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I'm working on a problem for an hour and I wanted to get some hints. Suppose:

$y_1, y_2, y_3, y_4 \sim Dir(\alpha_1, \alpha_2, \alpha_3, \alpha_4)$

what is the distribution of $\frac{y_1}{y_1 + y_2}$ ?

My guess is that distribution should be $Beta(\alpha_1, \alpha_2)$

Could you guys give me some hints on how to show it?

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  • $\begingroup$ What is the beta distribution specified by four parameters? $\endgroup$
    – Mankind
    Commented May 7, 2015 at 17:26
  • $\begingroup$ @HowDoIMath, my bad! it's dirichlet. $\endgroup$
    – Linda
    Commented May 7, 2015 at 17:35

2 Answers 2

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Well, actually I'm also looking for this answer. I've found a tool in here: http://research.microsoft.com/en-us/um/cambridge/projects/infernet/codedoc/html/M_MicrosoftResearch_Infer_Distributions_Beta_op_Division.htm

I'm not sure how it can perform the Beta.Division operator, but I think it may help you. Besides, I find out a paper. It also seems helpful.

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By the story of the dirichlet distribution, given $X_{1} \sim \Gam(\alpha_{1}, 1)$, it follows that the distribution of $y_{1}$ can be expressed as: \begin{equation*}\begin{aligned} y_{1} &\sim \frac{X_{1}}{\sum_{i = 1}^{4} X_{i}} \end{aligned}\end{equation*}

Therefore, we can write the distribution of $\alpha = \frac{y_{1}}{y_{1} + y_{2}}$ as: \begin{equation*}\begin{aligned} \alpha &\sim \frac{y_{1}}{y_{1} + y_{2}}\\ &\sim \frac{X_{1}/\sum_{i = 1}^{4} X_{i}}{X_{1}/\sum_{i = 1}^{4} X_{i} + X_{2}/\sum_{i = 1}^{4} X_{i}}\\ &\sim \frac{X_{1}}{X_{1} + X_{2}}\\ &\sim \Dir(\alpha_{1}, \alpha_{2})\\ &\sim \Beta(\alpha_{1}, \alpha_{2}) \end{aligned}\end{equation*}

Which shows that $\alpha \sim \Beta(\alpha_{1}, \alpha_{2})$.

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