I want to set up a model that will rely on something similar to a zero-inflated Dirichlet distribution. As such, I'm trying to figure out how a zero-inflated Dirichlet distribution is set up from the start.
The Dirichlet distribution exists as the normalization of a product of gamma RV's to the unit simplex. $$ X \sim \prod_{\ell = 1}^d\mathcal{G}(x_{\ell}\mid\alpha_{\ell}, 1) $$ Let $r = \lVert X\rVert_{1}$, $y_{\ell} = x_{\ell} / \lVert X \rVert_1$ for $\ell = 1$ to $d-1$, and $y_d = 1 - \sum_{\ell = 1}^{d-1}y_{\ell}$. Then, $$ (r,Y) \sim \prod_{\ell = 1}^d \mathcal{G}(ry_{\ell}\mid\alpha_{\ell}, 1)\times r^{d-1} $$ $r^{d-1}$ comes from the determinant of the Jacobian of the transformation from $X$ to $r,Y$. Then $f(Y) = \int_0^{\infty}f(r,Y)dr = \text{Dir}(Y\mid\alpha)$
I'm trying to figure out the above process in the presence of zero-inflation.
Something like $$ x_{\ell}\mid\alpha_{\ell}, \pi_{\ell} \sim (1 - \pi_{\ell})\delta_{(0)} + \mathcal{G}(x_{\ell}\mid\alpha_{\ell}, 1) $$ Or $$ x_{\ell}\mid\alpha_{\ell},\nu_{\ell} \sim \mathcal{G}(x_{\ell}\mid\alpha_{\ell}, 1)^{\nu_{\ell}}\delta^{1 - \nu_{\ell}} $$
Is there any literature describing how the zero-inflated Dirichlet distribution is created? I want to be able to reduce it back to the component gammas for Bayesian analysis.
