Thesis - wip

content/appendixs/appendix_kde_bisection.tex

@@ -0,0 +1,30 @@
+We describe the bisection procedure used to compute $\tau = \hat{F}_0^{-1}(1-\alpha;\mathcal{B}_0)$
+in the KDE-ROLL forward pass (\Cref{sec:roll-kde-forward}).
+
+\paragraph{Bracket initialisation.}
+Because $\sigma(u;\,v) \to 0$ exponentially as $u \to -\infty$ and $\sigma(u;\,v) \to 1$
+as $u \to +\infty$, with tails decaying within roughly $10/v_0$ of the data range, the
+interval
+\[
+  \mathrm{lo} = \min(\mathbf{s}^{(0)}) - \tfrac{10}{v_0},
+  \qquad
+  \mathrm{hi} = \max(\mathbf{s}^{(0)}) + \tfrac{10}{v_0}
+\]
+is guaranteed to satisfy $\hat{F}_0(\mathrm{lo}) < 1-\alpha < \hat{F}_0(\mathrm{hi})$.
+
+\paragraph{Bisection iterations.}
+At each step we evaluate the midpoint,
+\[
+  \tau_{\mathrm{mid}} = \tfrac{1}{2}(\mathrm{lo} + \mathrm{hi}),
+\]
+and update the bracket:
+\[
+  \begin{cases}
+    \mathrm{lo} \leftarrow \tau_{\mathrm{mid}} & \text{if } \hat{F}_0(\tau_{\mathrm{mid}};\,\mathcal{B}_0) < 1-\alpha, \\
+    \mathrm{hi} \leftarrow \tau_{\mathrm{mid}} & \text{otherwise.}
+  \end{cases}
+\]
+Iteration continues until $\lvert\hat{F}_0(\tau_{\mathrm{mid}};\,\mathcal{B}_0)-(1-\alpha)\rvert < \varepsilon$
+(with $\varepsilon = 10^{-4}$), or until a maximum iteration count is reached.
+The method converges unconditionally and requires no derivative evaluation, in contrast
+to the Newton--Raphson alternative documented in Appendix~\ref{appendix:kde-nr}.