# 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 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 $$(u'_1-0, u'_2-u'_1, u'_3-u'_2,\dots,u'_n-u'_{n-1})$$

## Simulating Dirichlet distributions

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

## References

Friedman, Greg. 2008. arXiv:0809.4221 [Math], September.

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