Effects of approximate representation in belief propagation for inference in wireless sensor networks

Loopy belief propagation (BP) is a low-complexity inference algorithm that has found widespread use. In this paper, we consider employing BP for joint ground-truth detection and corruption parameter estimation in wireless sensor networks. We cast this problem using a hybrid factor graph involving both discrete ground-truth variables and continuous parameter variables. Since BP messages incident to a continuous variable are functions of the continuous variable, compact representation of these messages is desirable for reducing the message exchange cost. We adopt a message approximation scheme based on projection onto a finite-dimensional L₂ space, and investigate the effects of this message approximation. Specifically, we derive a sufficient condition for convergence of loopy BP on the class of graphs of interest, and we provide an upper bound on the iterative evolution of message error. Under certain contractivity conditions, the error bound is shown to consist of a transient term and a steady-state term. We showcase the theoretical results on a cognitive radio example with a simple discretization scheme.