Generating random draws from the Dir(1, 1, …, 1) distribution using the uniform distribution

Fix an integer K \geq 2 and parameters \alpha_1, \dots, \alpha_K > 0, commonly written as a vector \boldsymbol{\alpha} = (\alpha_1, \dots, \alpha_K). \boldsymbol{X} = (X_1, \dots, X_K) has the Dirichlet distribution (we write \boldsymbol{X} \sim \text{Dir}(\boldsymbol{\alpha})) if it has support on the probability simplex \left\{ (x_1, \dots, x_K) \in \mathbb{R}^K: x_i \geq 0 \text{ and } \sum_{i=1}^K  = 1 \right\} and its probability density function (PDF) satisfies

\begin{aligned} f(x_1, \dots, x_K) &\propto \prod_{i=1}^K x_i^{\alpha_i - 1}. \end{aligned}

The \text{Dir}(1, 1, \dots, 1) distribution is special because its PDF is constant over its support:

\begin{aligned} f(x_1, \dots, x_K) \propto 1. \end{aligned}

The following lemma demonstrates one way to generate random draws from the \text{Dir}(1, 1, \dots, 1) distribution:

Lemma. Let U_1, \dots, U_{K-1} \sim \text{Unif}[0,1], and let U_{(1)}, \dots, U_{(K-1)} denote the order statistics for U_1, \dots, U_{K-1}. Then

\begin{aligned} \left( U_{(1)} - 0, U_{(2)} - U_{(1)}, \dots, 1 - U_{(K-1)} \right)  \sim \text{Dir}(1, 1, \dots, 1). \end{aligned}

(See Section 2 of Reference 1 for various ways to generate samples from the Dirichlet distribution for arbitrary \boldsymbol{\alpha}.)

The proof of the lemma starts with the following theorem, which gives the joint density of all the order statistics:

Theorem. Let X_1, \dots, X_n be i.i.d. random variables with PDF f. Then the joint density of the order statistics X_{(1)}, \dots, X_{(n)} is

\begin{aligned} f_{X_{(1)}, \dots, X_{(n)}}(y_1, \dots, y_n) = n! f(y_1) \dots f(y_n) 1 \{ y_1 < y_2 < \dots < y_n \}. \end{aligned}

(This is Theorem 6.1 of Reference 2; you can find a proof for this theorem there.) Applying this theorem to U_1, \dots, U_n, we have

\begin{aligned} f_{U_{(1)}, \dots, U_{(K-1)}}(y_1, \dots, y_{K-1}) = (K-1)! 1 \{ 0 \leq y_1 < y_2 < \dots < y_n \leq 1 \}. \end{aligned}

The lemma then follows by applying a change of variables to the PDF (see this link for the steps one needs to carry out for the change of variables).


  1. Frigyik, B. A., et. al. (2010). Introduction to the Dirichlet Distribution and Related Processes.
  2. DasGupta, A. Finite Sample Theory of Order Statistics and Extremes.

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