how about this idea for sampling in the simplex?

[Paul Johnson]

At some point, we will just need to master simulation of multivariate
pdfs on generic domains :)

Until then, I wonder what you think about this.  Consider the simplex
that is the triangle with vertices (0 0 1) (0 1 0) (1 0 0).

We want points selected in there in an equally likely way.  As noted in
this thread, it is obvious that x3=1-x1-x2, so this reduces to selecting
x1 and x2.  

Suppose we know x1=X1.  Then it is necessary that x2 be chosen uniformly
on 1-X1. Correct?  Hence, the conditional probability density function
for x2 has to be 

 p(x2|x1)= 1/(1-x1).

Now we need to find a probability model for x1 such that the joint pdf
is a constant, or

  p(x1,x2)= p(x1)*p(x2|x1)=C.  

That's the goal, since we want sampling that is uniform. Solving for
p(x1), I get

  p(x1)= (1-x1)*C.

In other words, the probability of getting a large value of x1 is
negatively proportional to the value of x1. C is the reciprocal of the
area of the triangle.  On the same envelope I've done these other
calculations, I get the area of the triangle is sqrt(3).

So, how do we get a draw from this distribution, p(x1)=(1-x1)/sqrt(3),
from tools in swarm? .  

Well, as McCoy says on Star Trek,"dammit, Jim, I'm just a country
doctor," but consider this possibility. At first, when I forgot to write
down the C part, it was obvious. If we just need a draw from p(x1)=1-x1,
then

Step 1. Draw a uniform variate s0 from [0,1].

Step 2. Draw a uniform variate s1 from [s0,1].   

But we have to take into account C?  Well, I stared at this a long time
and concluded that we don't have to, since it is a constant, it does not
alter the relative likelihood of the outcomes.

So I believe that this draw s1 has the desired distribution.

I usually make some colossal mistake when I try to solve these problems
for which I'm not well prepared, but in this thread I won't be the first
;)

-- 
Paul E. Johnson                       email: address@hidden
Dept. of Political Science            http://lark.cc.ukans.edu/~pauljohn
University of Kansas                  Office: (785) 864-9086
Lawrence, Kansas 66045                FAX: (785) 864-5700

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