Machine Learning

15.7.3 The Sum-Product Algorithm

We will develop the algorithm in a “bottom-up” approach, via the use of an example. Once the rationale is understood, the generalization can readily be obtained. Let us consider the factor tree of Figure 15.26. The factor nodes are denoted by capital letters and squares, and each one is associated with a potential function. The rest are variable nodes, denoted by circles. Assume that we want to compute the marginal P(x₁). Node x₁, being a variable node, is connected to factor nodes only. We split the graph in as many (tree) subgraphs as the factor nodes connected to x₁ (three in our case). In the figure, each one of these subgraphs is encircled, having as roots the nodes A, H, and G, respectively. Recall that ...