Partition Functions of Normal Factor Graphs

One of the most common types of functions in mathematics, physics, and engineering is a sum of products, sometimes called a partition function. After “normalization,” a sum of products has a natural graphical representation, called a normal factor graph (NFG), in which vertices represent factors, edges represent internal variables, and half-edges represent the external variables of the partition function. In physics, so-called trace diagrams share similar features. We believe that the conceptual framework of representing sums of products as partition functions of NFGs is an important and intuitive paradigm that, surprisingly, does not seem to have been introduced explicitly in the previous factor graph literature. Of particular interest are NFG modifications that leave the partition function invariant. A simple subclass of such NFG modifications offers a unifying view of the Fourier transform, tree-based reparameterization, loop calculus, and the Legendre transform.