The simplex


The space of convex combinations of things. Hacks for it.

Simulating uniformly from the simplex

This one is apparently “folk wisdom”.

But say you wish to simulate a vector drawn uniformly from the \(n\)-simplex.

  • simulate \(n\) random uniform variables on the unit interval, \((u_1,_u_2,\dots,u_n)\)

  • Sort them in decreasing order, \((u'_1,_u'_2,\dots,u'_n)\)

  • Your random vector is \((u'_1-0, u'_2-u'_1, u'_3-u'_2,\dots,u'_n-u'_{n-1})\)

This surprisingly relates to Dirichlet variables. Can you see how?

References

Friedman, Greg. 2008. “An Elementary Illustrated Introduction to Simplicial Sets.” September 24, 2008. http://arxiv.org/abs/0809.4221.

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