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I want to find the moment generating function (or the Laplace transform) of the Dirichlet distribution. I know the moments can be found using the gamma functions as follows :$$E\left[\prod_{i=1}^K x_i^{\beta_i}\right]=\frac{B\left(\boldsymbol{\alpha}+\boldsymbol{\beta}\right)}{B\left(\boldsymbol{\alpha}\right)}=\frac{\Gamma\left(\sum_{i=1}^{n}\alpha_{i}\right)}{\Gamma\left(\sum_{i=1}^{n}\alpha_{i}+\beta_{i}\right)}\times\prod_{i=1}^{n}\frac{\Gamma\left(\alpha_{i}+\beta_{i}\right)}{\Gamma\left(\alpha_{i}\right)},$$ but what I am really interested in is the functional form of the MGF (or the Laplace transform) so that it can be used to find other sampling distributions thereof or any other transformation of the Dirichlet also.

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2 Answers 2

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You may find the MGF of the Dirichlet by consulting pages 15 and 16, here:

https://mast.queensu.ca/~communications/Papers/msc-jiayu-lin.pdf

This work is not mine. It is simply a reference I found online.

Original, now dead: http://www.mast.queensu.ca/~web/Papers/msc-jiayu-lin.pdf

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    $\begingroup$ the link is dead by now $\endgroup$
    – deemel
    Aug 14, 2019 at 10:29
  • $\begingroup$ @deemel, I found a new link to the document online. I hope but can't guarantee this is permanent. $\endgroup$
    – RMurphy
    Aug 15, 2019 at 1:01
  • $\begingroup$ Looks like it has moved again, to here: mast.queensu.ca/~communications/Papers/msc-jiayu-lin.pdf. For future reference, it's "On The Dirichlet Distribution," a 2016 Master of Science report by Jiayu Lin. $\endgroup$
    – Tom Loredo
    Aug 27, 2021 at 4:08
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You may compute it along the following lines: let $E_n$ be the set: $$E_n = \left\{(x_1,\ldots,x_n): x_i\geq 0, \sum_{i=1}^{n}x_i=1\right\}\subset\mathbb{R}^n.$$ Then: $$ \int_{E_n} x_1^{\alpha_1-1}\cdot\ldots\cdot x_{n}^{\alpha_n-1}\,d\mu = \frac{\prod_{i=1}^{n}\Gamma(\alpha_i)}{\Gamma(\alpha_1+\ldots+\alpha_n)}$$ for any choice of the $\alpha_i$ such that $\text{Re}(\alpha_i)>0$.

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  • $\begingroup$ I can compute the moments, but what I really need is the moment generating function M(t1,t2,t3,...,tn). $\endgroup$
    – Udita
    Jun 2, 2015 at 15:53
  • $\begingroup$ If you are able to compute moments you are able to compute the moment generating function too, @Udita. $\endgroup$ Jun 2, 2015 at 15:55
  • $\begingroup$ Hi Jack, yes, that is true. The moments can be used to know the value of differential of MGF at 0. But I want to know if there is a nice mathematical form of M(t1,t2,..,tn) for arbitrary t's. $\endgroup$
    – Udita
    Jun 2, 2015 at 16:03
  • $\begingroup$ @Udita: again, if you know the Taylor coefficients in zero of an entire function, you know the values of such a function everywhere. $\endgroup$ Jun 2, 2015 at 16:06
  • $\begingroup$ That's true. Thanks. $\endgroup$
    – Udita
    Jun 2, 2015 at 16:08

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