Representation for the survival function of the multinomial distribution in terms of the Dirichlet density? Let $K_p\sim \text{Binomial}(n,p)$ where $n\in \mathbb{N}$ and $0 \leq p \leq 1$.
Simple computations show that, for $1 \leq k \leq n-1$ and $p_0\in (0,1)$,
\begin{align}
    \mathbb{P}(K_{p_0} \geq k)
    &= \int_0^{p_0} \frac{d}{d p} \mathbb{P}(K_p \geq k) d p \notag \\[1mm]
    &= \int_0^{p_0} \left[\frac{d}{d p} \sum_{i=k}^n \binom{n}{i} p^i (1 - p)^{n - i}\right] d p \notag \\[1mm]
    &= \int_0^{p_0} \left[\hspace{-1mm}
        \begin{array}{l}
            n \sum_{i=k}^n \binom{n-1}{i-1} \, p^{i-1} (1 - p)^{(n-1) - (i-1)} \\[1mm]
            - n \sum_{i=k}^{n-1} \binom{n-1}{i} \, p^i (1 - p)^{(n-1) - i}
        \end{array}
        \hspace{-1mm}\right] d p \notag \\[1mm]
    &= \int_0^{p_0} \left[\hspace{-1mm}
        \begin{array}{l}
            n \sum_{i=k}^n \binom{n-1}{i-1} \, p^{i-1} (1 - p)^{(n-1) - (i-1)} \\[1mm]
            - n \sum_{j=k+1}^n \binom{n-1}{j-1} \, p^{j-1} (1 - p)^{(n-1) - (j-1)}
        \end{array}
        \hspace{-1mm}\right] d p \quad (\text{with } j = i + 1) \notag \\
    &= \int_0^{p_0} \left[n \binom{n-1}{k-1} \, p^{k-1} (1 - p)^{(n-1) - (k-1)}\right] d p \notag \\
    &= \frac{n!}{(k-1)! (n - k)!} \int_0^{p_0} p^{k-1} (1 - p)^{n-k} d p.
\end{align}
This turns the survival function of the binomial distribution into an integral over the density of the beta distribution.
Is it possible to derive an analogous representation for the survival function of the multinomial distribution in terms of an integral over the density function of the Dirichlet distribution ?
 A: For any $d\in \mathbb{N}$, the $d$-dimensional simplex and its interior are defined by
\begin{equation}\label{eq:def.simplex}
    \mathcal{S}_d := \big\{\mathbf{s}\in [0,1]^d: \|\mathbf{s}\|_1 \leq 1\big\} \quad \text{and} \quad \mathrm{Int}(\mathcal{S}_d) := \big\{\mathbf{s}\in (0,1)^d: \|\mathbf{s}\|_1 < 1\big\},
\end{equation}
where $\|\mathbf{s}\|_1 := \sum_{i=1}^d |s_i|$ denotes the $\ell_1(\mathbb{R})$ norm.
Given a set of probability weights $\mathbf{p}\in \mathrm{Int}(\mathcal{S}_d)$, the $\mathrm{Multinomial}\hspace{0.2mm}(N,\mathbf{p})$ probability mass function is defined by
\begin{equation}
    p_N(\mathbf{k}) = \frac{N!}{(N - \|\mathbf{k}\|_1)! \prod_{i=1}^d k_i!} \cdot q^{N - \|k\|_1} \prod_{i=1}^d p_i^{k_i}, \quad \mathbf{k}\in \mathbb{N}_0^d \cap N \mathcal{S}_d,
\end{equation}
where $q := 1 - \|\mathbf{p}\|_1 > 0$ and $N\in \mathbb{N}$.
If $U_1,U_2,\dots,U_N$ are i.i.d. $\mathrm{Uniform}(0,1)$'s and $I_k := (p_{j-1}, p_j]$ with the convention $p_0 := 0$, then $\mathbf{K} := (\sum_{i=1}^N 1_{\{U_i\in I_j\}})_{j=1}^d\sim \mathrm{Multinomial}(N,\mathbf{p})$, so that
\begin{align*}
    &\mathbb{P}(K_i\geq k_i, ~\forall i\in [d]) \\
    &\quad= \mathbb{P}(U_{(k_i)} \leq p_1 + p_2 + \dots + p_i, ~\forall i\in [d]) \\
    &\quad= \int_0^{p_1} \int_{u_1}^{p_1+p_2} \dots \int_{u_{d-1}}^{p_1+p_2+\dots+p_d} N! \prod_{i=1}^{d+1} \frac{(u_i - u_{i-1})^{k_i-k_{i-1}-1}}{(k_i - k_{i-1} - 1)!} \mathrm{d} u_1 \mathrm{d} u_2 \dots \mathrm{d} u_d,
\end{align*}
where $u_0 := 0$, $u_{d+1} := 1$, $k_0 := 0$, $k_{d+1} := N+1$.
After the change of variables $v_i = u_i - u_{i-1}, ~i\in [d+1]$ and $j_i = k_i - k_{i-1}, ~i\in [d+1]$, the above is
\begin{align*}
    &\mathbb{P}(K_i\geq k_i, ~\forall i\in [d]) \\
    &\quad= \int_0^{p_1} \int_0^{(p_1 - v_1) + p_2} \dots \int_0^{\sum_{k=1}^{d-1} (p_k - v_k) + p_d} N! \prod_{i=1}^{d+1} \frac{v_i^{j_i-1}}{(j_i - 1)!} \mathrm{d} v_1 \mathrm{d} v_2 \dots \mathrm{d} v_d \notag \\
    &\quad= \int_{\substack{v_1\in p_1\mathcal{S}_1 \\ (v_1,v_2)\in (p_1+p_2)\mathcal{S}_2 \\ \vdots \\ (v_1,v_2,\dots,v_d)\in (\sum_{i=1}^d p_i)\mathcal{S}_d}} \frac{\Gamma(N+1)}{\prod_{i=1}^{d+1} \Gamma(j_i)} \prod_{i=1}^{d+1} v_i^{j_i-1} \mathrm{d} v_1 \mathrm{d} v_2 \dots \mathrm{d} v_d.
\end{align*}
