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. In this talk, I will describe a generalization of undirected factor graphs to plated factor graphs, and a corresponding generalization of the variable elimination algorithm that exploits efficient tensor algebra in graphs with plates of variables. This tensor variable elimination algorithm has been integrated into the Pyro probabilistic programming language, enabling scalable, automated exact inference in a wide variety of deep generative models with repeated discrete latent structure. I will discuss applications of such models to polyphonic music modeling, animal movement modeling, and unsupervised word-level sentiment analysis, as well as algorithmic applications to exact subcomputations in approximate inference and ongoing work on extensions to continuous latent variables.