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.
[Math] the moment generating function of Dirichlet distribution
probabilitystatistics
Best Answer
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