Implement the zero-inflated dirichlet distribution

Dirichlet regression is possible in brms using the dirichlet family. But this requires that outcomes be non-zero.

Election outcomes are one case where outcomes are distributed according to the dirichlet distribution (on the probability scale and sum to 1), but that exhibit considerable zero-inflation. This is often true where parties do not stand for whatever reason.

Rather than fudge this by replacing 0 with a tiny number, it would be good to be able to model the zero-inflation. Effectively, this would be a multinomial extension of the zero-inflated beta distribution in the same way that the dirichlet distribution is a multinomial extension of the standard beta distribution.

At present, this is possible using the zadr() function in the Compositional package. But this lacks a lot of functionality and is Frequentist, not Bayesian.

An accompanying paper for the Compositional package is available here. There is also another good paper detailing zero-inflated dirichlet regression in the context of microbiome data here.