\section{Theoretical Bounds and Normalization for Evidential Uncertainties}\label{ap:limits}

We derive here the analytic extrema used to normalize all uncertainties to $[0,1]$.  
Let $\boldsymbol{\alpha}(x)$ be a $K$-class Dirichlet with total evidence $S(x)=\sum_k \alpha_k(x)$.  
To obtain meaningful and comparable bounds, we consider the extreme configurations of $\boldsymbol{\alpha}$ that maximise (or minimise) each uncertainty while remaining consistent with the evidential interpretation of the Dirichlet.

\begin{itemize}
    \item \emph{Bounds for aleatoric uncertainty.}
Aleatoric uncertainty
\[
u_{\mathrm{ale}}(x)
  = \sum_{k=1}^K \frac{\alpha_k(S-\alpha_k)}{S(S+1)}
\]
quantifies intrinsic class ambiguity \emph{given fixed evidence}.  
It is maximised when the classifier assigns equal expected class probabilities, i.e.\ when the Dirichlet is symmetric:
\[
\alpha_1=\cdots=\alpha_K=c,\qquad S=Kc.
\]
Substituting,
\[
u_{\mathrm{ale}}
  = \sum_{k=1}^K \frac{c(Kc-c)}{Kc(Kc+1)}
  = \frac{Kc(K-1)c}{Kc(Kc+1)}
  = \frac{K-1}{K}\cdot \frac{c}{c+\tfrac{1}{K}}.
\]
As $c\rightarrow\infty$ (high-evidence but fully ambiguous), this converges to
\[
u_{\mathrm{ale}}^{\max} = \frac{K-1}{K}.
\]

Regarding its minimum, it is $0$, achieved when one class dominates ($\alpha_j\!\to\!S$, others $\to 0$).

\item \emph{Bounds for epistemic uncertainty.}
Epistemic uncertainty
\[
u_{\mathrm{epi}}(x)
  = \sum_{k=1}^K \frac{\alpha_k(S-\alpha_k)}{S^2(S+1)}
\]
captures the variability of the Dirichlet mean under uncertain evidence.  
It is maximised when the model expresses \emph{complete ignorance}, i.e.\ the Dirichlet concentration is at its weakest:
\[
\alpha_1=\cdots=\alpha_K=1,\qquad S=K.
\]
Substituting,
\[
u_{\mathrm{epi}}
  = \sum_{k=1}^K \frac{1(K-1)}{K^2(K+1)}
  = K\,\frac{K-1}{K^2(K+1)}
  = \frac{K-1}{K(K+1)}.
\]
Any increase in evidence (larger $\alpha_k$) monotonically decreases $u_{\mathrm{epi}}$, hence this is the theoretical maximum. Regarding its minimum, it is $0$, achieved when one class dominates ($\alpha_j\!\to\!S$, others $\to 0$).

\item \emph{Bounds for vacuity.}
Vacuity
\[
u_{\mathrm{vac}}(x) = \frac{K}{S(x)}
\]
reflects the \emph{absence of evidence}.  
Its minimum occurs when evidence is arbitrarily large ($S\!\to\!\infty$):
\[
u_{\mathrm{vac}}^{\min}=0.
\]
Its maximum occurs when the evidence is minimal, i.e.\ $\alpha_k=1$ for all classes, $S=K$:
\[
u_{\mathrm{vac}}^{\max} = \frac{K}{K} = 1.
\]

\end{itemize}

Given any uncertainty value $u(x)$ with theoretical interval $[u_{\min},u_{\max}]$, we map it to a common $[0,1]$ scale via
\[
\tilde{u}(x) = \frac{u(x)-u_{\min}}{u_{\max}-u_{\min}}.
\]
This yields aligned and interpretable uncertainty scores across pixels, instances, uncertainty types, and datasets.
