From a theoretical point of view, statistical inference is an attractive model of brain operation. However, it is unclear how to implement these inferential processes in neuronal networks. We offer a solution to this problem by showing in detailed simulations how the belief propagation algorithm on a factor graph can be embedded in a network of spiking neurons. We use pools of spiking neurons as the function nodes of the factor graph. Each pool gathers "messages" in the form of population activities from its input nodes and combines them through its network dynamics. Each of the various output messages to be transmitted over the edges of the graph is computed by a group of readout neurons that feed in their respective destination pools. We use this approach to implement two examples of factor graphs. The first example, drawn from coding theory, models the transmission of signals through an unreliable channel and demonstrates the principles and generality of our network approach. The second, more applied example is of a psychophysical mechanism in which visual cues are used to resolve hypotheses about the interpretation of an object's shape and illumination. These two examples, and also a statistical analysis, demonstrate good agreement between the performance of our networks and the direct numerical evaluation of belief propagation.