  Neural networks and other machine learning models compute continuous
representations, while humans communicate mostly through discrete symbols.
Reconciling these two forms of communication is desirable for generating
human-readable interpretations or learning discrete latent variable models,
while maintaining end-to-end differentiability. Some existing approaches (such
as the Gumbel-Softmax transformation) build continuous relaxations that are
discrete approximations in the zero-temperature limit, while others (such as
sparsemax transformations and the Hard Concrete distribution) produce
discrete/continuous hybrids. In this paper, we build rigorous theoretical
foundations for these hybrids, which we call "mixed random variables." Our
starting point is a new "direct sum" base measure defined on the face lattice
of the probability simplex. From this measure, we introduce new entropy and
Kullback-Leibler divergence functions that subsume the discrete and
differential cases and have interpretations in terms of code optimality. Our
framework suggests two strategies for representing and sampling mixed random
variables, an extrinsic ("sample-and-project") and an intrinsic one (based on
face stratification). We experiment with both approaches on an emergent
communication benchmark and on modeling MNIST and Fashion-MNIST data with
variational auto-encoders with mixed latent variables.
