  We consider the decentralized binary hypothesis testing problem in networks
with feedback, where some or all of the sensors have access to compressed
summaries of other sensors' observations. We study certain two-message feedback
architectures, in which every sensor sends two messages to a fusion center,
with the second message based on full or partial knowledge of the first
messages of the other sensors. We also study one-message feedback
architectures, in which each sensor sends one message to a fusion center, with
a group of sensors having full or partial knowledge of the messages from the
sensors not in that group. Under either a Neyman-Pearson or a Bayesian
formulation, we show that the asymptotically optimal (in the limit of a large
number of sensors) detection performance (as quantified by error exponents)
does not benefit from the feedback messages, if the fusion center remembers all
sensor messages. However, feedback can improve the Bayesian detection
performance in the one-message feedback architecture if the fusion center has
limited memory; for that case, we determine the corresponding optimal error
exponents.
