Why do you parameterize the second Dirichlet (\bar \omega_c) with a Dirichlet? The alpha parameter isn’t required to be constrained between 0 and 1. I’d just put a normal hierarchical prior as usual then apply a function (exp or softplus) to ensure the output to ensure it’s always larger than 0. Something similar came up once in another discussion of hierarchical estimation of the Dirichlet alpha parameter.
The log transformation at the end also strikes me as odd. It seems like you lose all the properties that make one reach for a Dirichlet in the first place (between 0-1, sum to 1). Do you need to constrain \bar{\nu}_c to be negative? Couldn’t you just directly model what you want in the end rather than going through all the hassle?