We instantiate the general ROLL framework (\Cref{sec:roll-formulation}) with Beta-distributed
scores.  Beta ROLL applies when model scores are constrained to the open interval $(0,1)$,
as is the case when $f_\theta$ uses a sigmoid output activation.  The Beta distribution is
supported on $(0,1)$ and its two shape parameters allow it to capture a wide range of
unimodal and skewed score distributions.

\paragraph{Forward pass.}

Given score vectors $\mathbf{s}^{(k)} \in (0,1)^{|\mathcal{B}_k|}$ for class $k$, let
$\mathbf{t}^{(k)} = \mathbf{1} - \mathbf{s}^{(k)}$ denote the complementary scores.
For any vector $\mathbf{v}$, write $\bar{v} = \frac{1}{|\mathbf{v}|}\sum_i v_i$.
We estimate the shape parameters using the closed-form log-moment estimator
of~\cite{chen2022beta}: define for each class $k$,

\begin{equation}
  D_k = \overline{s^{(k)}\!\ln s^{(k)}} - \bar{s}^{(k)}\cdot\overline{\ln s^{(k)}}
      + \overline{t^{(k)}\!\ln t^{(k)}} - \bar{t}^{(k)}\cdot\overline{\ln t^{(k)}}
  \label{eq:beta-denom}
\end{equation}

\begin{equation}
  \hat{a}_k = \frac{\bar{s}^{(k)}}{D_k}, \qquad \hat{b}_k = \frac{\bar{t}^{(k)}}{D_k}
  \label{eq:beta-params}
\end{equation}

These estimators follow from two simultaneous moment conditions:
$\mathbb{E}[X] = a/(a+b)$ and
$\operatorname{Cov}(X,\ln X) + \operatorname{Cov}(Y,\ln Y) = 1/(a+b)$,
where $Y=1-X\sim\operatorname{Beta}(b,a)$.
Replacing expectations by sample means and solving in closed form yields~\eqref{eq:beta-params}.

The CDF of $\operatorname{Beta}(\hat{a}_k,\hat{b}_k)$ at $x\in(0,1)$ is the
regularised incomplete beta function:
\begin{equation}
  \hat{F}_k(x) = I_x(\hat{a}_k,\hat{b}_k)
               = \frac{B(x;\,\hat{a}_k,\hat{b}_k)}{B(\hat{a}_k,\hat{b}_k)},
  \qquad B(x;\,a,b) = \int_0^x u^{a-1}(1-u)^{b-1}\,\mathrm{d}u
  \label{eq:beta-cdf}
\end{equation}

Substituting into the general ROLL objective \eqref{eq:roll-tpr-at-fpr}:
\begin{equation}
  \mathcal{L}_{\text{ROLL-TPR@FPR}}^{\text{BETA}}(f_\theta(\mathcal{B});\,\alpha)
  = I_{\hat\tau}(\hat{a}_1,\hat{b}_1),
  \qquad
  \hat\tau = I^{-1}_{1-\alpha}(\hat{a}_0,\hat{b}_0)
  \label{eq:roll-tpr-at-fpr-beta}
\end{equation}
where $I^{-1}_{1-\alpha}(a,b)$ denotes the $(1-\alpha)$-quantile of $\operatorname{Beta}(a,b)$.

\paragraph{Gradient.}

Unlike the Gaussian case, we do not derive closed-form gradients for Beta ROLL.
The regularised incomplete beta function $I_x(a,b)$ and its inverse are smooth
functions of all their arguments and are available as differentiable operations in
PyTorch (\texttt{torch.special.betainc} and \texttt{torch.special.betaincinv}).
Gradients of \eqref{eq:roll-tpr-at-fpr-beta} with respect to $f_\theta(\mathbf{x}_i)$
are therefore propagated via automatic differentiation through the estimator
\eqref{eq:beta-params}, the CDF evaluation, and the quantile inversion.