  Simplex-valued data appear throughout statistics and machine learning, for
example in the context of transfer learning and compression of deep networks.
Existing models for this class of data rely on the Dirichlet distribution or
other related loss functions; here we show these standard choices suffer
systematically from a number of limitations, including bias and numerical
issues that frustrate the use of flexible network models upstream of these
distributions. We resolve these limitations by introducing a novel exponential
family of distributions for modeling simplex-valued data - the continuous
categorical, which arises as a nontrivial multivariate generalization of the
recently discovered continuous Bernoulli. Unlike the Dirichlet and other
typical choices, the continuous categorical results in a well-behaved
probabilistic loss function that produces unbiased estimators, while preserving
the mathematical simplicity of the Dirichlet. As well as exploring its
theoretical properties, we introduce sampling methods for this distribution
that are amenable to the reparameterization trick, and evaluate their
performance. Lastly, we demonstrate that the continuous categorical outperforms
standard choices empirically, across a simulation study, an applied example on
multi-party elections, and a neural network compression task.
