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 Post subject: Integral over standard n-simplexPosted: Tue, 28 Sep 2010 13:20:09 UTC
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Joined: Tue, 28 Sep 2010 12:52:16 UTC
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Given a standard n-1 simplex in :

where n is a positive integer, and a vector with n elements ,

I need to evaluate the integral

(1)

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 with respect to .

Similar integrals that have well-known solutions are:

(2)

and

(3)

where 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).

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 Post subject: Re: Integral over standard n-simplexPosted: Tue, 28 Sep 2010 13:24:58 UTC
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BayesianGuy wrote:
The integral is taken over the standard n-1 simplex with respect to .

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

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 Post subject: Re: Integral over standard n-simplexPosted: Tue, 28 Sep 2010 18:37:58 UTC
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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 with respect to .

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

Sorry, I meant that the integral is taken with respect to the to Lebesgue measure on , e.g. . 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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