IEEE Signal Processing - May 2018 - 118

lecture notes
Marcelo G.S. Bruno and Stiven S. Dias

A Bayesian Interpretation of
Distributed Diffusion Filtering Algorithms

I

n this lecture note, we use a Bayesian
methodology to formulate the optimal
solution to the problem of cooperative
tracking of a time-varying signal over a
partially connected network of multiple
agents, where each agent has sensing,
processing, and communication capabilities of its own. Subsequently, assuming a
general state-space model for the hidden
state vectors and the agents' observations,
we present, also from a Bayesian perspective, the general form of the adaptthen-combine (ATC) and the random
exchange (RndEx) distributed diffusion
filters, contrasting them with the ideal,
optimal network filter.
We then discuss how the general ATC
and RndEx diffusion filters reduce to
modified versions of existing linear ATC
and RndEx diffusion filters found in the
literature [1], [2] when the underlying
posterior probability distribution of the
unknown tracked states is Gaussian.

Relevance
In modern engineering systems, multiple
agents dispersed over remote nodes of a
network often cooperate with each other
to execute a common task such as, e.g.,
tracking an unknown time-varying signal
[3]. Fully distributed solutions to the cooperative signal tracking problem, where
nodes typically have access to local measurements only but exchange messages
Digital Object Identifier 10.1109/MSP.2018.2791632
Date of publication: 26 April 2018

118

were already known, but in a different
with their neighbors over partially concontext, from the literature on joint mulnected networks, have been extensively distitarget detection and tracking under the
cussed in recent literature and include the
name of Kullback-Leibler (KL) fusion;
well-known ATC linear diffusion Kalman
see, e.g., [5]. Simultaneously, [6] shed
filter [1] introduced by Cattivelli and Sayed
some light on the Bayesian interpretaand the RndEx diffusion (or "gossip") Kaltion of RndEx diffusion.
man filter [2] proposed by Kar and Moura.
The discussion in
Compared to earlier
this lecture note further
consensus-based soluWe use a Bayesian
improves the undertions, diffusion algomethodology to formulate
standing of existing
rithms no longer try to
the optimal solution to the linear distributed filterforce all network nodes
ing algorithms and also
to agree at each time
problem of cooperative
provides guidance to
instant on the same
tracking of a time-varying
design more advanced
state vector estimate.
signal over a partiallyalgorithms that are
Accordingly, under a
connected network
needed for more comdiffusion strategy, iterof multiple agents.
plex signal models and
ative internode comare the object of ongomunication between
ing original research.
consecutive data measurements is no longer
required or is kept at a minimum, thus making that class of algorithms more suitable for
Prerequisites
real-time applications.
This lecture note assumes a background
However, neither [1] nor [2] was writin the theory of probability and random
ten from a Bayesian perspective, and
vectors taking values in 0 N. Familiarity
it was, therefore, unclear what kind
with Kalman filtering is also assumed,
of posterior probability distribution
especially how to derive the Kalman filof the unknown states those methods
ter from first principles such as the total
were implicitly computing at each
probability theorem and Bayes law. Such
network node and how it compared to
derivations, whenever required, are omitthe optimal posterior distribution of the
ted due to space constraints; if the reader
states conditioned at each time instant
is unfamiliar with those topics, we suggest
on all present and past network data. In
a review of references such as [7].
particular, the seminal work by Dedecius and Djuric´ [4] has recently proProblem statement
vided a clear Bayesian interpretation of
Throughout the lecture note, we use boldATC diffusion, building on results that
faced uppercase italic letters, e.g., X, to
IEEE Signal Processing Magazine

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May 2018

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1053-5888/18©2018IEEE



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