Integral over standard n-simplex

BayesianGuy · **Joined:** Tue, 28 Sep 2010 12:52:16 UTC **Posts:** 2

Given a standard n-1 simplex $\Omega$ in $\mathbb{R}^n$ :
$\Omega = \left\{ {{\textbf{y}}:\,\,\,\,{y_i} \ge 0,\,\,\,\,1 = \sum\limits_{i = 1}^n {{y_i}} } \right\}$
where n is a positive integer, and a vector $\textbf{a}$ with n elements a_i>1

,

I need to evaluate the integral

(1) $G({\textbf{a}},m,s,t) = \int\limits_{\textbf{y} \in \Omega } {{{\left( {\sum\limits_{i = 1}^m {{y_i}} } \right)}^s}{{\left( {\sum\limits_{i = m}^n {{y_i}} } \right)}^t}\left( {\prod\limits_{i = 1}^n {{y_i}^{{a_i} - 1}} } \right)dy}$

where t and s are real non-negative numbers, and m is an integer between 1 and n.

The integral is taken over the standard n-1 simplex $\Omega$ with respect to dy = dy_1 dy_2...dy_n

.

Similar integrals that have well-known solutions are:

(2) $\int\limits_{y \in \Omega } {\prod\limits_{i = 1}^n {{y_i}^{{a_i} - 1}} dy} = \frac{{\prod\limits_{i = 1}^n {\Gamma ({a_i})} }}{{\Gamma \left( {\sum\limits_{i = 1}^n {{a_i}} } \right)}}$

and

(3) $\int\limits_{y \in \Omega } \left(\sum\limits_{i = 1}^m y_i \right)^s \left( \sum\limits_{i = m + 1}^n y_i \right)^t \left( \prod\limits_{i = 1}^n y_i^{a_i - 1} \right)dy$ $= \frac{\prod\limits_{i = 1}^n {\Gamma ({a_i})} } {\Gamma \left( {s + t + \sum\limits_{i = 1}^n {{a_i}} } \right)} \frac{\Gamma \left( {s + \sum\limits_{i = 1}^m {{a_i}} } \right)} {\Gamma \left( {\sum\limits_{i = 1}^m {{a_i}} } \right)} \frac{\Gamma \left( {t + \sum\limits_{i = m + 1}^n {{a_i}} } \right)} {\Gamma \left( {\sum\limits_{i = m + 1}^n {{a_i}} } \right)}$

where $\Gamma$ is the Gamma function.

Note that the only difference between (1) and (3) is that the second sum in (3) goes from m+1 to n instead of from m to n. This difference is however important, surely making the solution to (1) more complicated than that of (3).

outermeasure · **Posted:** Tue, 28 Sep 2010 13:24:58 UTC

BayesianGuy wrote:

The integral is taken over the standard n-1 simplex $\Omega$ with respect to dy = dy_1 dy_2...dy_n

.

That is impossible. You don't have a nontrivial n-dimensional volume measure on an (n-1)

-simplex.

BayesianGuy · **Joined:** Tue, 28 Sep 2010 12:52:16 UTC **Posts:** 2

outermeasure wrote:

BayesianGuy wrote:

The integral is taken over the standard n-1 simplex $\Omega$ with respect to dy = dy_1 dy_2...dy_n

.

That is impossible. You don't have a nontrivial n-dimensional volume measure on an (n-1)

-simplex.

Sorry, I meant that the integral is taken with respect to the to Lebesgue measure on $\mathbb{R}^{n-1}$ , e.g. $dy = dy_1 dy_2...dy_{n-1}$ . This is the most common measure with respect to which the PDF of a Dirichlet distribution is defined and my integral is an attempt to calculate an expectation with respect to such a distribution (although a normalization constant has been omitted).

Thanks for spotting the error.

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Integral over standard n-simplex

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