\section{Sensitivity to Hyperparameter $\beta$}
\begin{figure}
    \centering
    \includegraphics[width=0.49\linewidth]{8_appendix/figures/beta_all_patients_vs_rotation_error.png}
    \includegraphics[width=0.49\linewidth]{8_appendix/figures/beta_all_patients_vs_translation_error.png}
    \caption{Sensitivity of continuous weighting to the fall-off parameter $\beta$. Left: rotation error. Right: translation error. Solid lines denote the mean and median across all held-out test images, while shaded regions denote the 25th--75th and 10th--90th percentiles. Performance is best in the low-$\beta$ regime ($0.001$--$0.01$), while larger $\beta$ values increase both the central error and the upper tail of the error distribution.}
    \label{fig:beta_sweep}
\end{figure}
Continuous weighting is defined by converting normalized landmark uncertainty $\tilde{u}_i$ into reliability weights via
\[
w_i = \exp(-\beta \tilde{u}_i),
\]
where $\beta$ controls the rate at which uncertain landmarks are suppressed. To assess the robustness of our method to this design choice, we performed a sensitivity analysis over $\beta \in \{0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1.0\}$ while keeping all other settings fixed.

Figure~\ref{fig:beta_sweep} summarizes the resulting rotation and translation error distributions across all held-out test images. For each $\beta$, we report the mean and median together with the 25th--75th and 10th--90th percentile bands in order to characterize both central tendency and spread.

The results show that performance is best and most stable in the low-$\beta$ regime ($0.001$--$0.01$). In this range, both rotation and translation errors remain low across the median and upper percentiles. As $\beta$ increases, both the central error and the spread of the distribution grow, with especially clear degradation for $\beta \geq 0.1$. This trend indicates that overly aggressive fall-off suppresses not only unreliable landmarks, but also landmarks that still provide useful geometric information for pose estimation.

% These findings are consistent with the discussion in the main text that fixed hyperparameters may not be optimal across varying image conditions. In particular, the $\beta$ sweep suggests that continuous weighting is robust within a small low-$\beta$ regime, but becomes increasingly brittle as uncertainty is mapped too aggressively into near-zero weights. 