Belief propagation generalizes backpropagation

The two most important algorithms in artificial intelligence are backpropagation and belief propagation. In spite of their importance, the connection between them is poorly characterized. We show that when an input to backpropagation is converted into an input to belief propagation so that (loopy) belief propagation can be run on it, then the result of belief propagation encodes the result of backpropagation; thus backpropagation is recovered as a special case of belief propagation. In other words, we prove for apparently the first time that belief propagation generalizes backpropagation. Our analysis is a theoretical contribution, which we motivate with the expectation that it might reconcile our understandings of each of these algorithms, and serve as a guide to engineering researchers seeking to improve the behavior of systems that use one or the other.