# The simplex

October 25, 2016 — October 25, 2016

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

## 1 Simulating uniformly from the simplex

This one is apparently “folk wisdom”.

But say I 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, \((u'_1,_u'_2,\dots,u'_n)\)
- The random vector is \((u'_1-0, u'_2-u'_1, u'_3-u'_2,\dots,u'_n-u'_{n-1})\)

## 2 Simulating Dirichlet distributions

See Dirichlet variables, which are distributions over the simplex. C&C gumbel max tricks.

## 3 References

Friedman. 2008. “An Elementary Illustrated Introduction to Simplicial Sets.”

*arXiv:0809.4221 [Math]*.