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).

References:

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.