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I want to find a simpler form or closed form for the following integral: $$ \int_A \,\prod_{t=1}^T f_{\Gamma(a,\theta_t)}(x_t) \,d\mathbf{x} $$ where $A$ is the simplex $$ A = \{\mathbf{x} \in \mathbb{R}^T : \sum_{t=1}^T x_t = 1 \hbox{ and } x_t \geq 0\} $$ and $f_{\Gamma(a,\theta_t)}$ is the Gamma density $$ f_{\Gamma(a,\theta_t)}(x_t) = \theta_t^a x_t^{a-1} \exp(-x_t \theta_t) / \Gamma(a) . $$ We require that $a$ and $\theta_t$ are positive real numbers.

It seems that this shouldn't be too hard, but I've really been stumped. I have tried various methods from the paper by Wolpert and Wolf mentioned in this blog post - but when I follow the algorithm described in that paper, I end up running into even harder analytical difficulties at the end, when it is required to take an inverse Laplace transform of some product.

I feel that there must be a lower-tech way to do this! Thanks in advance for any help.

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1 Answer 1

Indeed, your integral is $g(1)$ where the Laplace trnsform $L(g)$ of $g$ is such that, for every positive $s$, $$ L(g)(s)=\prod_{t=1}^T\left(\frac{\theta_t}{\theta_t+s}\right)^a. $$ For some simple values of $a$, one can decompose this rational fraction of $s$ into simple elements and inverse each term. If I am not mistaken, when $a=1$ and all the $\theta_t$ are distinct, the result is $$ g(1)=\sum_t\theta_t\mathrm{e}^{-\theta_t}\prod_{i\ne t}\frac{\theta_i}{\theta_i-\theta_t}. $$ Unfortunately, I do not know a simple way of inverting $L(g)$ in the general case.

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Thank you for this answer! Unfortunately this is exactly the approach I was taking to solve the problem myself, but so far I've been stuck at the same spot. At least now I know I'm probably not missing something totally obvious. –  N F May 31 '11 at 18:36
Yes, I wrote this at least partly as a kind of note to myself... (By the way, why the name Not Durrett? Is this a kind of personal motto?) –  Did May 31 '11 at 18:57
:) Well, I started using "Not Durrett" online in a couple of math forums last year when I was going through Durrett's probability textbook in a class. He seems to have a bit better grasp of probability than I do, so I was "Not Durrett". His textbook is certainly frustrating at times for someone who doesn't already know everything though... –  N F Jun 2 '11 at 3:53

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