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From: Chip Eastham on 9 Aug 2010 19:48 On Aug 9, 7:31 pm, Chip Eastham <hardm...(a)gmail.com> wrote: > 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 So now we need to consider how the probability density function for W*P relates to that of W. But since its a linear transformation, the change of variables involves a constant Jacobian of the tranformation. --c
From: Ray Koopman on 10 Aug 2010 04:07 On Aug 9, 4:42 pm, Paul <paul_ru...(a)att.net> 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." > > 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 Thank-you, Paul and Chip. That's what I needed.
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