Hopefully final draft

content/appendixs/appendix_gauss_grad.tex

@@ -1,6 +1,11 @@
 We instantiate the general ROLL framework (\Cref{sec:roll-formulation}) with
 Gaussian-distributed scores.  Each class's scores are modelled as
-$\mathcal{N}(\mu_k, \sigma_k^2)$, fitted by maximum-likelihood estimation.
+$\mathcal{N}(\mu_k, \sigma_k^2)$, fitted by maximum-likelihood
+estimation.\footnote{If a class's scores are literally constant, $\sigma_k=0$ and the
+CDF and gradient expressions throughout this appendix are undefined (division by
+$\sigma_k$ or $\sigma_k^2$). This edge case is unsupported in the current
+implementation; it does not arise with real, continuously-valued model outputs, but
+supporting it in general would require an explicit floor on $\sigma_k$.}
 
 \paragraph{Forward pass.}
 
@@ -27,6 +32,31 @@ Substituting into the general ROLL objective \eqref{eq:roll-tpr-at-fpr}:
   \label{eq:roll-tpr-at-fpr-gaussian}
 \end{align}
 
+\paragraph{Closed-form AOC.}
+
+Unlike the general (KDE) case, the full AOC integral admits a closed form for the
+Gaussian instantiation, bypassing the 99-point quadrature of \Cref{sec:roll-aoc}
+entirely. Writing $\Phi$ for the standard normal CDF
+($\Phi(x) = \tfrac{1}{2}[1+\operatorname{erf}(x/\sqrt{2})]$) and $\varphi$ for its
+density, substitute \eqref{eq:roll-tpr-at-fpr-gaussian} into
+$\mathcal{L}_{\text{AOC}} = \int_0^1 \mathcal{L}_{\text{ROLL-TPR@FPR}}(\alpha)\,d\alpha$
+and change variables via $u = \Phi^{-1}(1-\alpha)$, so $\tau = \mu_0 + \sigma_0 u$:
+\[
+  \mathcal{L}_{\text{AOC}}^{\text{Gaussian}}
+  = \int_{-\infty}^{\infty} \Phi\!\left(\frac{\mu_0-\mu_1+\sigma_0 u}{\sigma_1}\right)\varphi(u)\,du.
+\]
+Applying the standard identity
+$\int_{-\infty}^{\infty}\Phi(a+bu)\,\varphi(u)\,du = \Phi\!\bigl(a/\sqrt{1+b^2}\bigr)$
+with $a = (\mu_0-\mu_1)/\sigma_1$, $b = \sigma_0/\sigma_1$:
+\begin{equation}
+  \mathcal{L}_{\text{AOC}}^{\text{Gaussian}} = \Phi\!\left(\frac{\mu_0-\mu_1}{\sqrt{\sigma_0^2+\sigma_1^2}}\right).
+  \label{eq:gauss-aoc-closed-form}
+\end{equation}
+(Verified numerically against direct quadrature of the integral.) We nonetheless use
+the same 99-point average as every other instantiation in this thesis, for
+implementation uniformity across distribution families rather than necessity
+(\Cref{sec:roll-aoc}).
+
 \paragraph{Gradient derivation.}
 
 We derive the gradient of \eqref{eq:roll-tpr-at-fpr-gaussian} with respect to