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From: Ray Koopman on 9 Aug 2010 16:54
I've used the following theorem in numeric work for some time,
and it certainly seems to be true, but I've never seen a proof.
Can anyone point to one (or give it, if it's easy)?

"If P is a fixed (n+1) by n matrix whose rows are the coordinates of
the vertices of an n-simplex, and W is a random (n+1)-vector (row)
whose density is Dirichlet(1,...,1), then W*P is uniform in the
simplex."
From: Chip Eastham on 9 Aug 2010 17:29
On Aug 9, 4:54 pm, Ray Koopman <koop...(a)sfu.ca> wrote:
> I've used the following theorem in numeric work for some time,
> and it certainly seems to be true, but I've never seen a proof.
> Can anyone point to one (or give it, if it's easy)?
>
> "If P is a fixed (n+1) by n matrix whose rows are the coordinates of
>  the vertices of an n-simplex, and W is a random (n+1)-vector (row)
>  whose density is Dirichlet(1,...,1), then W*P is uniform in the
>  simplex."

Perhaps it's a simple application of barycentric
coordinates, but I don't know what "density is
Dirichlet(1,...,1)" means in this context.

regards, chip
From: Ray Koopman on 9 Aug 2010 18:19
On Aug 9, 2:29 pm, Chip Eastham <hardm...(a)gmail.com> wrote:
> On Aug 9, 4:54 pm, Ray Koopman <koop...(a)sfu.ca> wrote:
>
>> I've used the following theorem in numeric work for some time,
>> and it certainly seems to be true, but I've never seen a proof.
>> Can anyone point to one (or give it, if it's easy)?
>>
>> "If P is a fixed (n+1) by n matrix whose rows are the coordinates of
>> the vertices of an n-simplex, and W is a random (n+1)-vector (row)
>> whose density is Dirichlet(1,...,1), then W*P is uniform in the
>> simplex."
>
> Perhaps it's a simple application of barycentric
> coordinates, but I don't know what "density is
> Dirichlet(1,...,1)" means in this context.

I mean that W has a Dirichlet distribution (e.g.,
http://en.wikipedia.org/wiki/Dirichlet_distribution) with all alpha_i
= 1. W contains the coordinates of a random point that is uniformly
distributed in the regular n-simplex whose vertices are given by the
rows of an identity matrix of order n+1. It seems to me that W*P is
then uniform in the simplex whose vertices are in P, but I need
something stronger than just "it seems to me...".
From: Chip Eastham on 9 Aug 2010 19:31
On Aug 9, 6:19 pm, Ray Koopman <koop...(a)sfu.ca> wrote:
> On Aug 9, 2:29 pm, Chip Eastham <hardm...(a)gmail.com> wrote:
>
> > On Aug 9, 4:54 pm, Ray Koopman <koop...(a)sfu.ca> wrote:
>
> >> I've used the following theorem in numeric work for some time,
> >> and it certainly seems to be true, but I've never seen a proof.
> >> Can anyone point to one (or give it, if it's easy)?
>
> >> "If P is a fixed (n+1) by n matrix whose rows are the coordinates of
> >>  the vertices of an n-simplex, and W is a random (n+1)-vector (row)
> >>  whose density is Dirichlet(1,...,1), then W*P is uniform in the
> >>  simplex."
>
> > Perhaps it's a simple application of barycentric
> > coordinates, but I don't know what "density is
> > Dirichlet(1,...,1)" means in this context.
>
> I mean that W has a Dirichlet distribution (e.g.,http://en.wikipedia.org/wiki/Dirichlet_distribution) with all alpha_i
> = 1.  W contains the coordinates of a random point that is uniformly
> distributed in the regular n-simplex whose vertices are given by the
> rows of an identity matrix of order n+1. It seems to me that W*P is
> then uniform in the simplex whose vertices are in P, but I need
> something stronger than just "it seems to me...".

A continuous uniform distribution is one that
has a constant probability density function over
the region where it is positive.

The Wikipedia article, describing the Dirichlet
distribution on a {K-1}-simplex, gives a formula
for the pdf at (x_1,...,x_K) which is a constant
for components of vector alpha identically 1:

(1/B(alpha)) * PRODUCT x_i^{alpha_i - 1} [i=1,..K]

and gives the details for calculating the constant
of normalization B(alpha).

regards, chip
From: Paul on 9 Aug 2010 19:42
On Aug 9, 4:54 pm, Ray Koopman <koop...(a)sfu.ca> wrote:
> I've used the following theorem in numeric work for some time,
> and it certainly seems to be true, but I've never seen a proof.
> Can anyone point to one (or give it, if it's easy)?
>
> "If P is a fixed (n+1) by n matrix whose rows are the coordinates of
>  the vertices of an n-simplex, and W is a random (n+1)-vector (row)
>  whose density is Dirichlet(1,...,1), then W*P is uniform in the
>  simplex."

You'll want to check this, as my best days are behind me, but ... Let
P_1, ..., P_{n+1} be the rows of P, viewed as points in \Re^n, let
S_1, ..., S_{n+1} be the vertices of the regular simplex described in
your second message, and let A and b be n x n and n x 1 matrices
respectively such that the affine transformation y = A*x + b maps each
vertex of S to the corresponding vertex of P. Then W*P is the image
under the affine transformation of W*S, so the probability of any
subset of P occurring is the probability of its preimage occurring
when you weight the vertices of S using W. You already know that
distribution is uniform, so the probability of the preimage is the
volume of the preimage divided by the volume of S, and that ratio is
the same as the volume of the image divided by the volume of P since
the volume of the image is the determinant of A (don't care what that
is, just that it's not zero) times the volume of the preimage (and
ditto for P resp. S).

Hope that made sense -- USENET does not make expressing math easy.

/Paul
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