From: Grant Izmirlian <izmirlian_at_nih.gov>

Date: Mon 16 Oct 2006 - 14:57:58 GMT

Duncan Murdoch's definition is _the_ only one that I know. X is Uniform on A means E phi(X) = \int_A phi(x) dx / \int_A dx, so that the probability density is equal to 1/ \int_A dx everwhere on the set A.

for (k in 1:3) X[,k] <- rgamma(n, shape=A[k], rate=1) S <- X %*% rep(1, d)

Y <- X/S

R-help@stat.math.ethz.ch mailing list

https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. Received on Tue Oct 17 01:19:10 2006

Date: Mon 16 Oct 2006 - 14:57:58 GMT

So, Alberto, you didn't see my post? If Y has d independent components that are gamma distributed with common rate and shapes A_1, A_2, ..., A_d, then X, given by the components of Y divided by their sum has distribution Dirichlet(A_1, A_2, ..., A_d). If you want Uniform on the d-simplex, then use A_1 = A_2 = ... = A_d = 1 (just as Duncan said)

original message:

Duncan Murdoch's definition is _the_ only one that I know. X is Uniform on A means E phi(X) = \int_A phi(x) dx / \int_A dx, so that the probability density is equal to 1/ \int_A dx everwhere on the set A.

By the way, another way to simulate X ~ Dirichlet(A1, A2, ..., Ad) is to generate d independent gamma variables having equal rate parameter (doesn't matter, so why not 1) and shape parameters A1, A2, ..., Ad Then the vector of components divided by their sum is the desired Dirichlet:

n <- 100000 d <- 3 # for three numbers that add to one ( the unit simplex in R^3) A <- rep(1, 3) # for uniform X <- matrix(0, n, d)

for (k in 1:3) X[,k] <- rgamma(n, shape=A[k], rate=1) S <- X %*% rep(1, d)

Y <- X/S

Present example will simulate n independant 3 vectors, each having non-negative components summing to 1, and having a distribution assigning equal mass to every possible value.

Changing d and the components of A will provide an arbitrary Dirichlet on the unit simplex in R^d

Grant Izmirlian

**NCI
**

>> Duncan Murdoch wrote "Another definition of uniform is to have equal

*>> density for all possible vectors; the Dirichlet distribution with
**>> parameters (1,1,1) would give you that. "
*

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