The simplex

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