  Circuit representations are becoming the lingua franca to express and reason
about tractable generative and discriminative models. In this paper, we show
how complex inference scenarios for these models that commonly arise in machine
learning -- from computing the expectations of decision tree ensembles to
information-theoretic divergences of deep mixture models -- can be represented
in terms of tractable modular operations over circuits. Specifically, we
characterize the tractability of a vocabulary of simple transformations --
sums, products, quotients, powers, logarithms, and exponentials -- in terms of
sufficient structural constraints of the circuits they operate on, and present
novel hardness results for the cases in which these properties are not
satisfied. Building on these operations, we derive a unified framework for
reasoning about tractable models that generalizes several results in the
literature and opens up novel tractable inference scenarios.
