  A wide class of machine learning algorithms can be reduced to variable
elimination on factor graphs. While factor graphs provide a unifying notation
for these algorithms, they do not provide a compact way to express repeated
structure when compared to plate diagrams for directed graphical models. To
exploit efficient tensor algebra in graphs with plates of variables, we
generalize undirected factor graphs to plated factor graphs and variable
elimination to a tensor variable elimination algorithm that operates directly
on plated factor graphs. Moreover, we generalize complexity bounds based on
treewidth and characterize the class of plated factor graphs for which
inference is tractable. As an application, we integrate tensor variable
elimination into the Pyro probabilistic programming language to enable exact
inference in discrete latent variable models with repeated structure. We
validate our methods with experiments on both directed and undirected graphical
models, including applications to polyphonic music modeling, animal movement
modeling, and latent sentiment analysis.
