\section{Closed-Form Evidential Uncertainty for the Segmentation Head}
\label{ap:formulas}

\paragraph{Dirichlet–Categorical evidential model.}
For each pixel $x$, the segmentation head predicts Dirichlet concentration parameters 
$\boldsymbol{\alpha}(x) = (\alpha_1(x),\ldots,\alpha_K(x))$ with total evidence $S(x)=\sum_k \alpha_k(x)$.  
These induce a Dirichlet distribution $\mathbf{p}(x)\!\sim D(\mathbf{p}\mid\boldsymbol{\alpha}(x))$ over class probabilities, and a categorical prediction $\hat{y}(x)\!\sim\mathrm{Cat}(\mathbf{p}(x))$.  
Aleatoric and epistemic uncertainties are defined as
\begin{equation}
    u_{\mathrm{ale}}(x)
    = \mathbb{E}_{\text{Dir}}\!\left[
        \mathrm{Var}_{\text{Cat}}\big(\hat{y}\mid \mathbf{p}\big)
      \right],
    \qquad
    u_{\mathrm{epi}}(x)
    = \mathrm{Var}_{\text{Dir}}\!\left[
        \mathbb{E}_{\text{Cat}}\big[\hat{y}\mid \mathbf{p}\big]
      \right].
    \label{eq:ua-ue-def-app}
\end{equation}

\paragraph{Closed-form expressions.}
For completeness, we derive the closed forms in Eq.~\eqref{eq:epi_ale-app} starting from the definitions in Section \ref{sec:methods}.  
Let $\boldsymbol{\alpha}(x)=(\alpha_1(x),\ldots,\alpha_K(x))$ and $S(x)=\sum_{k=1}^K \alpha_k(x)$, and denote $p_k$ the $k$-th component of $\mathbf{p}$.

\begin{itemize}
    \item \emph{Aleatoric uncertainty.}
For a categorical variable with one-hot encoding, the conditional variance given $\mathbf{p}$ is
\[
  \mathrm{Var}_{\mathrm{Cat}}(\hat{y}\mid\mathbf{p})
  = \sum_{k=1}^K p_k(1-p_k)
  = \sum_{k=1}^K \big(p_k - p_k^2\big).
\]
Taking the expectation under the Dirichlet prior,
\[
  u_{\mathrm{ale}}(x)
  = \mathbb{E}_{\mathrm{Dir}}
    \Big[\mathrm{Var}_{\mathrm{Cat}}(\hat{y}\mid\mathbf{p})\Big]
  = \sum_{k=1}^K \Big(\mathbb{E}[p_k] - \mathbb{E}[p_k^2]\Big).
\]
Using standard Dirichlet moments,
\[
  \mathbb{E}[p_k] = \frac{\alpha_k(x)}{S(x)},
  \qquad
  \mathbb{E}[p_k^2] = \frac{\alpha_k(x)\big(\alpha_k(x)+1\big)}{S(x)\big(S(x)+1\big)},
\]
we obtain
\[
  \mathbb{E}[p_k] - \mathbb{E}[p_k^2]
  = \frac{\alpha_k(x)}{S(x)}
    - \frac{\alpha_k(x)\big(\alpha_k(x)+1\big)}{S(x)\big(S(x)+1\big)}
  = \frac{\alpha_k(x)\big(S(x)-\alpha_k(x)\big)}{S(x)\big(S(x)+1\big)}.
\]
Summing over $k$ gives
\begin{equation}
  u_{\mathrm{ale}}(x)
  = \sum_{k=1}^K
    \frac{\alpha_k(x)\big(S(x)-\alpha_k(x)\big)}
         {S(x)\big(S(x)+1\big)}.
\end{equation}

\item \emph{Epistemic uncertainty.}
The epistemic term is defined as the variability (under the Dirichlet prior) of the mean categorical prediction:
\[
  u_{\mathrm{epi}}(x)
  = \mathrm{Var}_{\mathrm{Dir}}
    \Big[\mathbb{E}_{\mathrm{Cat}}[\hat{y}\mid\mathbf{p}]\Big]
  = \sum_{k=1}^K \mathrm{Var}_{\mathrm{Dir}}(p_k),
\]
where we sum the component-wise variances of $p_k$. For the Dirichlet,
\[
  \mathrm{Var}_{\mathrm{Dir}}(p_k)
  = \mathbb{E}[p_k^2] - \mathbb{E}[p_k]^2
  = \frac{\alpha_k(x)\big(\alpha_k(x)+1\big)}{S(x)\big(S(x)+1\big)}
    - \left(\frac{\alpha_k(x)}{S(x)}\right)^2.
\]
Bringing to a common denominator $S(x)^2\big(S(x)+1\big)$,
\[
  \mathrm{Var}_{\mathrm{Dir}}(p_k)
  = \frac{\alpha_k(x)\big(\alpha_k(x)+1\big)S(x)
         - \alpha_k^2(x)\big(S(x)+1\big)}
         {S(x)^2\big(S(x)+1\big)}
  = \frac{\alpha_k(x)\big(S(x)-\alpha_k(x)\big)}
         {S(x)^2\big(S(x)+1\big)}.
\]
Summing over $k$ yields
\begin{equation}
  u_{\mathrm{epi}}(x)
  = \sum_{k=1}^K
    \frac{\alpha_k(x)\big(S(x)-\alpha_k(x)\big)}
         {S(x)^2\big(S(x)+1\big)}.
\end{equation}

\end{itemize}
Putting both together, we recover the compact expressions:
\begin{equation}
    u_{\mathrm{ale}}(x)
    = \sum_{k=1}^K
      \frac{\alpha_k(x)\left(S(x)-\alpha_k(x)\right)}
           {S(x)\left(S(x)+1\right)},
    \qquad
    u_{\mathrm{epi}}(x)
    = \sum_{k=1}^K
      \frac{\alpha_k(x)\left(S(x)-\alpha_k(x)\right)}
           {S(x)^2\left(S(x)+1\right)}.
    \label{eq:epi_ale-app}
\end{equation}

