The simplex

October 25, 2016 — October 25, 2016

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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].