Gumbel cumulative distirubtion function is
\begin{align*}
F(x | \mu, \eta) &= \exp \left[ - \exp\left( -\frac{x - \mu}{\eta} \right) \right] ,\quad -\infty < x < \infty ,
\end{align*}
and its probability distribution function is
\begin{align*}
f(x|\mu, \eta) &= \frac{1}{\eta} \exp\left( -\frac{x - \mu}{\eta} \right) \exp \left[ - \exp\left(- \frac{x-\mu}{\eta} \right) \right]
\end{align*}
Let l = \set{l_1, l_2, \dots, l_C} ,\quad l_c \in \mathbb{R} be logits.
- for c = \set{1, 2, \dots, C}
- g_c \sim f(G_c | \mu=0, \eta=1)
- z_c = l_c + g_c
then the probability of z_c > z_{\backslash c} matches \pi_c.
Proof p(z_c > z_{\backslash c}) = \pi_c .
\begin{align*}
p(z_c > z_{\backslash c} | G_c = g_c) &= \prod_{i \neq c} p(z_c > z_i | G_c = g_c) \\
&= \prod_{i \neq c} p(l_c + g_c > l_i + G_i) \\
&= \prod_{i \neq c} p(G_i < l_c + g_c - l_i) \\
&= \prod_{i \neq c} F(l_c + g_c - l_i | \mu=0, \eta=1) \\
&= \prod_{i \neq c} \exp \left[ - \exp\left( l_i - l_c - g_c \right) \right]
\end{align*}
\begin{align*}
p(G_c = g_k) &= f(g_k | \mu=0, \eta=1) \\
&= \exp(- g_c) \exp \left[ - \exp\left( -g_c \right) \right] \\
\end{align*}
\begin{align*}
p(z_c > z_{\backslash c}) &= \int p(z_c > z_{\backslash c} | G_c = g_c) p(G_c = g_c) dg_c \\
&= \int
\prod_{i \neq c} \exp \left[ - \exp\left( l_i - l_c - g_c \right) \right]
\exp(- g_c) \exp \left[ - \exp\left( -g_c \right) \right]
d g_c \\
&= \int
\prod_{i \neq c} \exp \left[ - \exp\left( l_i - l_c - g_c \right) \right]
\exp(- g_c) \exp \left[ - \exp\left(l_c - l_c -g_c \right) \right]
d g_c \\
&= \int \prod_{i=1}^C \exp \left[ - \exp\left( l_i - l_c - g_c \right) \right] \exp(- g_c) d g_c \\
&= \int \exp \left[ - \frac{\sum_{i=1}^C \exp\left( l_i \right)}{\exp(l_c)} \exp(-g_c) \right] \exp(- g_c) d g_c \\
&= \int \exp \left[ - \frac{\exp(-g_c)}{\pi_c} \right] \exp(- g_c) d g_c \\
&= \int \exp \left[ -g_c - \frac{\exp(-g_c)}{\pi_c} \right] d g_c \\
&= \left[ \pi_c \exp\left( \frac{-\exp(-g_c)}{\pi_c} \right) \right]_{-\infty}^{\infty} \\
&= \pi_c
\end{align*}
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