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Let$$X=(X_1,\cdots,X_k)\sim Dir(\alpha_1,\cdots,\alpha_k)$$

According to this reference, the independence of the two vectors, $$\bigg(\frac{X_1}{X_1+\cdots+X_j},\cdots,\frac{X_j}{X_1+\cdots+X_j}\bigg)$$and$$\bigg(\frac{X_{j+1}}{X_{j+1}+\cdots+X_k},\cdots,\frac{X_k}{X_{j+1}+\cdots+X_k}\bigg),$$ follows from the neutrality and aggregation property of the Dirichlet distribution.

I'm wondering to what extent can this result be generalized?

More specifically, for three-subvector cases, let $\{\pi_1,\pi_2,\pi_3\}$ be a partition over $\{1,\cdots,k\}$. e.g., $k=3, \pi_1=\{1\},\pi_2=\{2\}$, and $\pi_3=\{3\}$. Can we say the following normalized subvectors are mutually independent$$\bigg(\frac{X_k}{\sum_{j\in\pi_1}X_j}\bigg)_{k\in\pi_1},\bigg(\frac{X_k}{\sum_{j\in\pi_2}X_j}\bigg)_{k\in\pi_2},\bigg(\frac{X_k}{\sum_{j\in\pi_3}X_j}\bigg)_{k\in\pi_3}$$

Xi'an
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Andeanlll
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1 Answers1

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The property stems from the fact that a Dirichlet random variable $$X=(X_1,\cdots,X_k)\sim Dir_k(\alpha_1,\cdots,\alpha_k)$$ is distributed as$$X=(Y_1,\cdots,Y_k)/\sum_{i=1}^k Y_i$$ when $$Y_1,\cdots,Y_k\sim \prod_{i=1}^k \mathcal{Ga}(\alpha_i).$$ The result follows from the independence of the $Y_i$'s since $$\bigg(\frac{X_k}{\sum_{j\in\pi_1}X_j}\bigg)_{k\in\pi_1},\bigg(\frac{X_k}{\sum_{j\in\pi_2}X_j}\bigg)_{k\in\pi_2},\bigg(\frac{X_k}{\sum_{j\in\pi_3}X_j}\bigg)_{k\in\pi_3}$$is distributed as$$\bigg(\frac{Y_k}{\sum_{j\in\pi_1}Y_j}\bigg)_{k\in\pi_1},\bigg(\frac{Y_k}{\sum_{j\in\pi_2}Y_j}\bigg)_{k\in\pi_2},\bigg(\frac{Y_k}{\sum_{j\in\pi_3}Y_j}\bigg)_{k\in\pi_3}$$

Andeanlll
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Xi'an
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  • Thanks for the answer. I got your point. So, basically, the mutual independence holds for any partition, right? But what about the singleton cases? when $\pi_i=\{i\}$, can we say $X_i$ is independent of $X_j$ for all $i\neq j$? I don't get how this is possible because $X_i$'s should sum up to 1... – Andeanlll Feb 20 '20 at 02:15
  • Oh, right :) Thanks! – Andeanlll Feb 20 '20 at 13:22