  In this paper, we investigate distributed inference schemes, over
binary-valued Markov random fields, which are realized by the belief
propagation (BP) algorithm. We first show that a decision variable obtained by
the BP algorithm in a network of distributed agents can be approximated by a
linear fusion of all the local log-likelihood ratios. The proposed approach
clarifies how the BP algorithm works, simplifies the statistical analysis of
its behavior, and enables us to develop a performance optimization framework
for the BP-based distributed inference systems. Next, we propose a blind
learning-adaptation scheme to optimize the system performance when there is no
information available a priori describing the statistical behavior of the
wireless environment concerned. In addition, we propose a blind threshold
adaptation method to guarantee a certain performance level in a BP-based
distributed detection system. To clarify the points discussed, we design a
novel linear-BP-based distributed spectrum sensing scheme for cognitive radio
networks and illustrate the performance improvement obtained, over an existing
BP-based detection method, via computer simulations.
