Simplex Point Picking: Distribution of x_i Over (0,1)

[noowutah]

I have an application where I need to pick a probability distribution (x_{1},\ldots,x_{n}) at random and uniformly from the simplex of all points for which the coordinates add up to 1, i.e. \sum_{i=1}^{n}x_{i}=1. Surprisingly, I didn't find much about simplex point picking on the internet, but http://en.wikipedia.org/wiki/User:Skinnerd/Simplex_Point_Picking appears to address this issue. Skinnerd suggests to pick individual members of (y_{1},\ldots,y_{n}) randomly from a uniform distribution over the interval (0,1) and then take x_{i}=\frac{\ln{}y_{i}}{\sum{}\ln{}y_{i}}. So far so good (although, why does he need the minus sign in his x_{i}=-\ln{}y_{i}?).

My question is: what is the distribution of x_{i} over the interval (0,1), i.e. what is the probability P(a<x<b) that one of these coordinates is in (a,b)\subseteq{}(0,1)?

 

[haruspex]

If X_i = x, that leaves a hyperpyramid $\Sigma_{i\neq i}X_j = 1 - x$. Can't you make the p.d.f of X_i proportional to the volume of that?

 

[noowutah]

volume of n-dimensional simplex

Great idea! I am a little confused about terminology. Hyperpyramid at http://physicsinsights.org/pyramids-1.html seems to mean that the height of the pyramid is the same as the side of the base -- which is not what we want here. We want something more like a generalization for n dimensions of a pentatope, see http://mathworld.wolfram.com/Pentatope.html. Mathworld advises on the volume of a simplex in n dimensions at http://mathworld.wolfram.com/Cayley-MengerDeterminant.html. What haruspex is suggesting, as I see it, is that

P(0<x<b)=S(\sqrt{2})-S(\sqrt{2}(1-b))

where S(z) is the volume of a simplex in n dimensions whose side length is z. In our case, z=\sqrt{2} because x_{1}+\ldots{}+x_{n}=1.

 

[haruspex]

Great idea! I am a little confused about terminology. Hyperpyramid at http://physicsinsights.org/pyramids-1.html seems to mean that the height of the pyramid is the same as the side of the base -- which is not what we want here. We want something more like a generalization for n dimensions of a pentatope, see http://mathworld.wolfram.com/Pentatope.html.

Seems that simplex is the word I should have used.

Mathworld advises on the volume of a simplex in n dimensions at http://mathworld.wolfram.com/Cayley-MengerDeterminant.html. What haruspex is suggesting, as I see it, is that

P(0<x<b)=S(\sqrt{2})-S(\sqrt{2}(1-b))

where S(z) is the volume of a simplex in n dimensions whose side length is z. In our case, z=\sqrt{2} because x_{1}+\ldots{}+x_{n}=1.

Not sure that's quite what I was saying. For a start, there should be a ratio of volumes in there.
I think I'm saying the p.d.f., f(x) = S_n-1((1-x)√2)/S_n(√2), or maybe the subscripts should be n, n+1. You'd then to integrate that to get the interval probability.

 

[noowutah]

Yes, indeed, it should be a ratio, not a difference. Thanks, haruspex!