Parameterising a latent simplex

Modeling
8

I’d try the first approach you had of just making it an additive model. You can basically turn anything that looks like a constant into a regression. That is, if y[n] ~ gamma(a[n], 1), then y / sum(y) ~ Dirichleta(a).

You can turn anything into a regression. In this case, just replace the a[n] with the result of a regression. Let’s say you have a matrix of predictors x of size N x K (i.e., K covariates per entry). Then you can set up a regression

data {
  int<lower=0> N;
  int<lower=0> K;
  matrix[N, K] x;
}
parameters {
  vector[K] beta;
  vector<lower=0>[N] phi;
}
transformed parameters {
  simplex[N] theta = phi / sum(phi);
}
model {
    vector[N] a = exp(x * beta);
    phi ~ gamma(a, 1);
    // implies theta ~ Dirichlet(exp(x * beta))