The Living Thing / Notebooks :

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.

  1. simulate \(n\) random uniform variables on the unit interval, \((u_1,_u_2,\dots,u_n)\)
  2. Sort them in decreasing order, \((u'_1,_u'_2,\dots,u'_n)\)
  3. Your random vector is \((u'_1-0, u'_2-u'_1, u'_3-u'_2,\dots,u'_n-u'_{n-1})\)

Refs

Frie08
Friedman, G. (2008) An elementary illustrated introduction to simplicial sets. arXiv:0809.4221 [Math].