# Uniformly distributed n-dimensional probability vectors over a simplex

What's the right way to generate a random probability vector $p={p_1,\ldots,p_n} \in {(0,1)}^n$ where $\sum_i p_i=1$, uniformly distributed over the $(n-1)$-dimensional simplex?

What I have is

Intervals = Table[{0, 1}, {i, n}]
RandomPoint := Block[{a},
a = RandomVariate[UniformDistribution[Intervals]];
a/Total[a]];


But I am unsure that this is correct. In particular, I'm unsure that it's any different from:

RandomPoint := Block[{a},
a = Table[Random[], {i, n}];
a/Total[a]];


And the latter clearly will not distribute vectors uniformly. Is the first code the right one?

• This question may be relevant. Commented Oct 8, 2013 at 11:21
• Thanks, @SjoerdC.deVries. That question seems to suggest that my first code is also incorrect? I'm assuming that that bunch of smart guys would have stumbled upon it. Commented Oct 8, 2013 at 11:42
• Perhaps DirichletDistribution might help?
– chuy
Commented Oct 8, 2013 at 14:03
• That question involved points on a sphere. Your constraint of $\sum{p_i}=1$ is different. Commented Oct 8, 2013 at 14:39
• Some folks here might find this StackOverflow duplicate useful (which is asking exactly the same question, but from a computer science perspective). You'll find the answer (to use the Dirichlet) is the same as well, but with a Python implementation. stackoverflow.com/questions/18659858/… Commented Feb 12, 2018 at 18:22

#/Total[#,{2}]&@Log@RandomReal[{0,1},{m,n}] will give you a sample of m points from a uniform distribution over an n-1-dimensional regular simplex. (An equilateral triangle is a 2-dimensional regular simplex.) Here's what m = 2000, n = 3 should look like, where {x,y} = {p[[2]]-p[[1]], Sqrt@3*p[[3]]} are the barycentric coordinates of the 3-element probability vector p:

Here's what you get if you omit the Log@ and normalize Uniform(0,1) variables, which is what both of the OP's examples do:

• Thanks a lot. Could you please explain in what respects does this behave differently from RandomVariate[UniformDistribution[]]? Commented Oct 9, 2013 at 8:10
• See for yourself. Try it with n = 2 and make a histogram of p[[1]]. Or use n = 3 and ListPlot the barycentric coordinates: {x,y} = {p[[2]]-p[[1]],Sqrt@3*p[[3]]}. Commented Oct 9, 2013 at 18:41
• Yes, the difference is clear -- see my answer below. Actually, I meant for you to explain the difference in algorithmic terms, or perhaps provide pointers to a textbook explanation of why your method is doing what it's doing. Commented Oct 11, 2013 at 10:39
• I generate a Dirichlet distribution in which all the concentration parameters are 1. See the link that Jacob provided, then scroll down to this section and remember that the log of a Uniform(0,1) variable is proportional to a Gamma variable with shape parameter 1. Commented Oct 11, 2013 at 13:55
• You can also use Mathematica's built-in DirichletDistribution: points = RandomVariate[DirichletDistribution[{1, 1, 1}], 2000] /. v_?VectorQ :> {v[[2]] - v[[1]], Sqrt[3] (1 - Total[v])}; and then ListPlot[points].
– chuy
Commented Oct 11, 2013 at 18:28

Starting in M10.2, you can just use RandomPoint:

pts=RandomPoint[Simplex[{{0,0,1},{0,1,0},{1,0,0}}], 1000];
Graphics3D[Point[pts]]


• That's a slick capability. Commented Jun 7, 2017 at 14:59

Old question, but I didn't see this method. Generates $n$ points uniformly randomly distributed on a simplex embedded in $d$ dimensions.

    genSimplex[n_, d_] :=
Table[Differences[Sort[Flatten[{0, RandomReal[1, d – 1], 1}]]], {n}];


The algorithm generates points that are randomly distributed on an outer face of a simplex. The way to generate them is, for a d-dimensional problem…

1. Generate d-1 uniformly distributed random values in the range [0,1]
2. Add a 0 and a 1 to the list
3. Sort the list
4. Extract a list of the differences between the elements

You now have a list of random values that sum to 1 (so they are on a plane that is defined by points that sum to one) and that are otherwise independent of each other, so their dispersion is uniform.

Updating the answer with a picture of example data with 1,000 points.

This topic is well-covered here... https://stackoverflow.com/questions/3010837/sample-uniformly-at-random-from-an-n-dimensional-unit-simplex

• This is more or less what I came up with on my own, but then I found this Question page while looking for a more efficient way! Commented Aug 14, 2021 at 6:13