# Deriving the marginal multivariate Dirichlet distribution

I am trying to understand how my professor (see derivation below) has derived the multivariate marginal distribution of a subvector of $$\theta_j$$´s from a Dirichlet distribution. I understand everything really, until the third line. I understand that he wants the integral to represent a unnormalised beta integral and hence he have to divide $$\theta^{\alpha_{k-1}-1}_{k-1}$$ and $$(1-\theta_1 ... - \theta_{k-1})^{\alpha_{k}-1}$$ by their sum, i.e $$\theta^{\alpha_{k-1}-1}_{k-1} + (1-\theta_1 ... - \theta_{k-1})^{\alpha_{k}-1}$$

If I understand correctly, the sum is equal to $$(1-\theta_1 ... - \theta_{k-1})^{\alpha_{k-2}+\alpha_{k-1}-2}$$, but how do we get that this is in fact the sum?

• Thanks for the edit @Xi'an Commented Nov 10, 2020 at 13:08

Since$$p(\theta_1,\ldots,\theta_{k-1})\propto (1-\theta_1-\cdots-\theta_{k-1})^{\alpha_{k}-1}\prod_{i=1}^{k-1} \theta_i^{\alpha_i-1}$$over the $$\mathbb R^k$$-simplex, $$\mathfrak S = \left\{(\theta_1,\ldots,\theta_{k-1})\in\mathbb R^{k-1}_+\,;\,\sum_{i=1}^{k-1} \theta_i\le 1\right\}$$ integrating out $$\theta_{k-1}$$ produces the marginal density of $$(\theta_1,\ldots,\theta_{k-2})$$: \begin{align} p(\theta_1,\ldots,\theta_{k-2})&\propto \int_0^{1-\theta_1-\cdots-\theta_{k-2}} (1-\theta_1-\cdots-\theta_{k-1})^{\alpha_k-1}\prod_{i=1}^{k-1} \theta_i^{\alpha_i-1}\,\text d\theta_{k-1}\\ &= \prod_{i=1}^{k-2} \theta_i^{\alpha_i-1}\, \int_0^{1-\theta_1-\cdots-\theta_{k-2}} (1-\theta_1-\cdots-\theta_{k-1})^{\alpha_{k}-1}\,\theta_{k-1}^{\alpha_{k-1}-1}\,\text d\theta_{k-1}\\ \end{align} Given the upper bound on $$\theta_{k-1}$$ found in the integral, one can rather naturally consider the change of variable $$\theta_{k-1}=(1-\theta_1-\cdots-\theta_{k-2})\eta$$ leading to \begin{align} p(\theta_1,\ldots,\theta_{k-2}) &= \prod_{i=1}^{k-2} \theta_i^{\alpha_i-1}\, \int_0^1 (1-\theta_1-\cdots-\theta_{k-2})^{\alpha_{k-1}-1}(1-\eta)^{\alpha_{k-1}-1}\\ &\qquad\times(1-\theta_1-\cdots-\theta_{k-2})^{\alpha_{k-1}-1}\eta^{\alpha_{k-1}-1}\, (1-\theta_1-\cdots-\theta_{k-2})\text d\eta\\ &=\prod_{i=1}^{k-2} \theta_i^{\alpha_i-1}\,(1-\theta_1-\cdots-\theta_{k-2})^{\alpha_{k-1}+\alpha_k-1}\, \underbrace{\int_0^1 (1-\eta)^{\alpha_{k}-1}\eta^{\alpha_{k-1}-1}\,\text d\eta}_\text{constant in \theta}\\ &\propto\prod_{i=1}^{k-2} \theta_i^{\alpha_i-1}\,(1-\theta_1-\cdots-\theta_{k-2})^{\alpha_{k-1}+\alpha_k-1}\, \end{align}
• +1, I wonder if there's a typo at the first line after "leading to..."; shouldn't it be $(1 - \eta)^{\alpha_{k}-1}$ rather than $(1 - \eta)^{\alpha_{k-1}-1}$? Commented Apr 26, 2022 at 6:29