IEEE Signal Processing - May 2018 - 123

What we have
learned

The formulation of the
diffusion filtering problem
in a Bayesian framework
as presented in this
lecture note improves
the understanding of
existing linear distributed
filtering algorithms.

In this lecture note, we
reviewed several preexisting algorithms
for distributed filtering over networks
using a Bayesian perspective. In the general
Bayesian formulation
of the ATC diffusion
filter as shown in [4], each node, in the socalled adapt step, updates its local posterior
pdf of the hidden states by assimilating
observations in that node's closed neighborhood; subsequently, in the combine
step, the updated pdfs are fused within
local neighborhoods using a minimum
weighted average KL divergence criterion. Under the linear, Gaussian model
assumption, the general Bayesian ATC
filter reduces to an ATC diffusion Kalman filter where, as shown again in [4],
the adapt step equations coincide with the
original diffusion Kalman filter in [1], but
the combine step is reminiscent of the
covariance intersection fusion rule in [9].
As an alternative to ATC filters, we
also reviewed the RndEx diffusion filters,
which also include an adapt step that
updates local posteriors at each node by
assimilating available observations in the
node's closed neighborhood. Afterwards,
however, rather than fusing the updated
pdfs in a combine step, the nodes exchange
them with randomly chosen neighbors
so that, over time, a given posterior pdf
travels along a random path across the
network assimilating, at different time
instants, local observations available at
each node that is visited along that path.
Under the linear, Gaussian model assumption, the update step at instant n at a given
node r again reduces to a Kalman filter
whose inputs, however, are a posterior
mean vector and a posterior covariance
matrix coming from another random
node s at instant n - 1, as in the gossip

Kalman filter previously introduced in [2].
The formulation
of the diffusion filtering problem in a
Bayesian framework
as presented in this
lecture note improves
the understanding of
existing linear distributed filtering algorithms. In addition and, more significantly,
it also provides guidance to design more
sophisticated algorithms, e.g., distributed
particle filters or distributed Bernoulli/
multi-Bernoulli filters, that are needed
for more complex signal models and are
detailed in other original research articles,
e.g., [6] and [10].

Acknowledgments
We would like to thank Prof. José M.F.
Moura, Prof, Petar Djuric´, and Dr. Kamil
Dedecius for discussions that greatly contributed to our understanding of the topics
in this lecture note.

Authors
Marcelo G.S. Bruno (bruno@ita.br)
received his bachelor's and master's
degrees in electrical engineering from the
University of São Paulo, Brazil, and his
Ph.D. degree in electrical and computer engineering from Carnegie Mellon
University, Pittsburgh, Pennsylvania.
Currently, he is an associate professor at
the Instituto Tecnológico de Aeronáutica,
Brazil. His research interests are in statistical signal processing, especially particle filters, Markov chain Monte Carlo,
statistics on manifolds, random finite
sets, and their applications in machine
learning, mobile robotics, telecommunications, and target detection and tracking
over networks.
Stiven S. Dias (stiven.dias@embraer
.com.br) received his bachelor's degree
in computer engineering from the

IEEE Signal Processing Magazine

May 2018

Universidade Federal do Espírito Santo,
Vitória, Brazil, in 2006, his master's degree
in aeronautical engineering from the
Instituto Tecnológico de Aeronáutica
(ITA), São José dos Campos, Brazil,
in 2008, and his Ph.D. degree in electrical
and computer engineering from ITA in
2014. His main research interests include
distributed estimation in sensor networks and multitarget, multisensor
tracking. He has been with Embraer
S.A. as a research and development engineer since February 2008 developing
advanced data fusion systems.

References

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1766-1784, Apr. 2011.
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[7] B. O. Anderson and J. B. Moore, "Optimal filtering,"
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