Fisher meets Feynman: score-based variational inference with a product of experts

We introduce a highly expressive yet distinctly tractable family for black-box variational inference (BBVI). Each member of this family is a weighted product of experts (PoE), and each weighted expert in the product is proportional to a multivariate $t$-distribution. These products of experts can model distributions with skew, heavy tails, and multiple modes, but to use them for BBVI, we must be able to sample from their densities. We show how to do this by reformulating these products of experts as latent variable models with auxiliary Dirichlet random variables. These Dirichlet variables emerge from a Feynman identity, originally developed for loop integrals in quantum field theory, that expresses the product of multiple fractions (or in our case, $t$-distributions) as an integral over the simplex.