We replace strong independence in credal networks with the weaker notion of
epistemic irrelevance. Focusing on directed trees, we show how to combine the
local credal sets in the networks into an overall joint model, and use this to
construct and justify an exact message-passing algorithm that computes updated
beliefs for a variable in the network. The algorithm, which is essentially
linear in the number of nodes, is formulated entirely in terms of coherent lower
previsions. We supply examples of the algorithm's operation, and report an
application to on-line character recognition that illustrates the advantages of
the model for prediction.