I imagine it quite sparse, but with rare categories that are small but not tend to absolute 0 (like in the case of a Dirichlet parameter < 1)
Do you mean
min(exp(X_beta))
because X_beta can have negative numbers and precision delta could become negative?
However I think the problem could be another one, I observed that a unique precision parameter does not fit all categories, and the dirichlet_multinomial is not flexible enough for this biological data (one precision for all categories may not be sufficient. Probably this is the reason why people model sequencing data with NB rather than multinomial, because the gene-wise overdispersion structure is quite complex). See, a dedicated post: