Sample uniformly at random from an n-dimensional unit simplex

Question

Sampling uniformly at random from an n-dimensional unit simplex is the fancy way to say that you want n random numbers such that

• they are all non-negative,

Answer 1

I'm with zdav: the Dirichlet distribution seems to be the easiest way ahead, and the algorithm for sampling the Dirichlet distribution which zdav refers to is also presented on the Wikipedia page on the Dirichlet distribution.

Implementationwise, it is a bit of an overhead to do the full Dirichlet distribution first, as all you really need is n random Gamma[1,1] samples. Compare below
Simple implementation

SimplexSample[n_, opts:OptionsPattern[RandomReal]] :=
  (#/Total[#])& @ RandomReal[GammaDistribution[1,1],n,opts]

Full Dirichlet implementation

DirichletDistribution/:Random`DistributionVector[
 DirichletDistribution[alpha_?(VectorQ[#,Positive]&)],n_Integer,prec_?Positive]:=
    Block[{gammas}, gammas = 
        Map[RandomReal[GammaDistribution[#,1],n,WorkingPrecision->prec]&,alpha];
      Transpose[gammas]/Total[gammas]]

SimplexSample2[n_, opts:OptionsPattern[RandomReal]] := 
  (#/Total[#])& @ RandomReal[DirichletDistribution[ConstantArray[1,{n}]],opts]

Timing

Timing[Table[SimplexSample[10,WorkingPrecision-> 20],{10000}];]
Timing[Table[SimplexSample2[10,WorkingPrecision-> 20],{10000}];]
Out[159]= {1.30249,Null}
Out[160]= {3.52216,Null}

So the full Dirichlet is a factor of 3 slower. If you need m>1 samplepoints at a time, you could probably win further by doing (#/Total[#]&)/@RandomReal[GammaDistribution[1,1],{m,n}].