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A distributed belief propagation protocol is developed to carry inference and decoding tasks using wireless sensor networks with high-dimensional, correlated observations. Statistical dependencies are modeled using factor graphs. The overall a-posteriori probability is factored so that its factor graph representation can be mapped to the actual communication network. Sum-product message passing updates over the graphical model can thus be mapped to messages among sensors. As an application scenario, distributed spectrum sensing is considered. Simulated tests show that exploiting the correlation present among sensor observations can considerably improve sensing performance.