  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.
