Hopefully final draft

content/appendixs/appendix_kde_bisection.tex

@@ -10,7 +10,8 @@ A fixed margin of $10/v_0 = 10h_0$ beyond the data range,
 \]
 only guarantees $\hat{F}_0(\mathrm{lo}) < 1-\alpha < \hat{F}_0(\mathrm{hi})$ for
 $1-\alpha \in (\sigma(-10;1),\,\sigma(10;1)) \approx (4.5\times10^{-5},\,1-4.5\times10^{-5})$,
-since that is the range the sigmoid tails have decayed into at distance $10h_0$ — not for
+since that is the range the sigmoid tails have decayed into at distance $10h_0$. This
+does not hold true for
 every $\alpha \in (0,1)$ as a fixed margin might suggest.
 
 Instead we use an exact, target-dependent bracket. For any $p \in (0,1)$, define
@@ -42,8 +43,8 @@ argument at $\max(\mathbf{s}^{(0)})$ gives $\hat{F}_0(\mathrm{hi}(p)) > p$.
 \end{proof}
 
 Setting $p = 1-\alpha$ gives a bracket valid for every $\alpha \in (0,1)$, unlike the
-fixed-margin version above. The degenerate case $\min(\mathbf{s}^{(0)}) = \max(\mathbf{s}^{(0)})$
-— a batch with constant negative-class scores — is excluded from
+fixed-margin version above. The degenerate case $\min(\mathbf{s}^{(0)}) = \max(\mathbf{s}^{(0)})$.
+This represents a batch with constant negative-class scores. I is excluded from
 \Cref{prop:bisection-bracket} above; it also breaks the Silverman bandwidth estimate
 itself ($h_0 = 0$), and is addressed as part of the degenerate-batch handling still
 outstanding for \Cref{sec:kde-bandwidth}.
@@ -68,7 +69,7 @@ certifies that the \emph{threshold} $\tau$ has been located accurately, but not
 of $\tau$ values can satisfy the residual bound while $\hat{F}_1(\tau)$ still varies
 substantially across that range.
 
-Instead, we exploit that $\hat{F}_1$ is itself a CDF — monotonically increasing — and
+Instead, we exploit that $\hat{F}_1$ is itself a CDF (monotonically increasing) and
 that the bracket invariant $\hat{F}_0(\mathrm{lo}) < 1-\alpha < \hat{F}_0(\mathrm{hi})$
 keeps the true root $\tau^\star$ inside $[\mathrm{lo},\mathrm{hi}]$ throughout. Hence
 \[