining the data from both agents,
$p_{i}(\chi_{i,k}|Z_{i,k}^{+})$,

$\begin{split}p_{i}(\chi_{i,k}|Z_{i,k}^{+})=\mathbb{F}(p_{i}(\chi_{i,k}|Z^{-}_{i,k}),p_{j}(\chi_{j,k}^{r}|Z^{-}_{j,k})).\end{split}$
(1)

Where $\chi_{j,k}^{r}$ is the subset of random states at agent $j$ that are
relevant to agent $i$ and assumed to be a non-empty set. $Z_{i,k}^{-}$
($Z_{j,k}^{-}$) is the local data agent $i$ ($j$) has prior to fusion with
agent $j$ ($i$), and $Z_{i,k}^{+}\equiv Z_{i,k}^{-}\cup Z_{j,k}^{-}$ is the
combined data after fusion. Notice that while the motivation is heterogeneous
fusion, the above statement is general and can be used for homogeneous and
heterogeneous fusion. For instance, in the target tracking example, if
$\chi_{i,k}=\chi_{j,k}=\chi_{j,k}^{r}=[x^{T},s_{i}^{T},s_{j}^{T}]^{T}$ then
(1) simplifies to a homogeneous fusion problem, which can be solved using
different exact or approximate fusion rules, as discussed in the introduction.
However, if $\chi_{i,k}=[x^{T},s_{i}^{T}]^{T}$ and
$\chi_{j,k}=[x^{T},s_{j}^{T}]^{T}$, then the ‘relevant’ random states in $j$
are $\chi_{j,k}^{r}=x$, and (1) describes heterogeneous fusion. In this case,
note that $i$’s knowledge about $s_{i}$, described by the marginal pdf
$p(s_{i}|Z_{i,k}^{+})$, should still be updated following fusion with respect
to $x$ (likewise for $j$’s marginal pdf over $s_{j}$). This key distinction
separates conventional homogeneous fusion from heterogeneous fusion: in the
latter, agents seek to update their posterior pdf over their entire local
random state vector, using data gained only from fusion over relevant subsets
of local random state vectors. Heterogeneous fusion thus encompasses the set
of problems where the set of relevant random states $\chi_{j,k}^{r}$ is a
subset of either agent $i$’s random states $\chi_{j,k}^{r}\subset\chi_{i,k}$,
or agent $j$’s random states $\chi_{j,k}^{r}\subset\chi_{j,k}$ or both.

The goal of this paper is to develop the theory and understanding of the
heterogeneous fusion problem defined above. To be able to track the
information flow in the system, the following assumptions are made:

  1. 1.

The $n_{a}$ agents all communicate in a bilateral tree-structured network
topology. This guarantees that information can only flow in one path between
any two agents, thus avoiding loops.

  2. 2.

Full rate communication and sequential processing of incoming messages from
other agents (where agent sends a message at time step $k$ before processing
messages received at the same time step $k$).

  3. 3.

The random state vector definitions between neighboring agents in the network
communication graph have at least one overlapping random state in their local
random state vector. Further, if agent $l$ is more than one hop away from
agent $i$ and has an overlapping random state with $i$, then all agents on the
tree path from $i$ to $l$ also share that common random state.

  4. 4.

At time step $k=0$, if there is a common prior distribution over the common
states of interest it is known to both agents, i.e.,

$p_{i}(\chi_{i,0}\cap\chi_{j,0})=p_{j}(\chi_{i,0}\cap\chi_{j,0})=p_{c}(\chi_{i,0}\cap\chi_{j,0})$

where here $p_{c}$ is the common prior pdf. If there is no common prior, then
$p_{c}(\chi_{i,0}\cap\chi_{j,0})=1$.

  5. 5.

The global problem structure (representing the full joint distribution over
$\chi_{k}$) for state estimation at the agents is such that it allows
conditional independence between random states that are local to each agent
given (conditioned on) the common random states between them.

An example for these assumptions is given in Fig. 2. Shown is a 5-agent,
6-target tracking problem, where the local measurements to targets
$x_{t}(t=1,..6)$ and landmarks to estimate local biases $s_{i}(i=1,..5)$ are
denoted by the full black arrows. The tree structured communication topology
is indicated by the dashed red arrows between the agents. It can be seen that
due to the bilateral communication between any two agents there are no loops
in the network and information can flow only in one path. Assumption 3 is also
demonstrated, for example, if agents 3 and 5 both estimate target 5’s random
states, agent 4 must estimate it as well. Finally, the conditional
independence holds, as for example agent 1’s local random state vectors
$x_{1}$ and $s_{1}$ are conditionally independent from agent 2’s local random
state vectors $x_{3}$ and $s_{2}$ given the common target random state vector
$x_{2}$ ($x_{1},s_{1}\perp x_{3},s_{2}|x_{2}$).

Figure 2: Target tracking example. Full black arrows denote local measurements
to targets $x_{t}$ and landmarks to estimate local biases $s_{i}$, red dashed
arrows indicate bi-directional communication channel between agents. The tree
structure topology can be seen, as there are no loops.

While assumption 1-4 can be easily enforced, assuming conditional independence
(assumption 5) is not trivial in dynamic problems. In [11] the full time
history is augmented to maintain conditional independence, but the problem of
regaining conditional independence, while keeping the local pdf consistent and
conservative when filtering (marginalizing out past random states) remains
unsolved. Thus, in addition to a fusion rule $\mathbb{F}$, a complementary
conservative filtering operation is needed to regain conditional independence.

###  II-B What is A Good Fusion Rule?

In the above formulation of the problem it is left to define a good fusion
rule $\mathbb{F}$ in (1) and how to evaluate it. Recently Lubold and Taylor
[13] claim that a fusion rule should provide a posterior pdf which is
conservative, i.e., overestimates the uncertainty relative to the true pdf.
They suggest new definitions for conservativeness, but to the best of our
knowledge it is not widely used. In the context of homogeneous fusion, common
definitions use the terms consistent and conservative interchangeably [9],
[14] and assume that the point estimate uncertainty can be described by mean
and covariance. In the following, the intuition regarding conservativeness
from [13] is combined with the common definitions of consistency from
homogeneous fusion to define a ‘good’ heterogeneous fusion rule $\mathbb{F}$,
firstly in terms of pdfs and then in the case of Bayesian point estimation.

From the standpoint of pdf fusion, a good heterogeneous fusion rule
$\mathbb{F}$ results in an updated local posterior pdf that: (i) overestimates
the uncertainty relative to the true pdf, and (ii) is conservative over the
agent’s random states of interest $\chi_{i,k}\subseteq\chi_{k}$ relative to
the marginal pdf over $\chi_{i,k}$ of a consistent centralized pdf.
Consistency here means that the fused result does not overestimate or
underestimate the uncertainty. The centralized pdf refers to the posterior pdf
over the full random state vector $\chi_{k}$ conditioned on all the available
data from all the agents up and including time step $k$,
$p(\chi_{k}|\bigcup_{i\in N_{a}}Z^{-}_{i,k})$.

Since consistency and conservativeness are often defined by the first two
moments of the pdf, i.e., their mean and covariance, the above definition can
be further narrowed in the context of Bayesian point estimation. A good
heterogeneous fusion rule $\mathbb{F}$ in this case is then one that when
forming a point estimate from its resulting local posterior
$p_{i}(\chi_{i,k}|Z_{i,k}^{+})$, for example by finding the minimum mean
squared error (MMSE) estimate, the estimate: (i) overestimates the uncertainty
relative to the true state error statistics, and (ii) is conservative relative
to the marginal error estimate of a consistent centralized point estimator.

For example, assume the means of the Gaussian random state vectors $\chi_{i}$
and $\chi$ are $\mu_{\chi_{i}}$ and $\mu_{\chi}$, and the covariances,
describing the mean squared error, are
$\Sigma_{\chi_{i}}=E[(\chi_{i}-\mu_{\chi_{i}})(\chi_{i}-\mu_{\chi_{i}})^{T}]$
and $\Sigma_{\chi}=E[(\chi-\mu_{\chi})(\chi-\mu_{\chi})^{T}]$, respectively,
where $E[\cdot]$ is the expectation operator. The actual values are unknown,
and the approximate estimate of them is given by $\bar{\mu}_{\chi_{i}}$,
$\bar{\mu}_{\chi}$, and $\bar{\Sigma}_{\chi_{i}}$, $\bar{\Sigma}_{\chi}$.
The definitions above then translate to the following:

  1. (i)

Overestimation of the uncertainty relative to the true error statistics
implies that $\bar{\Sigma}_{\chi_{i}}-\Sigma_{\chi_{i}}\succeq 0$, i.e., the
resulting matrix difference is positive semi-definite (PSD).

  2. (ii)

Conservativeness relative to the marginal estimate of the centralized
estimator implies that
$\bar{\Sigma}_{\chi_{i}}-\bar{\Sigma}_{\chi_{i}}^{cent}\succeq 0$, where
$\bar{\Sigma}_{\chi_{i}}^{cent}$ is the marginal covariance over $\chi_{i}$,
taken from the joint centralized covariance over $\chi$.

The centralized estimate in this case can be considered consistent if, for
example, it passes the NEES hypothesis test [15]. Note that since a consistent
centralized estimate neither overestimates nor underestimates the uncertainty,
a conservative (higher uncertainty) local estimate is expected to
underestimate the uncertainty. Thus requiring the local estimate to be
conservative relative to a consistent centralized estimate implicitly requires
it to not overestimate the uncertainty of the true error statistics.

###  II-C Related Work

The idea of splitting a global state vector into $N$ subsets of $N_{a}$
vectors ($N_{a}\ll N$) has been posed to solve two different problems. The
first problem, as presented for example in [16, 17, 18, 19], tries to
reconstruct an MMSE _point estimate_ of the global state vector, by fusing
$N_{a}$ local heterogeneous _point estimates_. The second problem, which is
the focus of this paper, tries to find $N_{a}$ different local _posterior
pdfs_ , given all the information locally available up to the current time.

The problem of locally fusing data from a non-equal state vectors dates back
to [20]. While this work offers a simple way for distributing into
heterogeneous local state vectors, it assumes that the local pdfs are
decoupled. In [21], Khan _et al._ relax that assumption, but ignore
dependencies between states in the fusion step and restrict the state
distribution to only states that are directly sensed by local sensors, where
in practice, an agent might be ‘interested’ in a larger set of local states
that are not all directly sensed. Similarly in cooperative localization [4],
[5] it is often assumed that agents are neighbors only if they take relative
measurement to each-other. Then, the local state is augmented with the other
agent’s position states to process the measurement, often by assuming or
approximating marginal independence. It can be shown that this approach
represents a subset of the heterogeneous fusion problems considered in the
paper.

In [22], Chong and Mori use conditional independence in Bayesian networks to
reduce the subset of states to be communicated. However, they assume
hierarchical fusion and for dynamic systems only consider the case of
deterministic state processes, omitting the important case of stochastic
processes which is considered in this paper. The work by Whitacre _et al._[2],
does not formally discuss conditional independence but introduces the idea of
marginalizing over common states to fuse Gaussian distributions, when cross
correlations of common states are known. However, they implicitly assume cross
correlations among the conditioned, or unique, states are small. While this
assumption might hold for their application, it does not offer a robust
solution. Reference [23] uses CI with Gaussian distributions to relax the
assumption of known correlations of the fused state. Reference [24] suggests a
similar solution to [23] but restrict it to systems where all agents have the
same set of common states. Although scalable and simple, such approaches do
not account for dependencies between locally exclusive states.

This work aims at gaining insight and understanding on how such issues should
be addressed by exploiting the structure of the underlying dynamical system in
such problems.

##  III Conditionally Factorized Channel Filters - CF2

The approach presented here assumes a tree-structured communication network
and starts with a general probabilistic channel filter (CF) formulation. Then,
as seen in Fig. 3, different fusion problems of interest are discussed.
Starting from the original homogeneous CF, where all agents keep and share
posterior pdf of the full global joint random state vector, cases are then
considered where an agent is only interested in and/or observes a subset of
the global joint distribution. It is shown that by leveraging conditional
independence, agents can communicate only marginal pdfs, leading to the
_Factorized CF (F-CF)_ and the _Bi-directional factorized CF (BDF-CF)_ ,
depending on whether marginal pdf is sent in one or two directions,
respectively. These extensions to the CF enable communication reduction by
sending only new and relevant data while maintaining the accuracy of the
original CF.

Additionally, a branch of approximate CF methods is introduced which includes
the _approximate BDF-CF_ and the _heterogeneous state CF (HS-CF)_. In these
methods, agents communicate only marginal pdf over common subsets, leaving the
conditional pdfs over the unique subset of variables local at each agent. The
main difference between these two methods is the size of the local joint
distributions, which influences the local processing requirements. In
approximate BDF-CF, each agent processes its unobserved variables in order to
get a rough estimate over less relevant random states. However, in the HS-CF,
the state space for the local joint distribution is minimized to hold only
locally relevant variables, which significantly reduces computation as shown
in Sec. VI-F.

Figure 3: Different CFs derived in this paper, where the first three blocks
describe exact fusion and the dashed blocks approximate, with local relevant
states $s=[s_{i}^{T},s_{j}^{T}]^{T}$ and common states of interest $x$ for
fusion shown.

###  III-A Decentralized Fusion and The Homogeneous CF

Let the full random state vector $\chi$ be defined as

$\chi=\begin{bmatrix}X\\\ S\\\ \end{bmatrix}\ \ \
\in\mathbb{R}^{(n_{X}+n_{S})\times 1},$ (2)

where $X$ and $S$ are vectors with $n_{X}$ and $n_{S}$ elements, respectively.
In the target tracking example (Fig. 2),
$X=[x_{1}^{T},x_{2}^{T},...,x_{n_{t}}^{T}]^{T}$ represent the random state
vectors of $n_{t}=6$ targets and $S=[s_{1}^{T},s_{2}^{T},...,s_{n_{a}}]^{T}$
represent the unknown random bias state vectors of the $n_{a}=5$ tracking
agents. For now it is assumed that the random state vector $\chi$ is static
and the time index notation is dropped (dynamic states will be revisited and
considered later). The state vector $\chi$ is a multivariate random variable
and can be described by the joint pdf, $p(\chi)=p(X,S)$.

The goal is to find the fused estimated underlying distribution of the state
$\chi$ given the joint data $Z^{+}_{k}=Z^{-}_{i,k}\cup Z^{-}_{j,k}$ at agents
$i$ and $j$. Using a distributed variant of Bayes’ rule, [8] shows that the
exact posterior pdf conditioned on joint data of $i$ and $j$ is given by:

$p_{f}(\chi|Z_{k}^{+})=\frac{1}{C}\cdot\frac{p_{i}(\chi|Z_{i,k}^{-})p_{j}(\chi|Z_{j,k}^{-})}{p_{c}(\chi|Z_{i,k}^{-}\cap
Z_{j,k}^{-})},$ (3)

where $C$ is a normalizing constant and $p_{c}$ is the posterior pdf
conditioned on the common data shared by agents $i$ and $j$ prior to the
current fusion.

In [7], Grime and Durrant-Whyte suggest that each agent add a filter on the
communication channel (hence channel filter) with a neighboring agent in a
tree network. The CF explicitly tracks $p_{c}$ between the two agents, thus
allowing exact removal of the common data to avoid double counting. While this
operation leads to exact fusion, i.e., equal to a centralized fusion center
(assuming full rate communication), the original CF formulation assumes that
both agents hold distributions over the full random state vector $\chi$. Thus
communication and local processing (inference) is with respect to (w.r.t.) the
full joint pdf $p(\chi)$. This approach is not scalable for large state-
spaces, which motivates the extension of the original CF into a family of
heterogeneous fusion methods that ‘break’ the joint pdf into smaller parts.
Intuitively, if agent $i$ only ‘cares’ about a subset $(x,s_{i})$ and $j$ only
‘cares’ about $(x,s_{j})$, we would like to enable communication of data only
regarding the common target $x$ as in the case of the heterogeneous state CF
(Fig. 3, right hand block). In the following, the structure of the underlying
estimation problem is exploited to conditionally factorize into relevant
subsets of the global random state vector, thereby enabling reduced
communication costs and allowing each agent to locally hold a smaller pdf,
i.e., reducing the computational cost of inference.

###  III-B Exact Factorized CF

From the law of total probability, the joint pdf over $\chi$ can be
conditionally factorized as

$p(\chi)=p(X)\cdot p(S|X)=p(S)\cdot p(X|S).$ (4)

Using this factorization and taking $X=x$, (3) can be expressed as

$\begin{split}&p_{f}(\chi|Z_{k}^{+})=\\\
&\frac{1}{C}\cdot\frac{p_{i}(x|Z_{i,k}^{-})p_{j}(x|Z_{j,k}^{-})}{p_{c}(x|Z_{i,k}^{-}\cap
Z_{j,k}^{-})}\cdot\frac{p_{i}(S|x,Z_{i,k}^{-})p_{j}(S|x,Z_{j,k}^{-})}{p_{c}(S|x,Z_{i,k}^{-}\cap
Z_{j,k}^{-})}.\end{split}$ (5)

In the following sections, this factorization of the fusion equation and
insights regarding conditional independence are used to derive new CF
variants. These differ in their assumptions and result in different
communication and local computation benefits.

####  III-B1 Factorized Channel Filter (F-CF)

First, a set of problems is considered where both agents take measurements to
a common random state vector $x$, but only one of them, for example agent $i$,
takes measurements to collect data on a local random state vector $S=s_{i}$.
It is further assumed that agent $j$ does not gain any data regarding $s_{i}$
from any other communicating agent. Thus $s_{i}$ and the data in $j$ that is
not available to agent $i$, noted by $Z_{j\setminus i,k}^{-}$, are
conditionally independent given $x$ ($s_{i}\perp Z_{j\setminus i,k}^{-}|x$).
This leads to the following important conclusion:

$p_{j}(s_{i}|x,Z_{j,k}^{-})=p_{c}(s_{i}|x,Z_{i,k}^{-}\cap
Z_{j,k}^{-})=p_{f}(s_{i}|x,Z_{k-1}^{+}),$ (6)

which means that the data agent $j$ has regarding $s_{i}$ is equal to the
common data with $i$, and is exactly the data it had available after the
previous fusion step ($k-1$). Eq. (5) can therefore be written as,

$\begin{split}p_{f}(\chi|Z_{k}^{+})=\frac{1}{C}\cdot\frac{p_{i}(x|Z_{i,k}^{-})p_{j}(x|Z_{j,k}^{-})}{p_{c}(x|Z_{i,k}^{-}\cap
Z_{j,k}^{-})}\cdot p_{i}(s_{i}|x,Z_{i,k}^{-}).\end{split}$ (7)

The fusion equation above has the following intuitive interpretation: if agent
$j$ does not gain any new local data regarding state $s_{i}$ (conditioned on
$x$), it should only communicate the marginal pdf $p_{j}(x|Z_{j,k}^{-})$,
describing the common random state vector $x$, to agent $i$.

This variant of the CF is dubbed _Factorized CF (F-CF)_ , since the
distribution is factorized into two contributions: 1) marginal pdf regarding
$x$, based on data gathered and shared by the two agents, and 2) conditional
pdf regarding the local random state vector $s_{i}$ based only on data from
agent $i$. The transition from the original CF to the factorized CF is shown
in Fig. 3, and is enabled by using the conditional independence of the problem
to separate common and local data. This fusion rule provides a possible
$\mathbb{F}$ sought in (1): here, the communicated distributions are not over
the same set of random states, as $i$ sends the joint distribution
$p_{i}(\chi|Z^{-}_{i,k})$ and $j$ sends only the marginal distribution
$p_{j}(x|Z^{-}_{j,k})$. Note that while the posterior fused pdf is the same as
can be achieved by the homogeneous fusion rule (3), the math behind this
heterogeneous fusion rule is fundamentally different. In the latter, the
conditional pdf $p_{i}(s_{i}|x,Z_{i,k}^{-})$, which is kept local at agent $i$
or is simply replaced in agent $j$, treats the common random state vector $x$
as a function parameter. Then, the fused marginal is recombined with the local
(or replaced) state conditional pdf in the joint distribution via the law of
total probability (4) at each agent. This weights the conditional pdf
differently as a function of $x$ and changes the overall joint pdf, thus
implicitly/indirectly updating $s_{i}$.

####  III-B2 Bi-directional Channel Filter (BDF-CF)

Consider now the case where both agents have different local random state
vectors $s_{i}$ and $s_{j}$ ($S=[s_{i}^{T},s_{j}^{T}]^{T}$). The global random
state vector is then $\chi=[x^{T},s_{i}^{T},s_{j}^{T}]^{T}$, and $\chi\sim
p(\chi)=p(x,s_{i},s_{j})$. If the local random state vectors are independent
given the common random state vector ($s_{i}\perp s_{j}|x$), then the
following holds:

$p(\chi)=p(x)\cdot p(s_{i}|x)\cdot p(s_{j}|x).$ (8)

Using the above factorization (5) can be split further,

$\begin{split}&p_{f}(\chi|Z^{+})=\frac{1}{C}\cdot\frac{p_{i}(x|Z_{i}^{-})p_{j}(x|Z_{j}^{-})}{p_{c}(x|Z_{i}^{-}\cap
Z_{j}^{-})}\cdot\\\
&\frac{p_{i}(s_{i}|x,Z_{i}^{-})p_{j}(s_{i}|x,Z_{j}^{-})}{p_{c}(s_{i}|x,Z_{i}^{-}\cap
Z_{j}^{-})}\cdot\frac{p_{i}(s_{j}|x,Z_{i}^{-})p_{j}(s_{j}|x,Z_{j}^{-})}{p_{c}(s_{j}|x,Z_{i}^{-}\cap
Z_{j}^{-})}.\end{split}$ (9)

As in the F-CF, assume that $s_{i}\perp Z_{j\setminus i,k}^{-}|x$ and
$s_{j}\perp Z_{i\setminus j,k}^{-}|x$, so (9) can be simplified:

$\begin{split}&p_{f}(\chi|Z^{+})=\\\
&\frac{1}{C}\cdot\frac{p_{i}(x|Z_{i}^{-})p_{j}(x|Z_{j}^{-})}{p_{c}(x|Z_{i}^{-}\cap
Z_{j}^{-})}\cdot p_{i}(s_{i}|x,Z_{i}^{-})\cdot
p_{j}(s_{j}|x,Z_{j}^{-}).\end{split}$ (10)

This CF variant is dubbed the _Bi-directional factorized CF (BDF-CF)_. The
symmetry between the two agents can be seen: the agents share marginal pdfs
regarding the random state vector $x$, and then each agent shares its unique
conditional pdf regarding the local states $s_{i}$ or $s_{j}$. As in the F-CF,
this gives another fusion rule $\mathbb{F}$, only now $i$ sends its marginal
pdf $p_{i}(x,s_{i}|Z_{i}^{-})=p_{i}(x|Z_{i}^{-})\cdot
p_{i}(s_{i}|x,Z_{i}^{-})$ and $j$ sends its marginal pdf
$p_{j}(x,s_{j}|Z_{j}^{-})$. The posterior fused pdfs are equal,
$p_{i,f}(x,s_{i},s_{j}|Z^{+})=p_{j,f}(x,s_{i},s_{j}|Z^{+})$.

Both the F-CF and the BDF-CF are a mathematically equivalent versions of the
original CF (for static systems). Their advantage though, is the considerable
reduction in communication requirements achieved by sending only new and
relevant data (given that the above assumptions are met). While the BDF-CF
(and F-CF) has the potential of saving communication costs, as shown later in
section VI-F, it still requires that every agent holds a local pdf over the
full global random state vector and communicates the conditional pdfs
regarding its local random states. By sacrificing part of the ‘exactness’ of
the CF over less relevant random states, significant reductions in both
computation and communication requirements can be gained. This leads to a new
family of approximate CF methods.

###  III-C Approximate Factorized CF

As noted above, mathematical equivalence to the original CF is achieved under
the assumption that agents do not locally gain data regarding each other’s
local states. This might still require considerable communication volume, for
example, when $s_{i}$ are agent $i$’s local states for a navigation filter,
which typically has 16 or more states [25]. In the following approximate
heterogeneous fusion algorithms, computation is further reduced by only
sending the marginal pdfs regarding common random states.

####  III-C1 Approximate BDF-CF

Consider the case where agents prioritize their random states of interest
(e.g., $[x^{T},s_{i}^{T}]$ for agent $i$) over random states local to other
agents (e.g. $s_{j}$), i.e., it is more important that their local pdf is
accurate in portions of the states of interest than in portions relevant to
other agents. The approximate fusion rule at $i$ can then be written as

$\begin{split}p_{i,f}&(\chi|Z_{i}^{+})=\\\
&\frac{1}{C}\cdot\frac{p_{i}(x|Z_{i}^{-})p_{j}(x|Z_{j}^{-})}{p_{c}(x|Z_{i}^{-}\cap
Z_{j}^{-})}\cdot p_{i}(s_{i}|x,Z_{i}^{-})\cdot p_{i}(s_{j}|x).\end{split}$
(11)

A similar expression can be written for agent $j$ by switching $i$ with $j$
for the conditional pdfs over $s_{i}$ and $s_{j}$. The terms
$p_{i}(s_{i}|x,Z_{i}^{-})$ and $p_{i}(s_{j}|x)$ represent the local
conditional distributions agent $i$ holds regarding the states not mutually
monitored by $i$ and $j$. Notice that conditioned on $x$, the data local to
agent $i$ is independent from $s_{j}$, i.e., agent $i$ doesn’t gain any data
directly influencing agent $j$’s local random state vector $s_{j}$. This
version of the heterogeneous CF is named the _Approximate BDF-CF_ , as it
approximates the posterior fused pdf that can be achieved by receiving the
conditional pdf $p_{j}(s_{j}|x,Z_{j}^{-})$ from agent $j$, as done in the BDF-
CF (10). The function formulation for $\mathbb{F}$ still holds, since this
fusion rule leads to different fused posterior results for agents $i$ and $j$.

Here the fundamental mathematical difference from homogeneous fusion is
highlighted again. In this heterogeneous fusion rule, fusion is over the
common random state vector alone, thus the local pdf is directly updated only
by the marginal pdf and this marginal fused pdf is the same at both agents.
However, when agent $i$ ($j$) merges back the updated marginal pdf into its
local joint pdf via the law of total probability, the conditional
probabilities over $s_{i}$ and $s_{j}$, which depend on local and non-equal
sets of data ($Z_{i}^{-}\neq Z_{j}^{-}$), will scale each of agent’s joint pdf
differently, leading to non-equal joint pdfs at agents $i$ and $j$.

####  III-C2 Heterogeneous State Channel Filter (HS-CF)

So far it has been assumed that all agents across the network hold a local pdf
over the same global random state vector $\chi$, which includes all locally
relevant random states. Thus, as the number of tasks in the network increases
so does the local computation load. However, if each agent to only hold a pdf
over its locally relevant subsets of states, the local computation then scales
with the agent tasks and not the global network tasks (or number of agents).
This motivates the last CF variant, the _Heterogeneous state CF (HS-CF)_.
Here, each agent holds its own pdf over heterogeneous subsets of random
states, e.g., $\chi_{i}=\begin{bmatrix}x^{T},s_{i}^{T}\end{bmatrix}^{T}$ and
$\chi_{j}=\begin{bmatrix}x^{T},s_{j}^{T}\end{bmatrix}^{T}$ for agents $i$ and
$j$, respectively. The fusion rule for each agent, over their locally relevant
random states, can be written as

$\begin{split}&p_{i,f}(\chi_{i}|Z_{i}^{+})=\frac{1}{C_{i}}\cdot\frac{p_{i}(x|Z_{i}^{-})p_{j}(x|Z_{j}^{-})}{p_{c}(x|Z_{i}^{-}\cap
Z_{j}^{-})}\cdot p_{i}(s_{i}|x,Z_{i}^{-})\\\
&p_{j,f}(\chi_{j}|Z_{j}^{+})=\frac{1}{C_{j}}\cdot\frac{p_{i}(x|Z_{i}^{-})p_{j}(x|Z_{j}^{-})}{p_{c}(x|Z_{i}^{-}\cap
Z_{j}^{-})}\cdot p_{j}(s_{j}|x,Z_{j}^{-}).\end{split}$ (12)

HS-CF fusion gives another fusion rule $\mathbb{F}$ for the problem statement
in (1), where agents fuse marginal pdfs regarding the common random state
vector $x$, and then update the joint pdf by merging the fused marginal with
the local relevant conditional pdf, as in the Approximate BDF-CF. Notice that
while here, as opposed to all previous heterogeneous fusion rules developed
above, the two pdfs are over different sets of random states, however, the
marginal pdfs over the common random state vector $x$ held by both agents will
still be equal, i.e, $p_{i,f}(x)=p_{j,f}(x)$.

###  III-D Log-Likelihood Representation

In multi-agent data fusion problems it is common to use the log of the pdf
instead the pdf itself. In addition to the connection that these log-
likelihoods have with formal definitions of quantities of information (e.g.,
Shanon information, Fisher information), their advantages for data fusion is
twofold: first, fusion is done by summation and subtraction instead of
multiplication and division. Second, when possible, it allows for the exposure
of the sufficient statistics, describing the pdf, e.g., the mean and
covariance for Gaussian distributions, as detailed in Sec. VI.

In the following, the fusion equations for the BDF-CF and HS-CF are expressed
in the log-likelihood representation, where the rest of the fusion rules can
be similarly translated and are omitted for brevity.

Taking natural logarithm of equation (10) and combining local log-likelihoods
via the law of total probability, the BDF-CF can be written as

$\begin{split}\log[p_{f}(\chi|Z^{+})]&=\log[p_{i}(x,s_{i}|Z^{-}_{i})]+\log[p_{j}(x,s_{j}|Z^{-}_{j})]-\\\
&\log[p_{c}(x|Z^{-}_{i}\cap Z_{j}^{-})]+\tilde{C}.\end{split}$ (13)

This representation demonstrates the intuitive interpretation of the BDF-CF
fusion rule. Here the fused posterior data $Z^{+}$ is built from the summation
of the contributions of local data available at each agent as it affects the
relevant subset of states, minus the contribution of the data they have in
common, where the latter only directly contributes to the marginal of the
common target $x$.

Similarly, the HS-CF fusion rule (12) representation in log-likelihood is
given by,

$\begin{split}\log[p_{i,f}&(\chi_{i}|Z^{+}_{i})]=\log[p_{i}(x,s_{i}|Z^{-}_{i})]+\\\
&\log[p_{j}(x|Z^{-}_{j})]-\log[p_{c}(x|Z^{-}_{i}\cap
Z_{j}^{-})]+\tilde{C}_{i}\\\
\log[p_{j,f}&(\chi_{j}|Z^{+}_{j})]=\log[p_{j}(x,s_{j}|Z^{-}_{j})]+\\\
&\log[p_{i}(x|Z^{-}_{i})]-\log[p_{c}(x|Z^{-}_{i}\cap
Z_{j}^{-})]+\tilde{C}_{j}.\\\ \end{split}$ (14)

In the rest of this paper, this representation is used to discuss the fusion
algorithm for general pdfs (Sec. V) and to derive a closed form fusion rule,
based on the sufficient statistics, for the special case where the underlying
distributions are Gaussian (Sec. VI).

##  IV Conditional Independence in Dynamic Systems

For a dynamic or partially-dynamic Bayesian network, as in Fig. 1(b), it is
generally not possible to claim conditional independence of local states (or
local data) based on the filtered dynamic state, $s_{i}\not\perp s_{j}|x_{k}$,
i.e., $s_{i}$ and $s_{j}$ are not conditionally independent given $x_{k}$ when
it is successively marginalized over time. Since this conditional independence
is required to allow heterogeneous fusion, there is a need to regain
conditional independence in dynamic stochastic systems. There are two
approaches to solve this problem: (i) by keeping a distribution over the full
time history $p(x_{k:0},s_{i},s_{j}|Z_{k})$, where $x_{k:0}$ denotes all
common dynamic states from $k=0$ until current time step $k$; (ii) by
enforcing conditional independence after marginalization by disconnecting the
dependencies between the relevant random states. In the following, these
solutions for the case of general pdfs are discussed. Then, Sec. VI derives
specific closed-form representations for Gaussian distributions.

###  IV-A Augmented State

The distribution over the full augmented state
$\chi_{k:0}=[x_{k:0}^{T},s_{i}^{T},s_{j}^{T}]^{T}$, given the data $Z_{k}^{-}$
can be recursively updated using the following formula [26]:

$p(\chi_{k:0}|Z_{k}^{-})=\frac{1}{C}\cdot
p(\chi_{k-1:0}|Z_{k-1}^{+})p(x_{k}|x_{k-1})p(Y_{k}|\chi_{k}),$ (15)

where here $Z_{k-1}^{+}$ is used to indicate an agent’s data after the
previous fusion step, $k-1$, $Y_{k}$ for the local sensor data gained at the
current time step $k$, and $Z_{k}^{-}$ is the data at time step $k$ prior to
fusion.

The augmented state approach leads to increase in the communication and
computation requirements as the size of the state vector $\chi_{k:0}$
increases. However, agents need to only send messages on the time window the
size of the network (as information propagates in the tree), which bounds the
communication requirement. Furthermore, due to the Markovian property of the
dynamic system, algorithms for efficient inference that factorize the
distribution by taking advantage of its structure can be designed. For
example, for Gaussian distributions, the information matrix structure is close
to block-diagonal (see Sec. VI-C), i.e., elimination can be done efficiently.

###  IV-B Conservative Filtering

Full knowledge over past system states has been assumed thus far, which
enables conditional independence between two agents local states and the
derivation of a family of heterogeneous fusion algorithms named CF2. However,
in many distributed fusion applications it is desirable to maintain only a
limited time window of recent state history. Thus, marginalizing out past
states into a smaller sliding window of recent time steps might be favored, as
maintaining the full accumulated state densities results in rapid state
dimension growth and yields computation and communication burden. While
marginalization is rather trivial for homogeneous fusion problems, in
heterogeneous fusion extra care must be taken to maintain conditional
independence. Without loss of generality, for the rest of the paper, a small
sliding window of only the current time step (as done in Kalman-filter (KF)
for example) will be used.

The main assumption is that local bias states $s_{i},s_{j}$ are conditionally
independent given the target states $x_{k:1}$. Since recursive marginalization
of the past target state $x_{k-1}$ results a coupling between $s_{i}$ and
$s_{j}$ (Fig. 1(b)), it is desirable to enforce conditional independence after
marginalization. A similar principle is also known in the graphSLAM literature
as conservative sparsification [27, 28], where Gaussian distributions are
discussed.

Here, the problem for general distributions is first described in a more
formal way, with consideration of an open research question regarding the term
‘conservative’ for general pdfs. Then, Sec. (VI-D) focuses on Gaussian
distributions and details the solution for conservative sparse marginalization
to enable conservative filtering.

Given a joint distribution $p(x_{1},x_{2},s_{i},s_{j})$, described by the PGM
of Fig. 1(b), marginalizing out $x_{1}$, as done in filtering, results in
coupling of all the variables in its Markov blanket, $x_{2},s_{i}$ and
$s_{j}$. Since conditional independence between $s_{i}$ and $s_{j}$ is a
fundamental assumption in the basis of the proposed methods, it is necessary
to retain it after marginalization. Thus, the goal is to approximate the dense
distribution $p(x_{2},s_{i},s_{j})$ by a sparse distribution such that

$\tilde{p}(x_{2},s_{i},s_{j})=\frac{1}{C}\cdot
p(x_{2})p(s_{i}|x_{2})p(s_{j}|x_{2}).$ (16)

For the pdf to be consistent, the approximation $\tilde{p}(x_{2},s_{i},s_{j})$
has to be conservative w.r.t. $p(x_{2},s_{i},s_{j})$, which, loosely speaking,
means the approximate distribution $\tilde{p}$ overestimates the uncertainty
of the true distribution $p$. See Sec. II-B for a more detailed discussion and
definitions of consistency and conservativeness as treated in this paper.

##  V Fusion Algorithm

Decentralized fusion algorithms, in general, are built out of two main steps:
sending out a message and fusion of incoming message. In the original
(homogeneous) channel filter (CF) algorithm [7], a message contains the
sufficient statistics of the Gaussian distribution, and fusion is done simply
by adding the received information and subtracting the common information
(here ‘information’ is used to mean information vector and information
matrix), where both actions are over the same full state vector $\chi$. In the
heterogeneous CF2, on the other hand, either the communicated or the local
distributions are over different random state vectors. Thus there is a need to
clarify how to locally construct and fuse messages to or from neighboring
agents, respectively.

First, different sets of random states are defined, allowing the separation of
the full global random state vector $\chi$ into smaller subsets as can be seen
in Fig. 4. Recall that $\chi_{i}$ ($\chi_{j}$) was previously defined as the
subset of locally relevant random states at agent _i_ (_j_), and let
$\chi_{c}^{ij}=\chi_{c}^{ji}$ be the set of common random states between
agents _i_ and _j_. Now, in some cases, as in the BDF-CF, agents might pass
through pdfs regarding random states in their random state vector, where the
data corresponding to that those random states were not gained by a local
observation, but from communication with a different neighboring agent. For
example, in the target tracking example of Fig. 2, agent 2 will passes a pdf
over target 1, $p(x_{1}|Z_{1})$, based on data collected by agent 1 and
communicated to agent 2 in previous fusion step, rather by taking a direct
local sensor measurement to the target. These sets of states are dubbed
‘passed through’ states and defined as $\chi_{\neg i}^{ij}$ for states that
are to be passed from _i_ to _j_ but are not local to _i_ and similarly
$\chi_{\neg j}^{ji}$. With these definitions the following holds,

$\chi=\chi_{i}+\chi_{j}-\chi_{c}^{ij}+\chi_{\neg i}^{ij}+\chi_{\neg j}^{ji}.$

It is now possible to discuss the content of messages and the expression for
local fusion for the BDF-CF and the HS-CF.

Figure 4: Diagram presenting the division of the full random state vector
$\chi$ into smaller subsets.

In the BDF-CF, an agent _i_ holds a posterior distribution over the full
global random state vector $\chi$. Assuming tree communication topology for
the network of agents, agent $i$ needs to communicate to its neighboring agent
_j_ a distribution over the set of local states $\chi_{i}$ and the “pass
through” set $\chi_{\neg i}^{ij}$. On the other hand, in the HS-CF, agent _i_
’s local distribution is only over the set of local relevant states
$\chi_{i}$. Agent _i_ thus sends agent _j_ the following distributions,

$\begin{split}&\text{BDF-CF:}\ \ p_{i}^{ij}(\chi_{i}\cup\chi_{\neg i}^{ij})\ \
\ \forall j\in N^{i}_{a}\\\ &\text{HS-CF:}\ \ p_{i}^{ij}(\chi_{c}^{ij})\ \ \ \
\ \ \ \ \ \ \ \forall j\in N^{i}_{a},\end{split}$ (17)

where $N^{i}_{a}$ is the set of all agents in _i_ ’s neighborhood.

The local fusion equations requires summing (and subtracting) log-likelihoods
over different sets of random states, depending on the sending agents, as each
pair of agents, in general, have different sets of random states in common, as
can be seen bellow:

$\begin{split}\text{BDF-CF:}\ \ \ \\\
\log[p_{i,f}(\chi)]&=\log[p_{i}(\chi_{i})]+\\\ &\sum_{j\in
N^{i}_{a}}\log[p_{j}^{ji}(\chi_{j}\cup\chi_{\neg
j}^{ji})]-\log[p_{c}^{ji}(\chi_{c}^{ji})]\\\ \text{HS-CF:}\ \ \ \ \\\
\log[p_{i,f}(\chi_{i}&)]=\log[p_{i}(\chi_{i})]+\\\ &\sum_{j\in
N^{i}_{a}}\log[p_{j}^{ji}(\chi_{c}^{ji})]-\log[p_{c}^{ji}(\chi_{c}^{ji})].\end{split}$
(18)

Notice that for a two agent, one common target problem, with the definitions:

$\begin{split}&\chi=[x^{T},s_{i}^{T},s_{j}^{T}]^{T},\ \ \
\chi_{i}=[x^{T},s_{i}^{T}]^{T},\\\ &\chi_{j}=[x^{T},s_{j}^{T}]^{T},\ \ \ \ \ \
\chi_{c}^{ji}=x,\ \ \ \chi_{\neg j}^{ji}=\varnothing\end{split}$

using the equations (17) and (18) result in the BDF-CF and HS-CF equations
given in (13) and (14), respectively. An example of the algorithm for a linear
Gaussian system, where agents communicate and fuse sufficient statistics
(information vector and information matrix) is given in the next section.

##  VI Gaussians - A Closed Form Fusion Rule

The goal here is to derive a closed-form fusion rule for the special case of
linear-Gaussian distributions to demonstrate how it works in practice and get
insight into the structure of the heterogeneous fusion problem. The
information (canonical) form of the Gaussian distribution will be used to this
end, as it is particularly convenient for deriving and describing key steps in
data fusion processing, for example in the information filter, the original CF
[7], the CI algorithm [9], and more. The use of the Gaussian information form
has two main advantages: (i) multiplication and division over pdfs become
summation and subtraction of the sufficient statistics (information vector and
information matrix) obtained from the log pdfs; and (ii) it gives insight into
the conditional independence structure of the problem, since zero off-diagonal
terms in the resulting information matrices indicate conditional independence
between corresponding random states, as discussed further and utilized for
conservative filtering in Sec. VI-D.

###  VI-A Preliminaries

Assume the full joint distribution over the random state vector $\chi$,
defined in (2), is a multivariate Gaussian with mean $\mu$ and covariance
matrix $\Sigma$,

$\mu=\begin{pmatrix}\mu_{X}\\\ \mu_{S}\\\ \end{pmatrix},\ \ \ \ \ \ \
\Sigma=\begin{pmatrix}\Sigma_{XX}&\Sigma_{XS}\\\
\Sigma_{SX}&\Sigma_{SS}\end{pmatrix}\\\ $ (19)

where $X$ and $S$ denote two correlated subsets of the joint random state
$\chi$. The information form of $\chi$ is given by

$\zeta=\Sigma^{-1}\mu=\begin{pmatrix}\zeta_{X}\\\ \zeta_{S}\\\ \end{pmatrix},\
\ \ \ \ \ \ \Lambda=\Sigma^{-1}=\begin{pmatrix}\Lambda_{XX}&\Lambda_{XS}\\\
\Lambda_{SX}&\Lambda_{SS}\end{pmatrix}\\\ $ (20)

The pdf in information form for the normally distributed state $\chi$, with
information vector $\zeta$ and information matrix $\Lambda$ is [29]:

$p(\chi;\zeta,\Lambda)=\frac{\exp(-\frac{1}{2}\zeta^{T}\Lambda^{-1}\zeta)}{\det(2\pi\Lambda^{-1})^{\frac{1}{2}}}\exp\big{(}{-\frac{1}{2}\chi^{T}\Lambda\chi+\zeta^{T}\chi\big{)}}.$
(21)

This pdf can also be expressed using factorization (4), where the marginal and
conditional distributions of a Gaussian are also Gaussian,

$\begin{split}p(X)&=\mathcal{N}^{-1}(X;\bar{\zeta}_{X},\bar{\Lambda}_{XX})\\\
p(S|X)&=\mathcal{N}^{-1}(S;\zeta_{S|X},\Lambda_{S|X})\end{split}$ (22)

where $\mathcal{N}^{-1}$ represents the information form of the Gaussian
distribution $\mathcal{N}$, and $(\bar{\zeta}_{X},\bar{\Lambda}_{XX})$ and
$(\zeta_{S|X},\Lambda_{S|X})$ are the sufficient statistics for the marginal
and conditional pdfs in information form, respectively, defined as [30]:

$\begin{split}\bar{\zeta}_{X}=\zeta_{X}-\Lambda_{XS}\Lambda^{-1}_{SS}\zeta_{S}&,\
\ \ \
\bar{\Lambda}_{XX}=\Lambda_{XX}-\Lambda_{XS}\Lambda^{-1}_{SS}\Lambda_{SX}\\\
\zeta_{S|X}=\zeta_{S}-\Lambda_{SX}X&,\ \ \ \
\Lambda_{S|X}=\Lambda_{SS}\end{split}$ (23)

###  VI-B Fusion

Starting with the original homogeneous fusion rule (3), by substituting linear
Gaussian distributions, taking logs and differentiating once for the fused
information vector ($\zeta^{f}=(\Sigma^{f})^{-1}\mu^{f}$) and twice for the
fused information matrix ($\Lambda^{f}=(\Sigma^{f})^{-1}$), [8] to obtain the
fusion equations,

$\begin{split}\zeta^{f}=\zeta^{i}+\zeta^{j}-\zeta^{c}\ ,\ \ \ \ \
\Lambda^{f}=\Lambda^{i}+\Lambda^{j}-\Lambda^{c}.\end{split}$ (24)

These equations are the basis of the original linear-Gaussian CF [7], which
explicitly tracks the ‘common information’ vector and matrix
$(\zeta_{c},\Lambda_{c})$, describing the pdf over $\chi$ conditioned on the
common data between communicating pairs of agents $i$ and $j$ ($Z^{-}_{i}\cap
Z^{-}_{j}$) in tree-structured communication networks.

Define $\bar{\zeta}^{f}_{X}$ and $\bar{\Lambda}^{f}_{XX}$ to be the _fused_
marginal information vector and matrix, respectively, over the common target
random state $X$ corresponding to $p_{f}(X)$, represented in information form.
Without loss of generality, the fused marginal information vector and matrix
can be achieved by using different fusion methods, exact and approximate (CI
[9] for example). This paper restricts attention to the CF for exact fusion.
Then using (24), the fused marginal information vector and matrix for Gaussian
distributions is given by

$\begin{split}\bar{\zeta}^{f}_{X}=\bar{\zeta}^{i}_{X}+\bar{\zeta}^{j}_{X}-\bar{\zeta}^{c}_{X},\
\ \
\bar{\Lambda}^{f}_{XX}=\bar{\Lambda}^{i}_{XX}+\bar{\Lambda}^{j}_{XX}-\bar{\Lambda}^{c}_{XX}\end{split}$
(25)

The HS-CF fusion rule, eq. (14), for Gaussian pdfs can be represented by the
simple closed form expression,

$\begin{split}\zeta^{i,f}=&\left(\begin{array}[]{c}\bar{\zeta}^{f}_{X}\\\
\hdashline[2pt/2pt]0\end{array}\right)+\left(\begin{array}[]{c}\Lambda^{i}_{xs_{i}}(\Lambda^{i}_{s_{i}s_{i}})^{-1}\zeta^{i}_{s_{i}}\\\
\hdashline[2pt/2pt]\zeta^{i}_{s_{i}}\end{array}\right)\\\
\Lambda^{i,f}=&\left(\begin{array}[]{c;{2pt/2pt}c}\bar{\Lambda}^{f}_{xx}&0\\\
\hdashline[2pt/2pt]0&0\end{array}\right)+\left(\begin{array}[]{c;{2pt/2pt}c}\Lambda^{i}_{xs_{i}}(\Lambda^{i}_{s_{i}s_{i}})^{-1}\Lambda^{i}_{s_{i}x}&\Lambda^i_{xs_i}\\\
\hdashline[2pt/2pt]\Lambda^{i}_{s_{i}x}&\Lambda^i_{s_is_i}\end{array}\right),\end{split}$
(26)

where $\bar{\zeta}^{f}_{x}$ and $\bar{\Lambda}^{f}_{xx}$ are given in (25) and
it is assumed $X=x$ and $S=[s_{i}^{T},s_{j}^{T}]^{T}$. An equivalent
expression for the fused information vector and matrix at agent $j$ is
achieved by switching $i$ with $j$.

It is important to note, as seen from (25), that the fused marginal pdf
$(\bar{\zeta}^{f}_{x},\bar{\Lambda}^{f}_{xx})$ is the same for agents $i$ and
$j$. However, the conditional part is kept local (the right part of (26)),
which means that after fusion, the _local joint_ distributions in $i$ (w.r.t.
$x$ and $s_{i}$) and $j$ (w.r.t. $x$ and $s_{j}$) are different. While agents
only update the information vector and matrix of the marginal pdf, over the
common random state $x$, the local joint distribution in moment representation
(e.g., $\mu^{i,f},\Sigma^{i,f}$) will be updated, thus also updating the local
states $s_{i}$ ($s_{j}$).

###  VI-C The Information Augmented State

For Gaussian distributions there are two similar but not equivalent filters:
the _augmented_ state (AS), [12], and the _accumulated_ state density (ASD),
[26], where they mostly differ in their retrodiction formulation. This is more
involved in the ASD and not imperative to our solution, so the attention here
is limited to the augmented state implementation [12]. Here the emphasis is on
Gaussian state distributions; a solution of the ASD for general distributions
is given in [26].

The augmented state for a sliding time window from time step $n$ to time step
$k$ (denoted by subscript $k:n$) given in [12] uses covariance formulation for
the prediction step, and information representation for the update step.
However, since the algorithms developed in this paper work in Gaussian
information space, it is advantageous to work with a full information filter
formulation. The derivation of the information AS (_iAS_) is given in Appendix
A, with the main results provided here, namely the prediction and update steps
in information form.

For a dynamic system, described by the discrete time equations

$\begin{split}&x_{k}=F_{k}x_{k-1}+Gu_{k}+\omega_{k},\ \ \ \
\omega_{k}\sim\mathcal{N}(0,Q_{k})\\\ &y_{k}=H_{k}x_{k}+v_{k},\ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ v_{k}\sim\mathcal{N}(0,R_{k}),\end{split}$ (27)

where $F_{k}$ is the state transition matrix, $G$ is the control effect matrix
and $H_{k}$ is the sensing matrix. $\omega_{k}$ and $v_{k}$ are the zero mean
white Gaussian process and measurement noise, respectively.

The predicted information vector and matrix for the time window $k:n$, given
all information up to and including time step $k-1$ are given by

$\begin{split}&\zeta_{k:n|k-1}=\begin{pmatrix}Q_{k}^{-1}Gu_{k}\\\
\zeta_{k-1:n|k-1}-\mathbf{F}^{T}Q_{k}^{-1}Gu_{k}\end{pmatrix}\\\
&\Lambda_{k:n|k-1}=\begin{pmatrix}Q_{k}^{-1}&-Q_{k}^{-1}\mathbf{F}\\\
-\mathbf{F}^{T}Q_{k}^{-1}&\Lambda_{k-1:n|k-1}+\mathbf{F}^{T}Q_{k}^{-1}\mathbf{F}\end{pmatrix},\\\
\end{split}$ (28)

where $\mathbf{F}=\big{[}F_{k-1}\ \ 0_{m\times m(k-n-2)}\big{]}$ and $m$ is
the size of the (not augmented) state vector. Notice the simplicity of the
above expression and its interesting interpretation: the predicted conditional
information matrix at the current time step, given previous time steps (upper
left block), depends inversely on process noise alone and is not affected by
the state dynamics.

For completeness, the measurement update in Gaussian information space is [12]

$\begin{split}&\zeta_{k:n|k}=\zeta_{k:n|k-1}+J_{k}i_{k}\\\
&\Lambda_{k:n|k}=\Lambda_{k:n|k-1}+J_{k}I_{k}J_{k}^{T},\end{split}$ (29)

where $J_{k}=\big{[}I_{m}\ \ 0_{m\times m(k-n-1)}\big{]}^{T}$,
$i_{k}=H_{k}^{T}R_{k}^{-1}y_{k}$ and $I_{k}=H_{k}^{T}R_{k}^{-1}H_{k}$.

For linear-Gaussian problems, the above _iAS_ can be used locally at each
agent, enabling conditional independence such that for $n=1$,

$\begin{split}p(x_{k:1},s_{i},s_{j})&=\\\ p(&x_{k:1}|Z_{k})\cdot
p(s_{i}|x_{k:1},Z_{k})\cdot p(s_{j}|x_{k:1},Z_{k}).\end{split}$ (30)

It can be seen that the full information matrix and vector given in eq. (28)
grows rapidly with the time step $k$, inducing high processing and
communication costs. Note that the block tri-diagonal structure of the updated
information matrix $\Lambda_{k:1|k-1}$, resulting from the Markov property of
the system dynamics can be utilized to reduce computation burden. But this
does not resolve the communication load problem which scales with the size of
the tree network, allowing messages to propagate through from one end of the
tree to the other. Instead, a filtering approach is taken to marginalize past
states to process only a sliding window $k:n$ ($n>1$) while maintaining
conditional independence. This requires conservative filtering, discussed in
the next section.

###  VI-D Conservative Filtering

The approach of Vial _et al._[27] is adopted for the special case of Gaussian
distributions. The sparse structure of the marginalized approximate
information matrix is enforced by removing the links between $s_{i}$ and
$s_{j}$. In other words, given a true dense Gaussian distribution
$\mathcal{N}(\zeta_{tr},\Lambda_{tr})$, a sparse approximate distribution
$\mathcal{N}(\zeta_{sp},\Lambda_{sp})$ is sought such that the mean is
unchanged and the approximation is conservative in the PSD sense,

$\begin{split}\Lambda_{tr}^{-1}\zeta_{tr}=\Lambda_{sp}^{-1}\zeta_{sp},\ \ \ \
\ \Lambda_{tr}-\Lambda_{sp}\succeq 0,\end{split}$ (31)

where again the information form of the Gaussian distribution is used.

Reference [27] minimizes the Kullback-Leibler Divergence (KLD) to find a lower
bound on the dense true information matrix $\Lambda_{tr}$. Along similar
lines, [31] suggests a method named ‘Uniform Pre-Transmission Eigenvalue-Based
Scaling’ to conservatively approximate a covariance matrix $\Sigma$ by
inflating a diagonal matrix $D$ built out of the diagonal entries of the full
matrix $\Sigma$. To achieve a conservative approximation $D_{c}$, $D$ is
inflated by multiplying by the largest eigenvalue of $Q=D^{-\frac{1}{2}}\Sigma
D^{-\frac{1}{2}}$. This results in $D_{c}=\lambda_{max}D$ such that
$D_{c}-\Sigma\succeq 0$.

This method is generalized here to find a lower bound sparse information
matrix $\Lambda_{sp}$ and regain conditional independence between $s_{i}$ and
$s_{j}$. This new generalized method differs from the one suggested in [31] in
two ways. Firstly, the approximation, $\Lambda_{sp}$, is allowed to be any
information matrix achieved by setting any off-diagonal elements of the true
dense information matrix $\Lambda_{tr}$ to zero, i.e., the resulting matrix is
in general not diagonal or even block-diagonal. Note that since the
information matrix (i.e. not the covariance) is changed, setting off-diagonal
elements to zero directly controls the conditional independence structure of
the underlying distribution. Specifically for the purpose of this paper, terms
relating local random states (e.g., $s_{i}$ and $s_{j}$) in $\Lambda_{tr}$ are
set to zero to regain conditional independence given common target states
(e.g., $x_{k:n}$).

The second change from the original method is that the information matrix is
approximated, as opposed to the covariance matrix. This means that a _lower_
bound is sought and not an _upper_ bound, i.e. ‘information’ must be deflated,
instead of uncertainty being inflated. This is achieved by choosing the
_minimal_ eigenvalue of
$\tilde{Q}=\Lambda_{sp}^{-\frac{1}{2}}\Lambda_{tr}\Lambda_{sp}^{-\frac{1}{2}}$,
resulting in

$\Lambda_{tr}-\lambda_{min}\Lambda_{sp}\succeq 0,$ (32)

where $\lambda_{min}\Lambda_{sp}$ is the sought of conservative marginal
sparse approximation of the dense information matrix $\Lambda_{tr}$. The new
information vector is computed to such that (31) holds

$\zeta_{sp}=(\lambda_{min}\Lambda_{sp})\Lambda_{tr}^{-1}\zeta_{tr}.$ (33)

###  VI-E Closed-Form Algorithm

This subsection provides a full summary description of the different steps
each agent $i$ takes to locally process, communicate and fuse data with its
neighbor $j$. While the steps described in the pseudo code in Algorithm 1 are
general in the sense that it can be applied with any pdf, it assumes linear-
Gaussian distributions and uses the theory developed above to provide equation
references (green arrows) for the closed-form expressions of the different
operations. Following Sec. V, the BDF-CF and HS-CF algorithms are used as an
example to detail the algorithm from the perspective of one agent $i$
communicating with a neighbor $j$.

Algorithm 1 BDF-CF / HS-CF algorithm

1:Define: $\chi_{i}$, $\chi_{j}$, $\chi_{c}^{ij}$ $\chi_{\neg i}^{ij}$,
$\chi_{\neg j}^{ji}$, Priors $\triangleright$ Fig. 4

2:for All time steps do

3: Propagate state local states $\triangleright$ Eq. 28

4: Propagate common states in the CF

5: if BDF-CF then

6: Conservative filtering $\triangleright$ Sec. VI-D

7: else if HS-CF then

8: Marginalize out past state

9: end if

10: Measurement update $\triangleright$ Eq. 29

11: Send message $\triangleright$ Eq. 17

12: Fuse received message $\triangleright$ Eq. 18

13: Update common information

14:end for

15:return

###  VI-F Calculation of Communication and Computation Savings

To highlight the potential gain of the proposed CF2 methods with respect to
communication and computation complexity and how they change with scale, three
numerical examples (small, medium and large) of a multi-agent multi-target
tracking problem are presented. Consider the problem introduced earlier of
tracking $n_{t}$ ground targets by $n_{a}$ agents (trackers), where each agent
computes a local KF estimate, i.e., the system dynamics are assumed to be
linear with additive Gaussian white noise (27). Each agent $i$ has 6 unknown
local position states described by the random vector $s_{i}$ and takes
measurements to $n_{t}^{i}$ targets, each having 4 position/velocity states
described by the random vector $x_{t}$ (e.g., east and north coordinates and
velocities). The full state vector then has $6n_{a}+4n_{t}$ random states.
Assume tree topology in ascending order, where each agent tracks 1/2/3
targets, corresponding to the small/medium/large examples, respectively, but
has only one target in common with its neighbor. Now, using the same logic as
before, assume that each agent is only concerned with the targets it takes
measurements to and its own position states. Each agent has only 10/14/18
local ‘relevant’ random states for tracking 1/2/3 targets, respectively.

Table I presents a comparison between the different channel filters for the
three different scale problems. The baseline for comparison is the original
(homogeneous) CF, with each agent estimating the full state vector. For the
communication data requirement, double precision (8 bytes per element) is
assumed. Since the matrices are symmetric covariances, agents only needs to
communicate $n(n+1)/2$ upper diagonal elements, where $n$ is the number of
random states. Each agent’s computation complexity is determined by the cost
of inverting an $n\times n$ matrix. It can be seen from the table that even
for the small scale problem, the communication data reduction is significant;
the BDF-CF requires about 42.7% of the original CF, while the approximate BDF-
CF and HS-CF requires only 9.2% as agents only communicate common targets
states information vectors and matrices. These gains then increase with scale,
for the medium and large problems the BDF-CF communication is about 33% of the
original CF and for the approximate methods is less than 1%.

Another important aspect is computation complexity. As seen from the table,
while the BDF-CF methods require each agent to process the full random state
vector, the HS-CF offers significant computational reduction. Since each agent
only processes the locally relevant random states, the HS-CF scales with
subset of states and not with number of agents and targets. In the medium and
large scale problems, as the size of the full system random vector states
increase, the HS-CF computation complexity is less than 1% of the other
methods, which can be critical in terms of computing power for resource-
constrained tracking platforms.

TABLE I: Data communication requirements and computational complexity for different fusion methods, for different problem scales. |  | Small | Medium | Large
---|---|---|---|---
| ($n_{a}$, $n_{t}$) | (2, 1) | (10, 11) | (25, 51)
| $n_{t}^{i}$ | 1 | 2 | 3
CF | Data req. $[$KB$]$ | 2.4 | 801 | 24300
Complexity | $O(16^{3})$ | $O(104^{3})$ | $O(354^{3})$
BDF-CF | Data req. $[$%CF$]$ | 42.7 | 33.7 | 33.2
Complexity | $O(16^{3})$ | $O(104^{3})$ | $O(354^{3})$
Approx. BDF-CF | Data req. $[$%CF$]$ | 9.2 | 0.25 | 0.02
Complexity | $O(16^{3})$ | $O(104^{3})$ | $O(354^{3})$
HS-CF | Data req. $[$%CF$]$ | 9.2 | 0.25 | 0.02
Complexity | $O(10^{3})$ | $O(14^{3})$ | $O(18^{3})$

##  VII Simulation Studies

Multi-agent multi-target tracking simulation scenarios where performed to
compare and validate the proposed algorithms, where the focus is on the BDF-CF
and HS-CF heterogeneous fusion algorithms. Since the dynamics and measurement
models are assumed to be linear with Gaussian noise, Algorithm 1 is used
together with the _iAS_ as the inference engine, i.e., agents estimate the
sufficient statistics (information vector and matrix) of the random state
vector. First, the algorithms are tested on a static target case, where
conditional independence of the local states can be easily guaranteed. This is
followed by a dynamic target test case with only two agents and one target, to
validate and compare the augmented state (smoothing) and the conservative
filtering approaches. Lastly, the conservative filtering approach is used for
a more interesting 4-agents 5-target scenario. Results for all the different
scenarios are based on Monte Carlo simulations and compare the new algorithms
to an optimal centralized estimator.

###  VII-A Example 1 - Static Case

A chain network, consisting of five agents connected bilaterally in ascending
order $(1\leftrightarrow 2\leftrightarrow 3\leftrightarrow 4\leftrightarrow
5)$,as depicted in (Fig. 2), attempts to estimate the position of six
stationary targets in a $2D$ space. Assume each tracking agent $i\ =1,...,5$
has perfect self position knowledge, but with constant agent-target relative
position measurement bias vector in the east and north directions
$s_{i}=[b_{e,i},b_{n,i}]^{T}$. In every time step $k$, each agent takes two
kinds of measurements: one for the target and one to collect data on the local
sensor bias random vector, which can be transformed into the linear pseudo-
measurements,

$\displaystyle\begin{split}y^{t}_{i,k}&=x^{t}+s_{i}+v^{i,1}_{k},\ \
v^{i,1}_{k}\sim\mathcal{N}(0,R^{1}_{i}),\\\ m_{i,k}&=s_{i}+v^{i,2}_{k},\ \
v^{i,2}_{k}\sim\mathcal{N}(0,R^{2}_{i}),\end{split}$ (34)

where $y^{t}_{i,k}$ is agent $i$ relative measurement to target $t$ at time
step $k$ and $m_{i,k}$ is a measurement to a known landmark at time step $k$
for bias estimation. $x^{t}=[e^{t},n^{t}]^{T}$ is the east and north position
of target $t\ =1,...,6$. The tracking assignments for each agent, along with
the measurements noise error covariances for the relative target ($R^{1}_{i}$)
and landmark ($R^{2}_{i}$) measurements are given in Table II and illustrated
in Fig. 2. The relative target measurement noise characteristics for different
targets measured by the same agent are taken to be equal. For example, agent
$1$ takes noisy measurements to targets $1$ and $2$ with $1\ m^{2}$ and $10\
m^{2}$ variances in the east and north directions, respectively, and $3\
m^{2}$ in both directions for the landmark.

Following the definitions from Sec. III, the full state vector includes 22
random states

$\chi=[{x^{1}}^{T},...,{x^{6}}^{T},s_{1}^{T},...,s_{5}^{T}]^{T},$ (35)

where for the HS-CF fusion, define the local random state vector at agent $i$

$\chi_{i}=[{X^{\mathcal{T}_{i}}}^{T},s_{i}^{T}]^{T}.$ (36)

Here $\mathcal{T}_{i}$ is the set of targets observed by agent $i$, and
$X^{\mathcal{T}_{i}}$ includes all target random state vectors $x^{t}$, s.t
$t\in\mathcal{T}_{i}$. In other words, the local random state vector at each
agent includes only locally relevant targets and the local biases. In the HS-
CF, where two agents $i$ and $j$ only share the marginal statistics regarding
common states, messages should only consist data regarding targets
$t\in\mathcal{T}_{i}\cap\mathcal{T}_{j}$. For example, according to Table II
and the network tree topology, for agents $1$ and $2$:
$\mathcal{T}_{1}\cap\mathcal{T}_{2}=T_{2}$.

The data communication requirements for this relatively small system were
calculated; similar to the results from Sec. VI-F, the BDF-CF and the HS-CF
requires about 38% and 2.6% of the original CF communication data
requirements, respectively.

TABLE II: Local platform target assignments and sensor measurement error covariances.  Agent | Tracked Targets | $R_{i}^{1}[m^{2}]$ | $R_{i}^{2}[m^{2}]$
---|---|---|---
1 | $T_{1},T_{2}$ | diag([1,10]) | diag([3,3])
2 | $T_{2},T_{3}$ | diag([3,3]) | diag([3,3])
3 | $T_{3},T_{4},T_{5}$ | diag([4,4]) | diag([2,2])
4 | $T_{4},T_{5}$ | diag([10,1]) | diag([4,4])
5 | $T_{5},T_{6}$ | diag([2,2]) | diag([5,5])

The BDF-CF and the HS-CF performance was tested with 500 Monte Carlo
simulations and compared to a centralized estimator. As mentioned before, in
the BDF-CF each platform processes the full random state vector (35), while in
the HS-CF each platform processes only the locally relevant random states
(36). In the simulations fusion occurs in every time step.

Figure 5: Example 1 (static) - NEES hypothesis test based on 500 Monte Carlo
simulations, where the dashed lines show bounds for 95$\%$ confidence level.
Shown are test results of agents 1 and 4 using for different fusion methods,
the results indicate all methods produce a consistent estimate.

Fig. 5 shows a NEES consistency test [15] results for agents 1 and 4 and a
centralized estimator. Results are based on 500 Monte Carlo simulations with
95% confidence level. It can be seen that the MMSE estimates of both agents,
with the BDF-CF and the HS-CF are consistent. Note that since the HS-CF only
estimates a subset of 6 states out of the full 22 state vector, the
consistency bounds are different for these methods. The second test to
determine the performance of a fusion method is whether it is conservative
relative to a centralized estimator (see Sec. II-B). To verify, the local
covariance matrix must be checked to see whether it is pessimistic relative to
the centralized covariance. One simple test is by computing the eigenvalues of
$\bar{\Sigma}_{\chi_{i}}-\bar{\Sigma}_{\chi_{i}}^{cent}$, where
$\bar{\Sigma}_{\chi_{i}}$ is the agent’s covariance and
$\bar{\Sigma}_{\chi_{i}}^{cent}$ is the centralized marginal covariance over
$\chi_{i}$. If the minimal eigenvalue is bigger or equal to zero, all
eigenvalue are bigger or equal to zero and the MMSE estimate is conservative
in the PSD sense. In the above simulations the minimal eigenvalues between all
agents and all simulations were 0, for both the BDF-CF and the HS-CF, thus
they are conservative in the PSD sense.

###  VII-B Example 2 - Dynamic Case

In dynamic systems, as discussed in Sec. IV, there is a challenge in
maintaining conditional independence. Two ways to solution are suggested, the
first using the _iAS_ , thus keeping a distribution over the full time history
over target random states, which is costly in both communication and
computation requirements. The second, more efficient solution, is to perform
conservative filtering by enforcing conditional independence in the
marginalization step and deflating the information matrix (Algorithm 1). Since
this process loses information due to deflation, the BDF-CF becomes an
approximate solution and is expected to be less accurate than the _iAS_
implementation. This is shown using a two agent, one (dynamic) target tracking
simulation. Here the target follows a linear dynamics model with time-varying
acceleration control,

$x_{k+1}=Fx_{k}+Gu_{k}+\omega_{k},\ \ \omega_{k}\sim\mathcal{N}(0,0.08\cdot
I_{n_{x}\times n_{x}}),$ (37)

where

$\begin{split}F=\begin{bmatrix}1&\Delta t&0&0\\\ 0&1&0&0\\\ 0&0&1&\Delta t\\\
0&0&0&1\end{bmatrix},\quad G=\begin{bmatrix}\frac{1}{2}\Delta t^{2}&0\\\
\Delta t&0\\\ 0&\frac{1}{2}\Delta t^{2}\\\ 0&\Delta
t\end{bmatrix}.\end{split}$ (38)

The acceleration input in the east and north directions is given by
$u_{k}=\begin{bmatrix}a_{e}\cdot\cos(d_{e}\cdot k\Delta t)\ \
a_{n}\cdot\sin(d_{n}\cdot k\Delta t)\end{bmatrix}^{T}$, where $a_{e}/a_{n}$
and $d_{e}/d_{n}$ define the east and north amplitude and frequency,
respectively. The measurement model is as in the static example, given in (34)
with noise parameters defined for agents 1 and 2 in Table II.

Results in Fig. 6 show a comparison between the _iAS_ with the full time
window ($k:1$) and filtering approaches (sliding window of size 1), using the
BDF-CF and the HS-CF for fusion. The plots in figure (a) show the NEES
consistency tests (75 simulations, 95% confidence level) for agent 1, where
the centralized (filtering results presented) and BDF-CF in the upper plot and
the HS-CF with its different bounds in the lower. The results are consistent,
with pessimistic behaviour of the BDF-CF due to the conservative filtering
approach. (b) Shows the root mean squared error (RMSE) results of the same
simulation, with agent 1 in the upper plot and agent 2 in the lower. The _iAS_
has better performance, with smaller RMSE and $2\sigma$ bounds, which is to be
expected due to its smoothing operation, but at the expense of much higher
computation and communication load.

The conservativeness of the fused estimate was checked again by computing the
minimal eigenvalues across 75 simulations and the two agents. For the _iAS_
approach, the BDF-CF and the HS-CF had small negative minimal eigenvalues of
$-0.002$ and $-0.0015$ respectively, thus slightly overconfident relative to
the centralized. For the conservative filtering approach, the BDF-CF was
conservative with minimal eigenvalue of $0.0008$ and the HS-CF was
overconfident with minimal eigenvalue of $-0.26$.

Figure 6: Results from a 75 Monte Carlo simulation of a 2 agent, 1 target
dynamic target tracking scenario. Shown is a comparison between _iAS_ and
filtering. (a) NEES test with the upper figure showing the BDF-CF compared to
the centralized (with filtering) and the lower showing the HS-CF. Here the
dashed black lines show bounds for 95$\%$ confidence level. (b) RMSE
comparisons between the BDF-CF and the HS-CF for _iAS_ and filtering
approaches for agent 1 (upper) and agent 2 (lower).

Conservative filtering allows to test the algorithms on a more interesting
dynamic simulation. As a test case, a simulation of a cooperative target
tracking task with 4 agents and 5 dynamic targets (full random state vector of
28 states) was performed. The dynamic model details are the same as in the 2
agent, 1 target dynamic simulation above, with measurement parameters defined
by the first 4 agents in Table II. The advantages of the BDF-CF and HS-CF
regarding communication and computation costs are highlighted again, as the
BDF-CF saves 58% in communication costs relative to the original CF, and the
HS-CF saves 94.5% in communication and 87.5% in computation complexity.

Results of 500 Monte Carlo simulation with filtering for agents 1 and 4 are
presented in Fig. 7. The plots in (a) show the NEES consistency test with 95%
confidence bound marked with black dashed lines. The upper plot shows the
centralized (black circles) and the BDF-CF for agents 1 (blue squares) and 4
(red x) NEES for the 28-state vector. The lower plot shows the HS-CF results,
which has a smaller 10-state random vector. (b) shows the corresponding RMSE
for agent 1 (upper plot) and 4 (lower plot). Note that the RMSE results for
the centralized estimate and BDF-CF, which hold distributions over the full
28-state vector, are marginalized and computed only over relevant local 10
agent random states for this comparison.

It is seen from the NEES plots that, as expected, the centralized estimator
produces a consistent MMSE estimate, and the BDF-CF overestimates the
uncertainty due to the information matrix deflation (covariance inflation) in
the conservative filtering step. The BDF-CF also produces a conservative MMSE
estimate relative to the centralized in the PSD sense for all agents, since
the minimum eigenvalue between the agents is positive ($3e-04$). The HS-CF is
slightly overconfident for both the consistency test and the PSD test, with
negative minimal eigenvalue of $-0.26$. However, the degree of non-
conservativeness in the HS-CF will in general be highly problem- and topology
dependent. Hence, the choice of whether to task agents with the full random
state vector, with either homogeneous DDF methods (e.g., classical CF and
conventional CI) or heterogeneous fusion with the BDF-CF, or to task them with
only a subset of relevant random states using the HS-CF, will hinge on the
desired trade-off in communication/computation complexity vs. resulting
overconfidence in state MMSE estimates, provided that the HS-CF allows for
stable convergence.

The HS-CF overconfidence is attributed to inaccurate removal of implicit and
hidden correlations due to marginalization in the filtering step (line 8 in
Algorithm 1). Correctly accounting for these dependencies is not in the scope
of this paper, but is the focus of ongoing work.

Figure 7: Results from a 500 Monte Carlo simulation of a cooperative target
tracking task, with filtering, consisting of 4 tracking agents and 5 dynamic
targets. Presented are results for agents 1 and 4. (a) NEES consistency test
results for the centralized and BDF-CF estimates of 28 random states (upper)
and the HS-CF estimates over 10 random states (lower). (b) Solid lines show
the RMSE over target and agent local states relevant to that agent, dashed
line shows the $2\sigma$ confidence bounds on logarithmic scale, for agent 1
(upper) and 4 (lower).

##  VIII Conclusions

Heterogeneous fusion defines a key family of problems for Bayesian DDF, as it
enables flexibility for large scale autonomous sensing networks. As shown in
this work, separating the global joint distribution into smaller subsets of
local distributions significantly reduces local agent computation and
communication requirements. The analysis and derivations presented in this
paper, while assuming tree structured networks for the purposes of exact
fusion via channel filtering, offers a basis for developing and analyzing
similar methods for more general heterogeneous problems involving exact or
approximate fusion in more complex networked fusion settings.

Probabilistic graphical models (PGMs) were used here to develop Bayesian DDF
algorithms. PGMs provided insight into the origin of the coupling between
random states not mutually tracked by two agents and enabled exploitation of
the conditional independence structure embedded in these graphs. This led to a
family of conditionally factorized channel filter (CF2) approaches for general
probabilistic and Gaussian pdfs that were demonstrated on static and dynamic
target tracking problems. The latter motivated the development and use of the
information augmented state (_iAS_) filter to regain conditional independence,
on the expense of increasing computation and communication costs. To overcome
this problem a conservative filtering approach was demonstrated to maintain
conditional independence over a small time window, without the need of the
full time history.

The DDF framework naturally enables sparse distributed estimation for high-
dimensional state estimation and the conditionally factorized-CF approach
represents a practical and theoretical shift in the state of the art, subject
to usual provisos and limitations of DDF. From a practical standpoint, the CF
strategies developed in this paper can already be used to improve scalability
in a variety of decentralized Bayesian estimation problems, such as
cooperative localization and navigation [25, 1], multi-agent SLAM [3] and
terrain height mapping [32], where a height distribution is estimated on a
grid map. In this case, for example, the BDF-CF can be used to reduce
communication in the network by dividing the the map into several overlapping
regions of interest, allowing agents to communicate only regarding those cells
in which they have new data to contribute. This scales the communication with
the number of locally observed grid cells instead of the entire map.

Indeed, some works are already leveraging heterogeneous DDF ideas for
robotics, [1, 2], despite the gap in theoretical guarantees and understanding
on the full nature of the problem and its limitations. This paper makes
progress by building theoretical foundations for future research and surfacing
a discussion on the assumptions and definitions of homogeneous DDF, as they
appear to be inadequate for real world robotics problems of heterogeneous
fusion. Heterogeneous fusion, as defined in this paper, requires a careful
revisit of the idea of ‘ideal’ centralized/decentralized Bayesian estimation
as well as the definitions of consistency and conservativeness for general
(non-Gaussian) pdfs and more specifically in the case of heterogeneous pdfs in
dynamic systems.

##  Appendix A Derivation of the iAS

The augmented state for a sliding time window from time step $n$ to time step
$k$ (denoted by subscript $k:n$) as shown in [12] is given by the following
equations:
_Prediction Step_

$X_{k:n|k-1}=\begin{pmatrix}F_{k-1}\chi_{k-1|k-1}+Gu_{k}\\\
X_{k-1:n|k-1}\end{pmatrix}$ (39)
$P_{k:n|k-1}=\begin{pmatrix}P_{k|k-1}&\mathbf{F}P_{k-1:n|k-1}\\\
P_{k-1:n|k-1}\mathbf{F}^{T}&P_{k-1:n|k-1}\end{pmatrix},$ (40)

where $\mathbf{F}=\big{[}F_{k-1}\ \ 0_{m\times m(k-n-2)}\big{]}$ and $m$ is
the size of the (not augmented) state vector.
_Update Step_
The measurement update in information space is as follows:

$P_{k:n|k}^{-1}=P_{k:n|k-1}^{-1}+J_{k}I_{k}J_{k}^{T}$ (41)
$P_{k:n|k}^{-1}X_{k:n|k}=P_{k:n|k-1}^{-1}X_{k:n|k-1}+J_{k}i_{k},$ (42)

where $J_{k}=\big{[}I_{m}\ \ 0_{m\times m(k-n-1)}\big{]}^{T}$,
$i_{k}=H_{k}^{T}R_{k}^{-1}z_{k}$ and $I_{k}=H_{k}^{T}R_{k}^{-1}H_{k}$.

Since the algorithms developed in this paper work in log space, it is
advantageous to work with an information filter, which is based on the log-
likelihood of the Gaussian distribution. Thus, a transformation of the
prediction step given in (39)-(40) from state space to the Gaussian
information space is needed.

First, define $P_{k:n|k-1}^{-1}$ to be the augmented predicted information
matrix:

$\begin{split}P_{k:n|k-1}^{-1}&=\begin{pmatrix}V_{11}&V_{12}\\\
V_{21}&V_{22}\end{pmatrix},\end{split}$ (43)

where from the matrix inversion lemma:

$\begin{split}V_{11}&=(P_{k|k-1}-\mathbf{F}P_{k-1:n|k-1}P_{k-1:n|k-1}^{-1}P_{k-1:n|k-1}\mathbf{F}^{T})^{-1}\\\
&=(P_{k|k-1}-\mathbf{F}P_{k-1:n|k-1}\mathbf{F}^{T})^{-1}.\end{split}$ (44)

The expression $\mathbf{F}P_{k-1:n|k-1}\mathbf{F}^{T}$ has the dimension
$m\times m$. From the definition of $\mathbf{F}$ above (44) can be simplified
by noticing that
$\mathbf{F}P_{k-1:n|k-1}\mathbf{F}^{T}=F_{k-1}P_{k-1|k-1}F_{k-1}^{T}$, i.e. it
depends only on the previous time step and not the full time history. Eq. (44)
is thus:

$V_{11}=(P_{k|k-1}-F_{k-1}P_{k-1|k-1}F_{k-1}^{T})^{-1}.$ (45)

Here $P_{k|k-1}^{-1}$ is the predicted information matrix at time step $k$,
given in the literature by:
$P_{k|k-1}^{-1}=(F_{k-1}P_{k-1|k-1}F_{k-1}^{T}+Q)^{-1}$, where $Q$ is the
process noise covariance. Taking the inverse and plugging in $P_{k|k-1}$, (45)
can be simplified to:

$V_{11}=Q^{-1}.$ (46)

Applying the matrix inversion lemma again the expressions for other terms are:

$V_{12}=V_{21}^{T}=-V_{11}\mathbf{F}P_{k-1:n|k-1}P_{k-1:n|k-1}^{-1}=-V_{11}\mathbf{F},$
(47)
$\begin{split}V_{22}=P_{k-1:n|k-1}^{-1}+\mathbf{F}^{T}V_{11}\mathbf{F}.\end{split}$
(48)

The predicted information matrix is then given by:

$\begin{split}P_{k:n|k-1}^{-1}=\begin{pmatrix}Q^{-1}&-Q^{-1}\Gamma^{T}\mathbf{F}\\\
-\mathbf{F}^{T}\Gamma Q^{-1}&P_{k-1:n|k-1}^{-1}+\mathbf{F}^{T}\Gamma
Q^{-1}\Gamma^{T}\mathbf{F}\end{pmatrix}\\\ \end{split}$ (49)

and the predicted information vector can now be derived:

$\begin{split}P_{k:n|k-1}^{-1}&X_{k:n|k-1}=\\\ &\begin{pmatrix}Q^{-1}Gu_{k}\\\
P_{k-1:n|k-1}^{-1}X_{k-1:n|k-1}-\mathbf{F}^{T}\Gamma
Q^{-1}Gu_{k}\end{pmatrix}.\end{split}$ (50)

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|  Ofer Dagan received the B.S. degree in aerospace engineering, in 2010, and
the M.S. degree in mechanical engineering, in 2015, from the Technion - Israel
Institute of Technology, Haifa, Israel. He is currently working toward the
Ph.D. degree in aerospace engineering with the Ann and H.J. Smead Aerospace
Engineering Sciences Department, University of Colorado Boulder, Boulder, CO,
USA. From 2010 to 2018 he was a research engineer in the aerospace industry.
His research interests include theory and algorithms for decentralized
Bayesian reasoning in heterogeneous autonomous systems.
---|---
|  Nisar R. Ahmed received the B.S. degree in engineering from Cooper Union,
New York City, NY,USA, in 2006 and the Ph.D. degree in mechanical engineering
from Cornell University, Ithaca, NY, USA, in 2012. He is an Associate
Professor of Autonomous Systems and H. Joseph Smead Faculty Fellow with Ann
and H.J. Smead Aerospace Engineering Sciences Department, University of
Colorado Boulder, Boulder, CO, USA. He was also a Postdoctoral Research
Associate with Cornell University until 2014. His research interests include
the development of probabilistic models and algorithms for cooperative
intelligence in mixed human–machine teams.
---|---