Asymptotic learning in feedforward networks with binary symmetric channels

Each of a large number of nodes takes a measurement in sequence to decide between two hypotheses about the state of the world. Each node also has available the decisions of some of its immediate predecessors and uses these and its own measurement to make its decision. Each node broadcasts its decision through a binary symmetric channel, which randomly flips the decision. The question treated here is whether there exists a decision strategy consisting of a sequence of likelihood ratio tests such that the decisions approach the true hypothesis as the number of nodes increases. We show that if each node learns from bounded number of predecessors, then the decisions cannot converge to the underlying truth. We show that if each node learns from all predecessors then the decisions converge in probability to the underlying truth when the flipping probabilities are bounded away from 1/2. We also derive, in the case when the flipping probabilities tend to 1/2, a condition on the convergence rate of the flipping probabilities that is required for the decisions to converge to the true hypothesis in probability.