CDF of Dirichlet Distribution

Question

We know that a random variable $p=(p_{1}, p_{2},..., p_{K})$ which follows a $\textit{Dirichlet}$ distribution with parameters $\textbf{a} = (a_{1}, a_{2},..., a_{K})$ has as pdf

$$f(p) = \frac{1}{B(\textbf{a})}\prod_{i=1}^{K}p_{i}^{a_{i}-1}$$

On https://en.wikipedia.org/wiki/Dirichlet_distribution there is no CDF expression for the $\textit{Dirichlet}$ distribution and here https://www.researchgate.net/post/What-is-the-CDF-of-Dirichlet-Distribution, they note that the expression of CDF is really complicated and they redirect you to Implementation of Dirichlet cdf? where they recommend using $\textit{Monte Carlo}$ approximation for the CDF.

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Suppose that $K=3$, then I assume the CDF of the distribution for $\epsilon >0$ would be

$$F(\epsilon)=\mathbb{P}(p\leq \epsilon ) = \mathbb{P}(\begin{bmatrix}p_{1}\\ p_{2}\\ p_{3} \end{bmatrix} \leq \begin{bmatrix} \epsilon\\ \epsilon\\ \epsilon \end{bmatrix}) = \mathbb{P}(p_{1}\leq \epsilon, p_{2}\leq \epsilon,p_{3}\leq \epsilon)= \frac{1}{B(\textbf{a})}\int_{0}^{\epsilon}\int_{0}^{\epsilon}\int_{0}^{\epsilon}p_{1}^{a_{1}-1}p_{2}^{a_{2}-1}p_{3}^{a_{3}-1}dp_{1}dp_{2}dp_{3} $$ $$=\frac{1}{B(\textbf{a})}\frac{\epsilon^{a_{1}}}{a_{1}}\frac{\epsilon^{a_{2}}}{a_{2}}\frac{\epsilon^{a_{3}}}{a_{3}}$$

I assume I do something totally wrong, because I do not take into account that $\sum_{i=1}^{K}p_{i}=1$. If I did then we would have to calculate the integral

$$\frac{\epsilon}{B(\textbf{a})}\int_{0}^{\epsilon}\int_{0}^{\epsilon}p_{1}^{a_{1}-1}p_{2}^{a_{2}-1}(1-p_{1}-p_{2})^{a_{3}-1}dp_{1}dp_{2} $$

where I assume that this is the integral that it is very complicated to be solved. So, the only way to compute it is through numerical approximations as pointed out here Implementation of Dirichlet cdf? ?

Answer 1

The Dirichlet distribution is either defined on the simplex of $\mathbb R^k$, $$\mathcal S_{k-1}=\big\{\mathbf x;\ x_i\in (0,1),~i=1,2,\ldots,k,~\sum_{i=1}^k x_i=1\big\}$$ in which case the density $$f(\mathbf x) = \frac{1}{B(\textbf{a})}\prod_{i=1}^{k}x_{i}^{a_{i}-1}$$ is with respect to the Lebesgue distribution over that simplex, or defined in $\mathbb R^{k-1}$, in which case the density $$f(\mathbf x) = \frac{1}{B(\textbf{a})}\prod_{i=1}^{k-1}x_{i}^{a_{i}-1}(1-x_1-\cdots-x_{k-1})^{a_k-1}$$ is with respect to the Lebesgue distribution over $\mathbb R^{k-1}$. The later is Wikipedia's definition albeit poorly written since written as a function of $k$ terms.

A particular instance of the later is the family of Beta distributions, which illustrates why it is not feasible to derive an closed form cdf, except for small integer values of the parameters $a_i$: $$\mathbb P_{\alpha,\beta}(X\le \epsilon)=\dfrac{B(\epsilon;\alpha,\beta)}{B(\alpha,\beta)}\quad0\le\epsilon\le 1$$ where $B(\epsilon;\alpha,\beta)$ is the so-called incomplete Beta function (and an acknowledgement of the absence of closed form!).

Both representations obviously lead to the same distribution, but writing events such as $\mathbb P(\mathbf X\in A)$ will depend on which representation is used for $A$, i.e., either $A\subset\mathcal S_{k-1}$ or $A\subset\mathbb R^{k-1}$. In the former case, $$\mathbb P(\mathbf X\in A)=\int_A \frac{1}{B(\textbf{a})}\prod_{i=1}^{k}x_{i}^{a_{i}-1}\,\text d\lambda_{\mathcal S_{k-1}}(\mathbf x)$$ and in the later $$\mathbb P(\mathbf X\in A)=\int_A \frac{1}{B(\textbf{a})}\prod_{i=1}^{k-1}x_{i}^{a_{i}-1}(1-x_1-\cdots-x_{k-1})^{a_k-1} \,\text dx_1\cdots\,\text dx_{k-1}$$