native/paper_source/text/7-appendix.tex

\newpage
\appendix
\onecolumn

\begin{center}
    {\Huge Appendix}
\end{center}

We firstly restate our contributions and demonstrate some additional related works, extended discussions, omitted preliminary and illustrations in Appendix~\ref{appendix:all in one}. Then we provide missing proofs in Appendix~\ref{appendix:Proof}.
In Appendix~\ref{appendix:More Details}, we illustrate the omitted experimental results. In Appendix~\ref{appendix:experiment_details}, we present implementation details of our experiments. 

% We firstly restate our contributions in Appendix~\ref{Appendix-restatement}. 
% Then in Appendix~\ref{appendix: related work}, we exhibit more related works. 
% We present more discussions in Appendix~\ref{appendix:discussions}.
% In Appendix~\ref{appendix:prelim}, we introduce the conformal prediction and 
% we provide missing proofs in Appendix~\ref{appendix:Proof}. In Appendix~\ref{appendix:Missing Illustration}, we exhibit the omitted illustrations in our paper. In Appendix~\ref{appendix:More Details}, we illustrate the omitted experimental results. In Appendix~\ref{appendix:experiment details}, we present implementation details of our experiments. 

\section{Additional Details and Discussion}
\label{appendix:all in one}
\input{text/appendix-contributions}
\input{text/appendix-related-works}
% \begin{figure*}[t]
%     \centering
%     \includegraphics[width=0.8\linewidth]{figure/PCP-new.pdf}
%     \caption{The illustration of Prejudicial Trick (PT). To obtain a $1-\alpha$ confidence interval, PT first assigns empty sets for a $1-p$ subset of the test points, and assigns $1-\alpha^\prime$ confidence interval for the remaining test points where $\alpha^\prime < \alpha$. The returned confidence interval still satisfies $\bP(Y \in \cC(X)) \geq 1-\alpha$ by setting a proper $\alpha^\prime$.}
%     \label{fig:enter-label}
% \end{figure*}


% \section{Omitted Proofs}
% \label{appendix:Proof}
% \textit{Proof to Theorem 1} Below, we theoretically derive the situation that arises above.  Recall that our goal is to find a triple tuple $(\cF,p,\Tilde{\alpha})$ s.t. $\frac{1}{p}\cF^{-1}(1-\Tilde{\alpha}) > \cF^{-1}(\frac{1-\Tilde{\alpha}}{p})$.
% First, we denote $\cF^{-1}(x)$ by $\cG(x)$ and assume that $\cG(x)$is differentiable(which just assumes that X is an almost surely continuous random variable):
% \begin{equation}
%     \frac{1}{p}\cG(1-\Tilde{\alpha}) > \cG(\frac{1-\Tilde{\alpha}}{p}),
% \end{equation}
% then we let $t=1-\Tilde{\alpha}$:

% \begin{equation}
%     \frac{1}{p}\cG(t)-\cG(t) > \cG(\frac{t}{p})-\cG(t),
% \end{equation}
% We deform the formula appropriately:
% \begin{equation}
%     \frac{1-p}{p}\cG(t) > \cG'(t)\frac{1-p}{p}t+o(t),
% \end{equation}
% namely:
% \begin{equation}
%     \cG(t) > \cG'(t)t.
% \end{equation}
% This equation implies that if we assume $\cG(0)=0$, the derivative of the inverse cumulative distribution function at point $1-\Tilde{\alpha}$ needs to be less than the slope of the line segment connecting this point to the origin, which is a sufficient and necessary condition for the situation we have described to be true. Figure~\ref{fig:illustration of the theorem} provides an illustration.

\subsection{More Discussions}
\label{appendix:discussions}
\textbf{Similarity between PT and method discussed in \citet{foygel2021limits}.} In Section~\ref{sec:PT}, we mention the similarity between our PT method and the randomness discussed in \citet{foygel2021limits}. While the mechanism in our work bears a structural resemblance to that in \citet{foygel2021limits}, our motivation and conclusion are fundamentally different. \citet{foygel2021limits} investigate the inherent trade-offs required to achieve conditional coverage, using randomization as a tool to explore theoretical limits. In contrast, our work focuses on the evaluation paradigm itself. We use PT not to achieve a desirable property~(like conditional coverage), but to demonstrate a failure mode of evaluating CP methods primarily through coverage and average length. Our primary contribution is to highlight this pitfall and to propose Interval Stability as a complementary diagnostic for the specific run-to-run variability induced by algorithmic randomness. This diagnostic perspective is orthogonal to the conditional-coverage tradeoffs studied by \citet{foygel2021limits}.


\subsection{Omitted Preliminary}
\label{appendix:prelim}
\textbf{Interval Prediction.} 
Interval prediction aims to construct a confidence interval that contains the true response value with a user-specified probability. Compared to traditional point estimation, interval prediction provides more comprehensive statistical information by quantifying the uncertainty using the interval length, which is often a more challenging goal. Definition~\ref{def:IP} presents the formal definition.
% We formally define the interval prediction in Definition~\ref{def:IP}.
\begin{definition}[Interval Prediction]
\label{def:IP}
Let $(\bm{X},Y)$  denote a feature-response pair.
% from a joint distribution $\cP_{\bm{X}Y}$. 
Given a miscoverage rate $\alpha$, interval prediction aims to construct a confidence interval $\cC_{1-\alpha}(\bm{X})$, such that
\begin{equation}\label{eqn: IP1}
\bP(Y \in \cC_{1-\alpha}(\bm{X}))\geq 1-\alpha.
\end{equation}
Given the coverage in Equation~\eqref{eqn: IP1}, a smaller confidence interval indicates a more precise estimate.
\end{definition}


\textbf{Conformal Prediction.} To construct an interval prediction, we introduce a widely used approach called vanilla conformal prediction. The VCP method is typically divided into four stages: dataset splitting, training, calibration, and construction. The whole procedure is presented in Algorithm~\ref{alg:vcp}.

\emph{Dataset Splitting.}
Let $\cD=\{(\bm{x}_i, y_i): {i\in \cI}\}$ denote the i.i.d. samples from a distribution $\cP_{\bm{X}Y}$ over the covariate $\bm{X} \in \bR^d$ and the response $Y\in \bR$. The VCP first randomly splits the dataset $\cD$ into two folds: a training fold $\cD_\text{tr}=\{(\bm{x}_i, y_i): {i\in \cI_\text{tr}}\}$ and a calibration fold $\cD_\text{ca}=\{(\bm{x}_i, y_i): {i\in \cI_\text{ca}}\}$, where $\cI_\text{tr}\cup \cI_\text{ca}=\cI$ and $\cI_\text{tr}\cap \cI_{ca}=\varnothing$.

\emph{Training Process.} We train a model denoted by $\hat{\mu}(\cdot)$ (\emph{e.g.}, a neural network) via the training fold \(\cD_\text{tr}\). 

\emph{Calibration Process.} Given the trained model $\hat{\mu}(\cdot)$, VCP calculates the non-conformity score on the calibration fold $\cD_{\text{ca}}$, denoted by $\cV = \{s(\bm{x}_i, y_i; \hat{\mu}) : i \in \cI_\text{ca}\}$. The non-conformity score $s(\cdot)$ measures how well the model $\hat{\mu}(\cdot)$ fits the ground truth. A commonly used non-conformity score in regression tasks is the absolute residual, defined as $s(\bm{x}_i, y_{i}; \hat{\mu})=\vert y_{i}-\hat{\mu}(\bm{x}_i)\vert$. 


\emph{Construction Process.} 
Finally, for a given miscoverage rate $\alpha$, we then compute a $(1-\Tilde{\alpha})$-th quantile $\hat{Q}_{1-\Tilde{\alpha}}(\cV)$ of the empirical distribution of the non-conformity score set $\cV$ calculated on the calibration set, where $1-\Tilde{\alpha}=(1-\alpha)(1+1/|\cV|)$. The prediction interval at a new point $\bm{x}^{\prime}$ is then given by 
\begin{equation}\label{eq:vcp_interval}
    \cC_{1-\alpha}(\bm{x}^\prime)=\{y:s(\bm{x}^\prime,y;\hat{\mu})\leq \hat{Q}_{1-\Tilde{\alpha}}(\cV)\}.
\end{equation}

\textbf{Coverage and Length.} 
To evaluate the performance of interval prediction, two commonly used metrics: \emph{coverage} and \emph{length}  are defined in Definition~\ref{cov&len}, as further illustrated in Figure~\ref{fig:coverage_interval_comparison}.

% \begin{figure}[t]
%     \centering
%     \begin{subfigure}[t]{0.42\linewidth}
%         \centering
%         \includegraphics[width=\linewidth]{figure/coverage5.png}
%         \caption{}
%         \label{fig:cov_len}
%     \end{subfigure}
%     % \hspace{1em}
%     % \begin{subfigure}[t]{0.42\linewidth}
%     %     \centering
%     %     \includegraphics[width=\linewidth]{figure/ablation_study_simulation_0.pdf}
%     %     \caption{}
%     %     \label{fig:motivating_experiment}
%     % \end{subfigure}
%     \caption{Left: Illustration of coverage and interval length. Right: PCP returns shorter intervals than VCP on synthetic data.}
%     \label{fig:coverage_interval_comparison}
% \end{figure}







\begin{definition}[Coverage and Length]
\label{cov&len}
Let $(\bm{X}, Y)$ denote a feature-response pair from a joint distribution $\cP_{\bm{X}Y}$, and let $\cC_{1-\alpha}(\bm{X})$ denote the confidence interval to be evaluated and let $|\cdot|$ denote a certain measure of $\cC_{1-\alpha}(\bm{X})$. The coverage and length of $\cC_{1-\alpha}(\bm{X})$ is given by:
\begin{equation}
\label{eqt:cov&len}
\begin{split}
    \text{Coverage} & :=  \bE \left[\bI(Y \in \cC_{1-\alpha}(\bm{X})) \right],  \\
    \text{Length} &:= \bE \ |\cC_{1-\alpha}(\bm{X})|.
\end{split}
\end{equation}
For example, the length of the prediction interval given by VCP in Equation~\eqref{eq:vcp_interval} is:
\begin{equation}
    \text{Length} = \bE \ \left[2\hat{Q}_{1-\Tilde{\alpha}}(\cV) \right].
\end{equation}
% where $V_i$ denotes the non-conformity score calculated in the calibration set in Algorithm~\ref{alg:vcp}.
%and $1-\alpha^\prime = \frac{(1+n_2)(1-\alpha)}{n_2}$ denotes the adjusted miscoverage rate defined in Algorithm~\ref{alg:vcp}.
\end{definition}

\begin{figure}[t]
    \centering
    \includegraphics[width=0.5\linewidth]{figure/coverage5.png}
    \caption{Illustration of coverage and interval length.}
    \label{fig:coverage_interval_comparison}
\end{figure}

Notably, the two metrics in Definition~\ref{cov&len} evaluate the quality of prediction intervals from different perspectives. 
Figure~\ref{fig:coverage_interval_comparison} illustrates the coverage and length given a distribution. 
Firstly, high coverage ensures that the true value falls within the interval with high probability. A valid confidence interval should guarantee that the coverage exceeds $1-\alpha$, as suggested in Equation~\ref{eqn: IP1}.
However, setting a sufficiently large interval always guarantees Equation~\eqref{eqn: IP1}, which is impractical and meaningless. Therefore, the length metric is required to ensure the interval's precision. 
Based on the above discussion, the gold standard in conformal prediction is \emph{making the length as small as possible, given that the coverage is larger than $1-\alpha$.} 

Following the gold standard, VCP ensures the coverage guarantee under mild exchangeability assumption (Proposition~\ref{def:exc}), but pays less attention to the length.
As a result, numerous works on improving the length of VCP from different perspectives \citep{papadopoulos2011regression,romano2019conformalized} use intuitively valid approaches. 

\begin{proposition}[Coverage Guarantee]
\label{def:exc}
The terms $\cU_i$ are exchangeable if arbitrary permutation leads to the same distribution, i.e., $(\cU_1,...,\cU_{|\cI_\text{\ca}|+1}) \stackrel{d}{=} (\cU_{\pi(1)},...,\cU_{\pi({|\cI_\text{ca}|+1})})$
%\begin{equation} \label{eq:exc}(\cU_1,...,\cU_{|\cI_\text{\ca}|+1}) \stackrel{d}{=} (\cU_{\pi(1)},...,\cU_{\pi({|\cI_\text{ca}|+1})}), \end{equation}
with arbitrary permutation $\pi$ over ${1,...,|\cI_\text{ca} +1|}$, where $\stackrel{d}{=}$ denotes equivalence in distribution.
Suppose that  the data pair $(\bm{x}_i,y_i),i \in \cI_{\text{ca}} $ and the test point $(\bm{x}^\prime,y^\prime)$ are exchangeable, then the confidence interval $\cC_{1-\alpha}(\bm{x}^\prime)$ returned by Algorithm~\ref{alg:vcp} satisfies 
\begin{center}
$\bP\left(y^{\prime} \in \cC_{1-\alpha}(\bm{x}^{\prime})\right) \geq 1-\alpha$.
\end{center}
\end{proposition}
% However, is the coverage-length standard still valid, if an implausible conformal prediction algorithm could still improve the length? This gives rise to the question: 

% \emph{Can we improve the length while maintaining the valid coverage  via an implausible conformal prediction algorithm?} 

% Specifically, this question challenges whether the coverage-length pair sufficiently evaluates the quality of the intervals returned by conformal prediction. 
% And if not, what course of action should be taken? We address this question in Section~\ref{sec3.2}.

\subsection{Missing Illustration}
\label{appendix:Missing Illustration}
In this section, we present the missing illustration of Example~\ref{example: PatientRecovery} in Section~\ref{sec1}, the illustration of PT~(Figure~\ref{fig:enter-label}) and VCP algorithm mentioned in Section~\ref{sec3.1}.

\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{figure/patient.drawio.png}
    \caption{Illustration of Example~\ref{example: PatientRecovery}. Doctor Alice and Bob both achieve $60\%$ accuracy. Bob is more precise regarding length, but the corresponding strategy is not practically valid.}
    \label{fig:patient}
\end{figure}

\begin{algorithm}[h]
\caption{Vanilla Conformal Prediction (VCP)}
\label{alg:vcp}
\begin{algorithmic}[1]
\STATE \textbf{Input:} miscoverage rate $\alpha$, dataset $\cD=\{(\bm{x}_i, y_{i}):{i\in \cI}\}$, test point $\bm{x}^\prime$, non-conformity score function $s(\bm{x}_i,y_i;\hat{\mu})$.
\STATE Randomly split $\cD$ into a training fold $\cD_{\tr}=\{(\bm{x}_i,y_i): i \in \cI_{\tr}\}$ and a calibration fold $\cD_{\ca}=\{(\bm{x}_i,y_i): i\in \cI_{\ca}\}$;\\
\STATE Train a model $\hat{\mu}$ based on the training fold $\cD_{\tr}$;\\
\STATE Calculate the non-conformity score on the calibration fold $\cD_\ca$, denoted by $\cV= \{s(\bm{x}_i,y_i,\hat{\mu}): i \in \cI_\ca \}$; \\
\STATE Compute the $(1- \Tilde{\alpha})$-th quantile $\hat{Q}_{1-\Tilde{\alpha}}(\cV)$ of the empirical distribution of the non-conformity score set $\cV$ calculated on the calibration set $\cD_{\ca}$, where $1-\Tilde{\alpha}=(1-\alpha)(1+1/|\cV|)$;
\STATE \textbf{Output:} Interval $\cC_{1-\alpha}(\bm{x}^\prime)=\{y:s(\bm{x}^\prime,y;\hat{\mu})\leq \hat{Q}_{1-\Tilde{\alpha}}(\cV)\}$.
\end{algorithmic}
\end{algorithm}

\begin{figure*}[t]
    \centering
    \includegraphics[width=0.8\linewidth]{figure/PCP-new.pdf}
    \caption{The illustration of Prejudicial Trick (PT). To obtain a $1-\alpha$ confidence interval, PT first assigns empty sets for a $1-p$ subset of the test points, and assigns $1-\alpha^\prime$ confidence interval for the remaining test points where $\alpha^\prime < \alpha$. The returned confidence interval still satisfies $\bP(Y \in \cC(X)) \geq 1-\alpha$ by setting a proper $\alpha^\prime$.}
    \label{fig:enter-label}
\end{figure*}


\section{Proofs for Theorems and Corollaries}
\label{appendix:Proof}

% \textit{Proof of Theorem 1.} We aim to identify a triple $(\mathcal{F}, p, \tilde{\alpha})$ such that
% \[
% \frac{1}{p} \mathcal{F}^{-1}(1 - \tilde{\alpha}) > \mathcal{F}^{-1}\left( \frac{1 - \tilde{\alpha}}{p} \right).
% \]

% Let us denote the quantile function (i.e., the inverse of the cumulative distribution function $\mathcal{F}$) by $\mathcal{G}(x) := \mathcal{F}^{-1}(x)$. Assume that $\mathcal{G}$ is differentiable, which holds if the random variable $X$ is almost surely continuous. The inequality becomes:
% \[
% \frac{1}{p} \mathcal{G}(1 - \tilde{\alpha}) > \mathcal{G}\left( \frac{1 - \tilde{\alpha}}{p} \right).
% \]

% Let $t = 1 - \tilde{\alpha}$. The inequality rewrites as:
% \[
% \frac{1}{p} \mathcal{G}(t) - \mathcal{G}(t) > \mathcal{G}\left( \frac{t}{p} \right) - \mathcal{G}(t),
% \]
% which simplifies to:
% \[
% \frac{1 - p}{p} \mathcal{G}(t) > \mathcal{G}\left( \frac{t}{p} \right) - \mathcal{G}(t).
% \]

% Using the differentiability of $\mathcal{G}$ and a first-order Taylor expansion of $\mathcal{G}(\frac{t}{p})$ around $t$, we obtain:
% \[
% \mathcal{G}\left( \frac{t}{p} \right) = \mathcal{G}(t) + \mathcal{G}'(t)\left( \frac{t}{p} - t \right) + o\left( \left| \frac{t}{p} - t \right| \right),
% \]
% which gives:
% \[
% \mathcal{G}\left( \frac{t}{p} \right) - \mathcal{G}(t) = \mathcal{G}'(t)\frac{1 - p}{p}t + o(t).
% \]

% Substituting back, we get:
% \[
% \frac{1 - p}{p} \mathcal{G}(t) > \mathcal{G}'(t)\frac{1 - p}{p}t + o(t).
% \]

% Cancelling the common factor $\frac{1 - p}{p}$ on both sides (noting that $p < 1$ so the factor is positive), and letting $t \to 0$ gives the condition:
% \[
% \mathcal{G}(t) > \mathcal{G}'(t) t \quad \text{for small } t.
% \]

% This condition implies that, assuming $\mathcal{G}(0) = 0$, the value of the quantile function at $t$ exceeds the tangent line approximation at that point. Geometrically, this means that the slope of the secant line from the origin to the point $(t, \mathcal{G}(t))$ is greater than the instantaneous slope $\mathcal{G}'(t)$. This is both necessary and sufficient for the original inequality to hold. An illustration of this condition is provided in Figure~\ref{fig:illustration of the theorem}.


% \begin{figure}[t]
%     \centering
%     \includegraphics[width=0.5\linewidth]{figure/algorithm.png}
%     \caption{An illustration of Theorem~\ref{thm:better_length}.}
%     \label{fig:illustration of the theorem}
% \end{figure}

\input{text/proofs}

% \subsection{Missing Illustration}
% \label{appendix:Missing Illustration}
% In this section, we present the missing illustration of Example~\ref{example: PatientRecovery} in Section~\ref{sec1}, the illustration of PT~(Figure~\ref{fig:enter-label}) and VCP algorithm mentioned in Section~\ref{sec3.1}.

% \begin{figure}[h]
%     \centering
%     \includegraphics[width=0.6\linewidth]{figure/patient.drawio.png}
%     \caption{Illustration of Example~\ref{example: PatientRecovery}. Doctor Alice and Bob both achieve $60\%$ accuracy. Bob is more precise regarding length, but the corresponding strategy is not practically valid.}
%     \label{fig:patient}
% \end{figure}

% \begin{algorithm}[h]
% \caption{Vanilla Conformal Prediction (VCP)}
% \label{alg:vcp}
% \begin{algorithmic}[1]
% \STATE \textbf{Input:} miscoverage rate $\alpha$, dataset $\cD=\{(\bm{x}_i, y_{i}):{i\in \cI}\}$, test point $\bm{x}^\prime$, non-conformity score function $s(\bm{x}_i,y_i;\hat{\mu})$.
% \STATE Randomly split $\cD$ into a training fold $\cD_{\tr}=\{(\bm{x}_i,y_i): i \in \cI_{\tr}\}$ and a calibration fold $\cD_{\ca}=\{(\bm{x}_i,y_i): i\in \cI_{\ca}\}$;\\
% \STATE Train a model $\hat{\mu}$ based on the training fold $\cD_{\tr}$;\\
% \STATE Calculate the non-conformity score on the calibration fold $\cD_\ca$, denoted by $\cV= \{s(\bm{x}_i,y_i,\hat{\mu}): i \in \cI_\ca \}$; \\
% \STATE Compute the $(1- \Tilde{\alpha})$-th quantile $\hat{Q}_{1-\Tilde{\alpha}}(\cV)$ of the empirical distribution of the non-conformity score set $\cV$ calculated on the calibration set $\cD_{\ca}$, where $1-\Tilde{\alpha}=(1-\alpha)(1+1/|\cV|)$;
% \STATE \textbf{Output:} Interval $\cC_{1-\alpha}(\bm{x}^\prime)=\{y:s(\bm{x}^\prime,y;\hat{\mu})\leq \hat{Q}_{1-\Tilde{\alpha}}(\cV)\}$.
% \end{algorithmic}
% \end{algorithm}

% \begin{figure*}[t]
%     \centering
%     \includegraphics[width=0.8\linewidth]{figure/PCP-new.pdf}
%     \caption{The illustration of Prejudicial Trick (PT). To obtain a $1-\alpha$ confidence interval, PT first assigns empty sets for a $1-p$ subset of the test points, and assigns $1-\alpha^\prime$ confidence interval for the remaining test points where $\alpha^\prime < \alpha$. The returned confidence interval still satisfies $\bP(Y \in \cC(X)) \geq 1-\alpha$ by setting a proper $\alpha^\prime$.}
%     \label{fig:enter-label}
% \end{figure*}



% \clearpage
\section{Omitted Experiments}
\label{appendix:More Details}
In this section, we present all the omitted experiments. In Appendix~\ref{appendix:Omitted Experimental Results}, we demonstrate the missing experimental results in Section~\ref{sec:coverage} and Section~\ref{sec:length}. In Appendix~\ref{appendix:More Ablation Studies}, we exhibit the ablation study results.




% \subsection{Model Details}
% \label{appendix:Model Details}
% In this section, we present the details of the structure of our model. Specifically, our model shares the same structure with\citep{romano2019conformalized}.\\
% \textbf{Neural Net.} Our neural network design includes three fully connected layers, with ReLU activation functions applied between each layer. The initial layer accepts an input feature vector $X$ of n dimensions and produces $64$ hidden units. The second layer mirrors this structure, generating another set of 64 hidden units. The final layer is a linear output layer that provides a pointwise prediction for the response variable $Y$. The network's parameters are optimized by minimizing a quadratic loss function. We used the Adam optimization algorithm with a constant learning rate of $5 \times 10^{-4}$, minibatch size of $64$, and a weight decay coefficient of $10^{-6}$. In addition, regularization of dropout is implemented, with a retention probability of $0.1$ for hidden units. To avoid overfitting, early stop is used and the number of training epochs is determined by cross-validation, with a maximum cap of $1000$ epochs.\\
% \textbf{CQR Neural Net.} We utilize neural networks to implement CQR for quantile regression. The network structure is consistent with the one described above, with the sole difference being that the output of the quantile regression network is a two-dimensional vector, which indicates the lower and upper conditional quantiles. Additionally, the training process remains the same, except that the pinball loss function in equation is employed instead of the quadratic loss.

\subsection{Omitted Experimental Results}
\label{appendix:Omitted Experimental Results}
\textbf{Classification Tasks.} We extend PT to classification tasks. We apply PT to the real-world IMAGENET-VAL dataset~\citep{Deng2009ImageNetAL} with several pre-trained models, a similar setting with~\citet{angelopoulos2020uncertainty}. To simulate model misspecification, a bias is introduced to the logits of several classes before the softmax operation. The magnitude of the bias is determined based on the scale of the outputs. The experimental results on classification tasks in Table~\ref{table:class} perform similarly to regression tasks. Specifically, PT-RAPS attains valid coverage across different models~(Theorem~\ref{thm:coverage_guarantee_pt}) while reducing the set size compared to RAPS~(Theorem~\ref{thm:non-smooth-length}).


\begin{figure}[t]
    \centering
    \includegraphics[width=0.5\linewidth]{figure/new_figure/ablation_study_simulation_guassian.pdf}
    \caption{PT fails to improve length when the conditions on the distribution of non-conformity score are not satisfied.}
    \label{fig:failure case}
\end{figure}

\begin{table}[t]
\caption{Comparison between RAPS and PT-RAPS in classification tasks across different models, with $\alpha=0.1, p=0.95$ and index range chosen as $300$.}
\label{class}
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lccccc}
\toprule
Method      & Bias & \multicolumn{2}{c}{RAPS}                     & \multicolumn{2}{c}{PT-RAPS} \\
\cmidrule(lr){3-4} \cmidrule(lr){5-6}
Model       &      & Coverage               & Length           & Coverage               & Length \\
\midrule
resnet18    & 40   & 0.90 \scriptsize{$\pm 0.000$} & 304.02 \scriptsize{$\pm 0.004$} & 0.90 \scriptsize{$\pm 0.000$} & \textbf{295.60} \scriptsize{$\pm 0.233$} \\
resnet50    & 40   & 0.90 \scriptsize{$\pm 0.000$} & 302.09 \scriptsize{$\pm 0.027$} & 0.90 \scriptsize{$\pm 0.000$} & \textbf{290.29} \scriptsize{$\pm 0.224$} \\
resnet101   & 40   & 0.90 \scriptsize{$\pm 0.000$} & 302.01 \scriptsize{$\pm 0.004$} & 0.90 \scriptsize{$\pm 0.000$} & \textbf{289.56} \scriptsize{$\pm 0.174$} \\
resnet152   & 40   & 0.89 \scriptsize{$\pm 0.004$} & 301.53 \scriptsize{$\pm 0.054$} & 0.90 \scriptsize{$\pm 0.000$} & \textbf{288.98} \scriptsize{$\pm 0.165$} \\
resnext101  & 40   & 0.90 \scriptsize{$\pm 0.000$} & 301.48 \scriptsize{$\pm 0.013$} & 0.90 \scriptsize{$\pm 0.000$} & \textbf{288.49} \scriptsize{$\pm 0.201$} \\
vgg16       & 40   & 0.90 \scriptsize{$\pm 0.004$} & 303.39 \scriptsize{$\pm 0.143$} & 0.90 \scriptsize{$\pm 0.000$} & \textbf{293.34} \scriptsize{$\pm 0.304$} \\
shufflenet  & 40   & 0.90 \scriptsize{$\pm 0.000$} & 304.05 \scriptsize{$\pm 0.040$} & 0.90 \scriptsize{$\pm 0.000$} & \textbf{295.79} \scriptsize{$\pm 0.282$} \\
inception   & 40   & 0.90 \scriptsize{$\pm 0.000$} & 304.10 \scriptsize{$\pm 0.013$} & 0.90 \scriptsize{$\pm 0.000$} & \textbf{297.25} \scriptsize{$\pm 0.228$} \\
densenet161 & 40   & 0.90 \scriptsize{$\pm 0.000$} & 302.03 \scriptsize{$\pm 0.009$} & 0.90 \scriptsize{$\pm 0.000$} & \textbf{289.29} \scriptsize{$\pm 0.197$} \\
\bottomrule
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vspace{-1em}
\label{table:class}
\end{table}



\textbf{Conformalized Quantile Regression.} We deploy PT into other variants of conformal prediction. Specifically, we choose CQR as a baseline~\citep{romano2019conformalized}. CQR inherits the advantages of both conformal prediction and classical quantile regression. We use the same datasets and evaluation metrics as in Section~\ref{sec:length}. To mimic the model misspecification, we add bias directly to the lower and upper quantiles obtained by the quantile regression. The experimental results on the real-world CQR tasks are exhibited in Table~\ref{table:CQR}. It illustrates that \textit{PT achieves shorter interval length while maintaining valid coverage on CQR.} 

% Similarly to Section~\ref{sec4.1}, using other datasets in addition to BIKE~\citep{bike_sharing_275} in Section~\ref{sec4.1}, we conduct ablation studies of the CQR tasks on the probability hyperparameter $p$ in PCP and the bias parameter $\mu$ added to the trained model, as illustrated in Appendix~\ref{appendix:More Ablation Studies}.

\begin{table}[t]
\caption{Comparison between CQR and PT-CQR in quantile regression task across different datasets.}
\label{CQR}
\begin{center}
\begin{small}
\begin{tabular}{lccccc}
\toprule
Method     & Bias & \multicolumn{2}{c}{CQR}                   & \multicolumn{2}{c}{PT-CQR} \\
\cmidrule(lr){3-4} \cmidrule(lr){5-6}
Dataset    &      & Coverage             & Length            & Coverage              & Length \\
\midrule
meps-19    & 1    & 0.91 \scriptsize{$\pm 0.000$} & 4.60 \scriptsize{$\pm 0.148$} & 0.91 \scriptsize{$\pm 0.246$} & \textbf{4.44} \scriptsize{$\pm 0.143$} \\
meps-20    & 1    & 0.91 \scriptsize{$\pm 0.000$} & 4.58 \scriptsize{$\pm 0.192$} & 0.91 \scriptsize{$\pm 0.179$} & \textbf{4.41} \scriptsize{$\pm 0.188$} \\
meps-21    & 1    & 0.91 \scriptsize{$\pm 0.000$} & 4.65 \scriptsize{$\pm 0.080$} & 0.91 \scriptsize{$\pm 0.161$} & \textbf{4.52} \scriptsize{$\pm 0.107$} \\
bike       & 1    & 0.91 \scriptsize{$\pm 0.000$} & 2.61 \scriptsize{$\pm 0.013$} & 0.90 \scriptsize{$\pm 0.268$} & \textbf{2.51} \scriptsize{$\pm 0.009$} \\
blog-data  & 1    & 0.91 \scriptsize{$\pm 0.000$} & 3.80 \scriptsize{$\pm 0.107$} & 0.93 \scriptsize{$\pm 0.116$} & \textbf{3.61} \scriptsize{$\pm 0.098$} \\
bio        & 1    & 0.91 \scriptsize{$\pm 0.000$} & 3.45 \scriptsize{$\pm 0.009$} & 0.90 \scriptsize{$\pm 0.112$} & \textbf{3.32} \scriptsize{$\pm 0.009$} \\
facebook-1 & 1    & 0.91 \scriptsize{$\pm 0.000$} & 3.38 \scriptsize{$\pm 0.022$} & 0.92 \scriptsize{$\pm 0.125$} & \textbf{3.22} \scriptsize{$\pm 0.027$} \\
facebook-2 & 1    & 0.91 \scriptsize{$\pm 0.000$} & 3.57 \scriptsize{$\pm 0.027$} & 0.92 \scriptsize{$\pm 0.085$} & \textbf{3.39} \scriptsize{$\pm 0.027$} \\
concrete   & 2    & 0.91 \scriptsize{$\pm 0.000$} & 4.39 \scriptsize{$\pm 0.022$} & 0.88 \scriptsize{$\pm 0.648$} & \textbf{4.23} \scriptsize{$\pm 0.018$} \\
star       & 2    & 0.91 \scriptsize{$\pm 0.000$} & 4.15 \scriptsize{$\pm 0.004$} & 0.90 \scriptsize{$\pm 0.349$} & \textbf{3.96} \scriptsize{$\pm 0.009$} \\
\bottomrule
\end{tabular}
\end{small}
\end{center}
\vspace{-1em}
\label{table:CQR}
\end{table}

\textbf{Relaxed PT without exactly null prediction sets.} The standard PT construction assigns a null prediction set to a fraction of test samples. To examine whether the phenomenon relies on exactly null outputs, we further consider a relaxed PT variant that replaces the null prediction set with a small nonempty prediction set controlled by a perturbation rate $\epsilon$. This construction is closer to localized CP in the sense that the returned set can be viewed as being generated by a very small but nonzero local scale estimate, rather than by an exactly zero scale. As shown in Table~\ref{table:relaxed_pt}, the relaxed PT-VCP still preserves comparable coverage while reducing the average interval length across all datasets. This suggests that the observed metric failure is not merely an artifact of returning exactly null prediction sets.

\begin{table}[t]
\caption{
Comparison of performance between VCP and localized-CP-like PT-VCP in regression tasks across different datasets at fixed $\alpha=0.1$, $p=0.96$, and perturbation rate $\epsilon=0.01$. Values are reported as mean $\pm$ std over 5 random seeds. Bold marks the smaller interval length between VCP and PT-VCP for each dataset.
}
\label{table:relaxed_pt}
\begin{center}
\begin{small}
\begin{tabular}{lccccc}
\toprule
Method     & Bias & \multicolumn{2}{c}{VCP}                   & \multicolumn{2}{c}{PT-VCP} \\
\cmidrule(lr){3-4} \cmidrule(lr){5-6}
Dataset    &      & Coverage             & Length            & Coverage              & Length \\
\midrule
meps-19    & 20 & 0.901 \scriptsize{$\pm 0.005$} & 42.34 \scriptsize{$\pm 0.51$} & 0.896 \scriptsize{$\pm 0.003$} & \textbf{41.73} \scriptsize{$\pm 0.68$} \\
meps-20    & 20 & 0.900 \scriptsize{$\pm 0.005$} & 41.98 \scriptsize{$\pm 0.26$} & 0.901 \scriptsize{$\pm 0.006$} & \textbf{41.40} \scriptsize{$\pm 0.42$} \\
meps-21    & 20 & 0.903 \scriptsize{$\pm 0.005$} & 42.28 \scriptsize{$\pm 0.25$} & 0.897 \scriptsize{$\pm 0.003$} & \textbf{41.62} \scriptsize{$\pm 0.48$} \\
bike       & 10 & 0.901 \scriptsize{$\pm 0.005$} & 20.46 \scriptsize{$\pm 0.04$} & 0.901 \scriptsize{$\pm 0.005$} & \textbf{19.79} \scriptsize{$\pm 0.08$} \\
blog-data  & 20 & 0.898 \scriptsize{$\pm 0.005$} & 41.67 \scriptsize{$\pm 0.75$} & 0.899 \scriptsize{$\pm 0.003$} & \textbf{41.04} \scriptsize{$\pm 0.86$} \\
bio        & 10 & 0.901 \scriptsize{$\pm 0.004$} & 21.13 \scriptsize{$\pm 0.05$} & 0.900 \scriptsize{$\pm 0.003$} & \textbf{20.55} \scriptsize{$\pm 0.06$} \\
facebook-1 & 10 & 0.902 \scriptsize{$\pm 0.003$} & 20.81 \scriptsize{$\pm 0.08$} & 0.899 \scriptsize{$\pm 0.003$} & \textbf{20.62} \scriptsize{$\pm 0.24$} \\
facebook-2 & 10 & 0.900 \scriptsize{$\pm 0.001$} & 20.97 \scriptsize{$\pm 0.15$} & 0.900 \scriptsize{$\pm 0.003$} & \textbf{20.87} \scriptsize{$\pm 0.30$} \\
concrete   & 5  & 0.895 \scriptsize{$\pm 0.030$} & 10.32 \scriptsize{$\pm 0.02$} & 0.899 \scriptsize{$\pm 0.021$} & \textbf{10.09} \scriptsize{$\pm 0.09$} \\
star       & 5  & 0.909 \scriptsize{$\pm 0.010$} & 10.14 \scriptsize{$\pm 0.01$} & 0.909 \scriptsize{$\pm 0.002$} & \textbf{9.78} \scriptsize{$\pm 0.02$} \\
\bottomrule
\end{tabular}
\end{small}
\end{center}
\vspace{-1em}
\end{table}

\textbf{Stronger misspecification.} We also evaluate PT-VCP under a larger bias setting to further investigate how model misspecification affects the length reduction induced by PT. This experiment complements the main regression results in Section~3.4, where the improvement in average length can be modest on some datasets. As reported in Table~\ref{table:stronger_bias}, increasing the bias amplifies the interval-length contrast between VCP and PT-VCP while maintaining comparable coverage. These results are consistent with our theoretical discussion that PT is more likely to appear favorable when the sufficient conditions for length reduction are more strongly satisfied.

\begin{table}[t]
\caption{
Comparison of performance between VCP and PT-VCP in regression tasks across different datasets at fixed $\alpha=0.1$ and $p=0.96$, using a larger bias setting to amplify the interval-length contrast. Values are reported as mean $\pm$ std over 5 random seeds. Bold marks the smaller interval length between VCP and PT-VCP for each dataset.
}
\label{table:stronger_bias}
\begin{center}
\begin{small}
\begin{tabular}{lccccc}
\toprule
Method     & Bias & \multicolumn{2}{c}{VCP}                   & \multicolumn{2}{c}{PT-VCP} \\
\cmidrule(lr){3-4} \cmidrule(lr){5-6}
Dataset    &      & Coverage             & Length            & Coverage              & Length \\
\midrule
meps-19    & 80 & 0.900 \scriptsize{$\pm 0.006$} & 162.32 \scriptsize{$\pm 0.51$} & 0.898 \scriptsize{$\pm 0.006$} & \textbf{156.99} \scriptsize{$\pm 0.65$} \\
meps-20    & 80 & 0.900 \scriptsize{$\pm 0.005$} & 161.96 \scriptsize{$\pm 0.26$} & 0.901 \scriptsize{$\pm 0.003$} & \textbf{156.62} \scriptsize{$\pm 0.65$} \\
meps-21    & 80 & 0.904 \scriptsize{$\pm 0.005$} & 162.27 \scriptsize{$\pm 0.25$} & 0.899 \scriptsize{$\pm 0.003$} & \textbf{156.88} \scriptsize{$\pm 0.75$} \\
bike       & 40 & 0.901 \scriptsize{$\pm 0.005$} & 80.46 \scriptsize{$\pm 0.04$} & 0.899 \scriptsize{$\pm 0.006$} & \textbf{77.29} \scriptsize{$\pm 0.18$} \\
blog-data  & 80 & 0.899 \scriptsize{$\pm 0.005$} & 161.64 \scriptsize{$\pm 0.73$} & 0.899 \scriptsize{$\pm 0.004$} & \textbf{156.10} \scriptsize{$\pm 0.92$} \\
bio        & 40 & 0.901 \scriptsize{$\pm 0.004$} & 81.13 \scriptsize{$\pm 0.05$} & 0.900 \scriptsize{$\pm 0.003$} & \textbf{78.11} \scriptsize{$\pm 0.14$} \\
facebook-1 & 40 & 0.902 \scriptsize{$\pm 0.002$} & 80.77 \scriptsize{$\pm 0.07$} & 0.901 \scriptsize{$\pm 0.003$} & \textbf{78.15} \scriptsize{$\pm 0.28$} \\
facebook-2 & 40 & 0.900 \scriptsize{$\pm 0.001$} & 80.94 \scriptsize{$\pm 0.15$} & 0.899 \scriptsize{$\pm 0.001$} & \textbf{78.33} \scriptsize{$\pm 0.32$} \\
concrete   & 20 & 0.895 \scriptsize{$\pm 0.030$} & 40.32 \scriptsize{$\pm 0.02$} & 0.892 \scriptsize{$\pm 0.021$} & \textbf{38.65} \scriptsize{$\pm 0.21$} \\
star       & 20 & 0.909 \scriptsize{$\pm 0.010$} & 40.14 \scriptsize{$\pm 0.01$} & 0.906 \scriptsize{$\pm 0.009$} & \textbf{38.47} \scriptsize{$\pm 0.27$} \\
\bottomrule
\end{tabular}
\end{small}
\end{center}
\vspace{-1em}
\end{table}

\textbf{Group Coverage.} In Section~\ref{sec:coverage}, we conduct experiments to evaluate the different performance of VCP and PT-VCP regarding group coverage. The experimental results shown in Table~\ref{table reg_group coverage} demonstrate that PT not only achieves shorter confidence intervals while maintaining overall coverage, \textit{but also improves the group coverage in regression tasks}\footnote{Group coverage is defined as the lowest coverage rate among all the groups.}.

\begin{table}[t]
\caption{Comparison of group coverage between VCP and PT-VCP on regression tasks across different datasets ($\alpha=0.1$).}
\label{table reg_group coverage}
\centering
\begin{small}
\begin{tabular}{l l cc}
\toprule
\textbf{Dataset} & \textbf{Group} & \textbf{VCP} & \textbf{PT-VCP} \\
\midrule
\multirow{3}{*}{bike}     
    & Day     & 0.878 \scriptsize{$\pm$ 0.007} & \textbf{0.884} \scriptsize{$\pm$ 0.010} \\
    & Month   & 0.826 \scriptsize{$\pm$ 0.010} & \textbf{0.857} \scriptsize{$\pm$ 0.011} \\
    & Year    & 0.851 \scriptsize{$\pm$ 0.005} & \textbf{0.871} \scriptsize{$\pm$ 0.004} \\
\midrule
\multirow{3}{*}{star}     
    & Gender  & 0.905 \scriptsize{$\pm$ 0.008} & \textbf{0.905} \scriptsize{$\pm$ 0.002} \\
    & Stark   & 0.890 \scriptsize{$\pm$ 0.005} & \textbf{0.895} \scriptsize{$\pm$ 0.007} \\
    & School1 & 0.902 \scriptsize{$\pm$ 0.008} & \textbf{0.899} \scriptsize{$\pm$ 0.022} \\
\midrule
\multirow{3}{*}{meps-19}  
    & SEX=1     & 0.883 \scriptsize{$\pm$ 0.004} & \textbf{0.895} \scriptsize{$\pm$ 0.001} \\
    & MARRY=1   & 0.901 \scriptsize{$\pm$ 0.004} & \textbf{0.901} \scriptsize{$\pm$ 0.003} \\
    & REGION=1  & 0.862 \scriptsize{$\pm$ 0.005} & \textbf{0.877} \scriptsize{$\pm$ 0.006} \\
\midrule
\multirow{3}{*}{meps-20}  
    & FTSTU=1   & 0.893 \scriptsize{$\pm$ 0.004} & \textbf{0.900} \scriptsize{$\pm$ 0.002} \\
    & ACTDTY=1  & 0.897 \scriptsize{$\pm$ 0.003} & \textbf{0.902} \scriptsize{$\pm$ 0.002} \\
    & HONRDC=1  & 0.792 \scriptsize{$\pm$ 0.010} & \textbf{0.846} \scriptsize{$\pm$ 0.008} \\
\midrule
\multirow{3}{*}{meps-21}  
    & RTHLTH=1  & 0.864 \scriptsize{$\pm$ 0.004} & \textbf{0.877} \scriptsize{$\pm$ 0.004} \\
    & MNHLTH=1  & 0.856 \scriptsize{$\pm$ 0.004} & \textbf{0.873} \scriptsize{$\pm$ 0.004} \\
    & HIBPDX=1  & 0.755 \scriptsize{$\pm$ 0.013} & \textbf{0.818} \scriptsize{$\pm$ 0.009} \\
\bottomrule
\end{tabular}
\end{small}
\label{tab:group_coverage}
\end{table}

\textbf{Additional p-value based efficiency criteria.} We further evaluate RAPS and PT-RAPS under the p-value based efficiency criteria proposed by ~\citep{10.1007/978-3-319-33395-3_2}. These criteria provide alternative summaries of predictive efficiency for classification. As shown in Tables~\ref{table:pvalue_main} and~\ref{table:pvalue_observed}, PT-RAPS obtains more favorable values on seven out of the ten criteria. This supports the conclusion that the PT failure mode is not restricted to the average set-size metric. However, this experiment should be interpreted as empirical evidence rather than a complete characterization of all possible efficiency criteria.


\begin{table*}[t]
\caption{
Main p-value criteria across models. Hyperparameters are fixed as $\alpha=\epsilon=0.1$, calibration size $=10{,}000$, seeds $=5$, $p=0.95$, \texttt{pt\_bias} $=40$, \texttt{pt\_index\_range} $=300$, with smoothed p-values enabled. Each entry is reported as RAPS / PT-RAPS. Bold indicates the better, i.e., smaller, value between RAPS and PT-RAPS for each metric/model pair.
}
\label{table:pvalue_main}
\begin{center}
\begin{small}
\resizebox{\textwidth}{!}{
\begin{tabular}{lcccccc}
\toprule
Model & S & N & U & F & M & E \\
\midrule
ResNet18
& \textbf{215.01} \scriptsize{$\pm 1.12$} / 241.49 \scriptsize{$\pm 0.89$}
& 304.02 \scriptsize{$\pm 0.01$} / \textbf{295.44} \scriptsize{$\pm 0.20$}
& 0.93 \scriptsize{$\pm 0.00$} / \textbf{0.89} \scriptsize{$\pm 0.00$}
& \textbf{214.05} \scriptsize{$\pm 1.12$} / 240.59 \scriptsize{$\pm 0.89$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 303.02 \scriptsize{$\pm 0.01$} / \textbf{294.49} \scriptsize{$\pm 0.20$} \\

ResNet50
& \textbf{212.77} \scriptsize{$\pm 1.04$} / 239.48 \scriptsize{$\pm 0.82$}
& 302.07 \scriptsize{$\pm 0.06$} / \textbf{290.17} \scriptsize{$\pm 0.27$}
& 0.94 \scriptsize{$\pm 0.00$} / \textbf{0.90} \scriptsize{$\pm 0.00$}
& \textbf{211.82} \scriptsize{$\pm 1.04$} / 238.57 \scriptsize{$\pm 0.82$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 301.07 \scriptsize{$\pm 0.06$} / \textbf{289.22} \scriptsize{$\pm 0.28$} \\

ResNet101
& \textbf{212.55} \scriptsize{$\pm 0.76$} / 239.27 \scriptsize{$\pm 0.57$}
& 301.96 \scriptsize{$\pm 0.07$} / \textbf{289.44} \scriptsize{$\pm 0.32$}
& 0.94 \scriptsize{$\pm 0.00$} / \textbf{0.90} \scriptsize{$\pm 0.00$}
& \textbf{211.59} \scriptsize{$\pm 0.76$} / 238.36 \scriptsize{$\pm 0.57$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 300.96 \scriptsize{$\pm 0.07$} / \textbf{288.49} \scriptsize{$\pm 0.32$} \\

ResNet152
& \textbf{212.48} \scriptsize{$\pm 0.82$} / 239.21 \scriptsize{$\pm 0.61$}
& 301.47 \scriptsize{$\pm 0.14$} / \textbf{288.93} \scriptsize{$\pm 0.32$}
& 0.94 \scriptsize{$\pm 0.00$} / \textbf{0.90} \scriptsize{$\pm 0.00$}
& \textbf{211.52} \scriptsize{$\pm 0.82$} / 238.30 \scriptsize{$\pm 0.61$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 300.47 \scriptsize{$\pm 0.14$} / \textbf{287.98} \scriptsize{$\pm 0.32$} \\

ResNeXt101
& \textbf{209.14} \scriptsize{$\pm 0.53$} / 236.20 \scriptsize{$\pm 0.46$}
& 301.40 \scriptsize{$\pm 0.04$} / \textbf{288.46} \scriptsize{$\pm 0.36$}
& 0.94 \scriptsize{$\pm 0.00$} / \textbf{0.90} \scriptsize{$\pm 0.00$}
& \textbf{208.18} \scriptsize{$\pm 0.53$} / 235.29 \scriptsize{$\pm 0.46$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 300.40 \scriptsize{$\pm 0.04$} / \textbf{287.51} \scriptsize{$\pm 0.36$} \\

VGG16
& \textbf{210.64} \scriptsize{$\pm 1.29$} / 237.56 \scriptsize{$\pm 1.04$}
& 303.26 \scriptsize{$\pm 0.15$} / \textbf{293.20} \scriptsize{$\pm 0.59$}
& 0.93 \scriptsize{$\pm 0.00$} / \textbf{0.88} \scriptsize{$\pm 0.00$}
& \textbf{209.70} \scriptsize{$\pm 1.29$} / 236.65 \scriptsize{$\pm 1.04$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 302.26 \scriptsize{$\pm 0.15$} / \textbf{292.25} \scriptsize{$\pm 0.59$} \\

ShuffleNet
& \textbf{216.70} \scriptsize{$\pm 0.82$} / 243.02 \scriptsize{$\pm 0.65$}
& 304.02 \scriptsize{$\pm 0.05$} / \textbf{295.74} \scriptsize{$\pm 0.28$}
& 0.94 \scriptsize{$\pm 0.00$} / \textbf{0.90} \scriptsize{$\pm 0.00$}
& \textbf{215.74} \scriptsize{$\pm 0.82$} / 242.11 \scriptsize{$\pm 0.65$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 303.02 \scriptsize{$\pm 0.05$} / \textbf{294.79} \scriptsize{$\pm 0.28$} \\

Inception
& \textbf{217.32} \scriptsize{$\pm 0.85$} / 243.58 \scriptsize{$\pm 0.73$}
& 304.09 \scriptsize{$\pm 0.03$} / \textbf{297.24} \scriptsize{$\pm 0.76$}
& 0.95 \scriptsize{$\pm 0.00$} / \textbf{0.90} \scriptsize{$\pm 0.00$}
& \textbf{216.36} \scriptsize{$\pm 0.85$} / 242.67 \scriptsize{$\pm 0.73$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 303.09 \scriptsize{$\pm 0.03$} / \textbf{296.29} \scriptsize{$\pm 0.76$} \\

DenseNet161
& \textbf{212.15} \scriptsize{$\pm 0.94$} / 238.91 \scriptsize{$\pm 0.75$}
& 302.02 \scriptsize{$\pm 0.02$} / \textbf{289.26} \scriptsize{$\pm 0.53$}
& 0.94 \scriptsize{$\pm 0.00$} / \textbf{0.90} \scriptsize{$\pm 0.00$}
& \textbf{211.19} \scriptsize{$\pm 0.94$} / 238.00 \scriptsize{$\pm 0.75$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 301.02 \scriptsize{$\pm 0.02$} / \textbf{288.31} \scriptsize{$\pm 0.53$} \\
\bottomrule
\end{tabular}
}
\end{small}
\end{center}
\vspace{-1em}
\end{table*}

\begin{table*}[t]
\caption{
Observed p-value criteria across models. Hyperparameters are the same as Table~\ref{table:pvalue_main}. Each entry is reported as RAPS / PT-RAPS. Bold indicates the better, i.e., smaller, value between RAPS and PT-RAPS for each metric/model pair.
}
\label{table:pvalue_observed}
\begin{center}
\begin{small}
\resizebox{\textwidth}{!}{
\begin{tabular}{lcccc}
\toprule
Model & OU & OF & OM & OE \\
\midrule
ResNet18
& 0.94 \scriptsize{$\pm 0.00$} / \textbf{0.90} \scriptsize{$\pm 0.00$}
& \textbf{214.57} \scriptsize{$\pm 1.13$} / 241.05 \scriptsize{$\pm 0.90$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 303.12 \scriptsize{$\pm 0.01$} / \textbf{294.54} \scriptsize{$\pm 0.20$} \\

ResNet50
& 0.95 \scriptsize{$\pm 0.00$} / \textbf{0.90} \scriptsize{$\pm 0.00$}
& \textbf{212.37} \scriptsize{$\pm 1.03$} / 239.06 \scriptsize{$\pm 0.81$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 301.17 \scriptsize{$\pm 0.06$} / \textbf{289.27} \scriptsize{$\pm 0.27$} \\

ResNet101
& 0.95 \scriptsize{$\pm 0.00$} / \textbf{0.90} \scriptsize{$\pm 0.00$}
& \textbf{212.12} \scriptsize{$\pm 0.76$} / 238.84 \scriptsize{$\pm 0.57$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 301.06 \scriptsize{$\pm 0.07$} / \textbf{288.54} \scriptsize{$\pm 0.32$} \\

ResNet152
& 0.95 \scriptsize{$\pm 0.00$} / \textbf{0.90} \scriptsize{$\pm 0.00$}
& \textbf{212.06} \scriptsize{$\pm 0.82$} / 238.79 \scriptsize{$\pm 0.61$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 300.58 \scriptsize{$\pm 0.13$} / \textbf{288.03} \scriptsize{$\pm 0.31$} \\

ResNeXt101
& 0.95 \scriptsize{$\pm 0.00$} / \textbf{0.90} \scriptsize{$\pm 0.00$}
& \textbf{208.72} \scriptsize{$\pm 0.53$} / 235.77 \scriptsize{$\pm 0.46$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 300.51 \scriptsize{$\pm 0.04$} / \textbf{287.56} \scriptsize{$\pm 0.36$} \\

VGG16
& 0.94 \scriptsize{$\pm 0.00$} / \textbf{0.90} \scriptsize{$\pm 0.00$}
& \textbf{210.19} \scriptsize{$\pm 1.29$} / 237.10 \scriptsize{$\pm 1.05$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 302.36 \scriptsize{$\pm 0.14$} / \textbf{292.30} \scriptsize{$\pm 0.58$} \\

ShuffleNet
& 0.95 \scriptsize{$\pm 0.00$} / \textbf{0.90} \scriptsize{$\pm 0.00$}
& \textbf{216.29} \scriptsize{$\pm 0.81$} / 242.61 \scriptsize{$\pm 0.65$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 303.13 \scriptsize{$\pm 0.05$} / \textbf{294.84} \scriptsize{$\pm 0.28$} \\

Inception
& 0.95 \scriptsize{$\pm 0.00$} / \textbf{0.91} \scriptsize{$\pm 0.00$}
& \textbf{216.90} \scriptsize{$\pm 0.85$} / 243.16 \scriptsize{$\pm 0.73$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 303.19 \scriptsize{$\pm 0.03$} / \textbf{296.34} \scriptsize{$\pm 0.76$} \\

DenseNet161
& 0.95 \scriptsize{$\pm 0.00$} / \textbf{0.90} \scriptsize{$\pm 0.00$}
& \textbf{211.72} \scriptsize{$\pm 0.94$} / 238.48 \scriptsize{$\pm 0.75$}
& 1.00 \scriptsize{$\pm 0.00$} / \textbf{0.95} \scriptsize{$\pm 0.00$}
& 301.11 \scriptsize{$\pm 0.02$} / \textbf{288.36} \scriptsize{$\pm 0.52$} \\
\bottomrule
\end{tabular}
}
\end{small}
\end{center}
\vspace{-1em}
\end{table*}

\textbf{Interval Stability.} In Section~\ref{sec:interval stability}, we introduce a new evaluation criterion, termed \emph{interval stability} and conduct several empirical evaluations using the datasets described in Section~\ref{sec:length}, the results of which are listed in Table~\ref{table:interval_stability_reg}. We further investigate the performance of the interval stability metric using CQR as the base algorithm in Table~\ref{table:interval_stability_cqr}. The results are similar to the results in Table~\ref{table:interval_stability_reg}. We also evaluate the interval stability metric on classification tasks. As shown in Table~\ref{table:interval_stability_cls}, interval stability successfully identifies the vacuous randomness in PT.


\begin{table}[t]
\caption{Comparison between CQR and PT-CQR regarding interval stability.}
\centering
\begin{small}
\begin{tabular}{l cc}
\toprule
\textbf{Dataset} & \textbf{CQR} & \textbf{PT-CQR} \\
\midrule
meps-19      & \textbf{0.00} \scriptsize{$\pm 0.000$}  & 0.13 \scriptsize{$\pm 0.005$}  \\
meps-20      & \textbf{0.00} \scriptsize{$\pm 0.000$}  & 0.13 \scriptsize{$\pm 0.006$}  \\
meps-21      & \textbf{0.00} \scriptsize{$\pm 0.000$}  & 0.14 \scriptsize{$\pm 0.003$}  \\
bike         & \textbf{0.00} \scriptsize{$\pm 0.000$}  & 0.08 \scriptsize{$\pm 0.001$}  \\
blog-data    & \textbf{0.00} \scriptsize{$\pm 0.000$}  & 0.11 \scriptsize{$\pm 0.003$}  \\
bio          & \textbf{0.00} \scriptsize{$\pm 0.000$}  & 0.10 \scriptsize{$\pm 0.000$}  \\
facebook-1   & \textbf{0.00} \scriptsize{$\pm 0.000$}  & 0.10 \scriptsize{$\pm 0.001$}  \\
facebook-2   & \textbf{0.00} \scriptsize{$\pm 0.000$}  & 0.10 \scriptsize{$\pm 0.001$}  \\
concrete     & \textbf{0.00} \scriptsize{$\pm 0.000$}  & 0.13 \scriptsize{$\pm 0.002$}  \\
star         & \textbf{0.00} \scriptsize{$\pm 0.000$}  & 0.12 \scriptsize{$\pm 0.001$}   \\
\bottomrule
\end{tabular}
\end{small}
\label{table:interval_stability_cqr}
\end{table}

\begin{table}[t]
\caption{Comparison between RAPS and PT-RAPS in classification tasks regarding interval stability, with $\alpha=0.1, p=0.95$ and index range chosen as $300$.}
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lccccc}
\toprule
\textbf{model} &bias  &\textbf{RAPS} & \textbf{PT-RAPS} \\
\midrule
resnet18    & 40   & \textbf{0.02} $\scriptsize{\pm 0.004}$   & 12.08 $\scriptsize{\pm 0.060}$  \\
resnet50    & 40   & \textbf{0.07} $\scriptsize{\pm 0.017}$   & 11.86 $\scriptsize{\pm 0.057}$  \\
resnet101   & 40   & \textbf{0.01} $\scriptsize{\pm 0.004}$   & 11.93 $\scriptsize{\pm 0.089}$  \\
resnet152   & 40   & \textbf{0.19} $\scriptsize{\pm 0.002}$   & 11.91 $\scriptsize{\pm 0.058}$  \\
resnext101  & 40   & \textbf{0.20} $\scriptsize{\pm 0.000}$   & 11.89 $\scriptsize{\pm 0.083}$  \\
vgg16       & 40   & \textbf{0.11} $\scriptsize{\pm 0.030}$   & 12.00 $\scriptsize{\pm 0.060}$  \\
shufflenet  & 40   & \textbf{0.03} $\scriptsize{\pm 0.025}$   & 12.08 $\scriptsize{\pm 0.055}$  \\
inception   & 40   & \textbf{0.07} $\scriptsize{\pm 0.010}$   & 12.19 $\scriptsize{\pm 0.080}$  \\
densenet161 & 40   & \textbf{0.03} $\scriptsize{\pm 0.006}$   & 11.90 $\scriptsize{\pm 0.057}$  \\
\bottomrule
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vspace{-1em}
\label{table:interval_stability_cls}
\end{table}



% \subsection{Interval Continuity}
% \label{appendix:Interval Continuity}
% In this section, we present another criterion \emph{interval continuty} to evaluate conformal prediction. The intuition of interval continuity comes from why PCP is not regarded as a good method in Example~\ref{example: Patient}: for two similar inputs, the returned confidence intervals perform pretty differently. 
% Therefore, the consistency of the interval regarding the feature might matter. 
% Based on this intuition, interval continuity is formally defined in Definition~\ref{def interval continuity}.

% The proposal of interval continuity originates from the in-depth analysis of the failure root cause of the PCP method in Example~\ref{example: Patient}. As shown in the example, when there are two observation samples with highly similar feature vectors in the input space, the confidence intervals generated by the PCP method exhibit significant differences. This contradictory phenomenon reveals the potential shortcomings of traditional interval estimation methods in terms of local feature stability, although the input features have adjacency in topological space, their corresponding prediction intervals fail to maintain corresponding continuity. This discontinuity not only damages the interpretability of the predicted results, but more importantly, may lead to reliability risks for the model in practical applications.

% From a mathematical perspective, the requirement for interval continuity is inherently isomorphic to the concept of function continuity. Just as continuous functions ensure the controllability of output changes corresponding to small input disturbances, interval continuity requires that neighboring points in the feature space should correspond to some convergence of the confidence interval set. Based on this intuition, interval continuity is formally defined in Definition~\ref{def interval continuity}.



% \begin{definition}[Interval Continuity]
% \label{def interval continuity}
% Let $\bm{X}$ denote a data point with a returned confidence interval $\cC(\bm{X})$. For regression task, $\cC(\bm{X})= [q_{\text{lo}}(\bm{X}), q_{\text{hi}}(\bm{X})]$.
% The interval continuity is defined as
% {
% \small
%     \begin{equation}\label{eq:interval_conti}
%     \begin{aligned}
%         \bE_{\bm{X}, \bm{\epsilon}} \max & \left( \frac{| q_{\text{lo}}(\bm{X}) - q_{\text{lo}}(\bm{X} + \bm{\epsilon}) |}{\| \bm{\epsilon}\|}, 
%         \frac{| q_{\text{hi}}(\bm{X}) - q_{\text{hi}}(\bm{X} + \bm{\epsilon}) |}{\| \bm{\epsilon}\|}  \right),
%     \end{aligned}
%     \end{equation}}
% where $\epsilon$ is drawn from a Gaussian distribution $\cN(0, \tau^2)$ with a small $\tau$.
% \\
% For classification task, the interval continuty is defined as
% {
% \small
%     \begin{equation}\label{eq:interval_conti_classification}
%     \begin{aligned}
%         \bE_{\bm{X}, \bm{\epsilon}} \left( \frac{|\cC(\bm{X}) \cup \cC(\bm{X+\bm{\epsilon}})|-|\cC(\bm{X}) \cap \cC(\bm{X+\bm{\epsilon}})|}{\| \bm{\epsilon}\|}
%   \right),
%     \end{aligned}
%     \end{equation}}
% where $\epsilon$ is drawn from a Gaussian distribution $\cN(0, \tau^2)$ with a small $\tau$.
% \end{definition}

% In practice, one may randomly utilize several test samples to estimate the interval continuity. The PCP and VCP methods exhibit different behaviors under this criterion due to randomness. The experimental results of interval continuity are listed in Table~\ref{table:interval_continuity_reg} and Table~\ref{table:interval_continuity_cqr}. Under this new metric, VCP performs better than PCP.
% Therefore, interval continuity serves as an effective metric for eliminating PCP-type methods.

% \textbf{Experimental evaluations.} We conduct experiments using datasets as in Section~\ref{sec4.1} with the metric interval continuity. Results~(Table~\ref{table:interval_continuity_reg}, Table~\ref{table:interval_continuity_cqr}) exhibit that PCP performs worse than VCP in terms of the interval continuity.

% \begin{table}[t]
% \caption{Comparison between VCP and PCP regarding interval continuity.}
% \centering
% \begin{small}
% \begin{tabular}{l cc}
% \toprule
% \textbf{Dataset} & \textbf{VCP} & \textbf{PCP} \\
% \midrule
% meps-19      & 0.05 \scriptsize{$\pm$ 0.01} & \textbf{29.02} \scriptsize{$\pm$ 2.77} \\
% meps-20      & 0.05 \scriptsize{$\pm$ 0.01} & \textbf{28.16} \scriptsize{$\pm$ 2.70} \\
% meps-21      & 0.05 \scriptsize{$\pm$ 0.01} & \textbf{30.71} \scriptsize{$\pm$ 2.11} \\
% bike         & 0.39 \scriptsize{$\pm$ 0.03} & \textbf{11.39} \scriptsize{$\pm$ 0.52} \\
% blog-data    & 0.06 \scriptsize{$\pm$ 0.01} & \textbf{19.88} \scriptsize{$\pm$ 1.46} \\
% bio          & 1.11 \scriptsize{$\pm$ 0.04} & \textbf{38.40} \scriptsize{$\pm$ 0.79} \\
% facebook-1   & 0.27 \scriptsize{$\pm$ 0.06} & \textbf{28.56} \scriptsize{$\pm$ 3.84} \\
% facebook-2   & 0.28 \scriptsize{$\pm$ 0.02} & \textbf{27.98} \scriptsize{$\pm$ 3.30} \\
% concrete     & 0.31 \scriptsize{$\pm$ 0.01} & \textbf{13.20} \scriptsize{$\pm$ 2.40} \\
% star         & 0.01 \scriptsize{$\pm$ 0.00} & \textbf{1.91}  \scriptsize{$\pm$ 0.43} \\
% \bottomrule
% \end{tabular}
% \end{small}
% \label{table:interval_continuity_reg}
% \end{table}


% \begin{table}[t]
% \caption{Comparison between VCP and PCP on CQR regarding interval continuity.}
% \centering
% \begin{small}
% \begin{tabular}{l cc}
% \toprule
% \textbf{Dataset} & \textbf{VCP} & \textbf{PCP} \\
% \midrule
% meps-19      & 0.11 \scriptsize{$\pm$ 0.05} & \textbf{37.09} \scriptsize{$\pm$ 10.16} \\
% meps-20      & 0.13 \scriptsize{$\pm$ 0.04} & \textbf{41.92} \scriptsize{$\pm$ 7.99}  \\
% meps-21      & 0.10 \scriptsize{$\pm$ 0.05} & \textbf{36.21} \scriptsize{$\pm$ 9.91}  \\
% bike         & 0.47 \scriptsize{$\pm$ 0.03} & \textbf{32.18} \scriptsize{$\pm$ 2.23}  \\
% blog-data    & 0.21 \scriptsize{$\pm$ 0.02} & \textbf{21.47} \scriptsize{$\pm$ 3.82}  \\
% bio          & 1.04 \scriptsize{$\pm$ 0.10} & \textbf{68.68} \scriptsize{$\pm$ 2.52}  \\
% facebook-1   & 0.70 \scriptsize{$\pm$ 0.08} & \textbf{24.54} \scriptsize{$\pm$ 1.66}  \\
% facebook-2   & 1.00 \scriptsize{$\pm$ 0.06} & \textbf{20.70} \scriptsize{$\pm$ 1.61}  \\
% concrete     & 0.33 \scriptsize{$\pm$ 0.02} & \textbf{55.47} \scriptsize{$\pm$ 4.38}  \\
% star         & 0.01 \scriptsize{$\pm$ 0.00} & \textbf{16.46} \scriptsize{$\pm$ 1.75}  \\
% \bottomrule
% \end{tabular}
% \end{small}
% \label{table:interval_continuity_cqr}
% \end{table}

\subsection{Ablation Studies}
\label{appendix:More Ablation Studies}
This section exhibits the ablation studies on the probability hyperparameter $p$ in PT and the bias parameter $\mu$ on different base algorithms~(Figure~\ref{fig:ablation_bike}-Figure~\ref{fig:ablation_star}). All the experiments are conducted based on various miscoverage rates $\alpha$. The experiment results demonstrate that, although not all the probability hyperparameters $p$ outperform the base algorithm, our goal is to show that \emph{there exist multiple~(at least one) probability hyperparameters such that PT-VCP outperforms VCP, which suffices to challenge the coverage-length gold standard.} Furthermore, we find that the bias parameter actually matters here, implying that PT-VCP performs better than VCP under misspecification, which validates Theorem~\ref{thm:non-smooth-length}.

\begin{figure*}[t]
    \centering
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_bike_0.pdf}
        \caption{VCP, Probability}
        \label{fig:ablation-vcp-prob_bike}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_bike_1.pdf}
        \caption{VCP, Misspecification}
        \label{fig:ablation-vcp-mis_bike}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_bike_2.pdf}
        \caption{CQR, Probability}
        \label{fig:ablation-cqr-prob_bike}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_bike_3.pdf}
        \caption{CQR, Misspecification}
        \label{fig:ablation-cqr-mis_bike}
    \end{subfigure}
    \caption{Ablation studies of dataset BIKE on different misspecification levels (b, d) and probability hyperparameters (a, c), including comparisons with VCP (a--b) and CQR (c--d).}
    \label{fig:ablation_bike}
\end{figure*}

%bio
\begin{figure}[t]
    \centering
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_bio_0.pdf}
        \caption{VCP, Probability}
        \label{fig:ablation-vcp-prob_bio}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_bio_1.pdf}
        \caption{VCP, Misspecification}
        \label{fig:ablation-vcp-mis_bio}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_bio_2.pdf}
        \caption{CQR, Probability}
        \label{fig:ablation-cqr-prob_bio}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_bio_3.pdf}
        \caption{CQR, Misspecification}
        \label{fig:ablation-cqr-mis_bio}
    \end{subfigure}
    
    \caption{Ablation studies of dataset BIO on different misspecification level (a, c) and probability hyperparameter (b, d), including the comparison with VCP (a-b) and CQR (c-d).}
    \label{fig:ablation_bio}
    
\end{figure}

%blog_data
\begin{figure*}[t]
    \centering
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_blog_data_0.pdf}
        \caption{VCP, Probability}
        \label{fig:ablation-vcp-prob_blog_data}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_blog_data_1.pdf}
        \caption{VCP, Misspecification}
        \label{fig:ablation-vcp-mis_blog_data}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_blog_data_2.pdf}
        \caption{CQR, Probability}
        \label{fig:ablation-cqr-prob_blog_data}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_blog_data_3.pdf}
        \caption{CQR, Misspecification}
        \label{fig:ablation-cqr-mis_blog_data}
    \end{subfigure}
    
    \caption{Ablation studies of dataset BLOGDATA on different misspecification level (a, c) and probability hyperparameter (b, d), including the comparison with VCP (a-b) and CQR (c-d).}
    \label{fig:ablation_blog_data}
    
\end{figure*}

%concrete
\begin{figure*}[t]
    \centering
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_concrete_0.pdf}
        \caption{VCP, Probability}
        \label{fig:ablation-vcp-prob_concrete}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_concrete_1.pdf}
        \caption{VCP, Misspecification}
        \label{fig:ablation-vcp-mis_concrete}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_concrete_2.pdf}
        \caption{CQR, Probability}
        \label{fig:ablation-cqr-prob_concrete}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_concrete_3.pdf}
        \caption{CQR, Misspecification}
        \label{fig:ablation-cqr-mis_concrete}
    \end{subfigure}
    
    \caption{Ablation studies of dataset CONCRETE on different misspecification level (a, c) and probability hyperparameter (b, d), including the comparison with VCP (a-b) and CQR (c-d).}
    \label{fig:ablation_concrete}
    
\end{figure*}

%facebook_1
\begin{figure*}[t]
    \centering
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_facebook_1_0.pdf}
        \caption{VCP, Probability}
        \label{fig:ablation-vcp-prob_facebook_1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_facebook_1_1.pdf}
        \caption{VCP, Misspecification}
        \label{fig:ablation-vcp-mis_facebook_1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_facebook_1_2.pdf}
        \caption{CQR, Probability}
        \label{fig:ablation-cqr-prob_facebook_1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_facebook_1_3.pdf}
        \caption{CQR, Misspecification}
        \label{fig:ablation-cqr-mis_facebook_1}
    \end{subfigure}
    
    \caption{Ablation studies of dataset FACEBOOK1 on different misspecification level (a, c) and probability hyperparameter (b, d), including the comparison with VCP (a-b) and CQR (c-d).}
    \label{fig:ablation_facebook_1}
    
\end{figure*}

%facebook_2
\begin{figure*}[t]
    \centering
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_facebook_2_0.pdf}
        \caption{VCP, Probability}
        \label{fig:ablation-vcp-prob_facebook_2}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_facebook_2_1.pdf}
        \caption{VCP, Misspecification}
        \label{fig:ablation-vcp-mis_facebook_2}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_facebook_2_2.pdf}
        \caption{CQR, Probability}
        \label{fig:ablation-cqr-prob_facebook_2}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_facebook_2_3.pdf}
        \caption{CQR, Misspecification}
        \label{fig:ablation-cqr-mis_facebook_2}
    \end{subfigure}
    
    \caption{Ablation studies of dataset FACEBOOK2 on different misspecification level (a, c) and probability hyperparameter (b, d), including the comparison with VCP (a-b) and CQR (c-d).}
    \label{fig:ablation_facebook_2}
    
\end{figure*}

%meps_19
\begin{figure*}[t]
    \centering
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_meps_19_0.pdf}
        \caption{VCP, Probability}
        \label{fig:ablation-vcp-prob_meps_19}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_meps_19_1.pdf}
        \caption{VCP, Misspecification}
        \label{fig:ablation-vcp-mis_meps_19}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_meps_19_2.pdf}
        \caption{CQR, Probability}
        \label{fig:ablation-cqr-prob_meps_19}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_meps_19_3.pdf}
        \caption{CQR, Misspecification}
        \label{fig:ablation-cqr-mis_meps_19}
    \end{subfigure}
    
    \caption{Ablation studies of dataset MEPS19 on different misspecification level (a, c) and probability hyperparameter (b, d), including the comparison with VCP (a-b) and CQR (c-d).}
    \label{fig:ablation_meps_19}
    
\end{figure*}

%meps_20
\begin{figure*}[t]
    \centering
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_meps_20_0.pdf}
        \caption{VCP, Probability}
        \label{fig:ablation-vcp-prob_meps_20}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_meps_20_1.pdf}
        \caption{VCP, Misspecification}
        \label{fig:ablation-vcp-mis_meps_20}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_meps_20_2.pdf}
        \caption{CQR, Probability}
        \label{fig:ablation-cqr-prob_meps_20}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_meps_20_3.pdf}
        \caption{CQR, Misspecification}
        \label{fig:ablation-cqr-mis_meps_20}
    \end{subfigure}
    
    \caption{Ablation studies of dataset MEPS20 on different misspecification level (a, c) and probability hyperparameter (b, d), including the comparison with VCP (a-b) and CQR (c-d).}
    \label{fig:ablation_meps_20}
    
\end{figure*}

%meps_21
\begin{figure*}[t]
    \centering
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_meps_21_0.pdf}
        \caption{VCP, Probability}
        \label{fig:ablation-vcp-prob_meps_21}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_meps_21_1.pdf}
        \caption{VCP, Misspecification}
        \label{fig:ablation-vcp-mis_meps_21}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_meps_21_2.pdf}
        \caption{CQR, Probability}
        \label{fig:ablation-cqr-prob_meps_21}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_meps_21_3.pdf}
        \caption{CQR, Misspecification}
        \label{fig:ablation-cqr-mis_meps_21}
    \end{subfigure}
    
    \caption{Ablation studies of dataset MEPS21 on different misspecification level (a, c) and probability hyperparameter (b, d), including the comparison with VCP (a-b) and CQR (c-d).}
    \label{fig:ablation_meps_21}
    
\end{figure*}

%star
\begin{figure*}[t]
    \centering
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_star_0.pdf}
        \caption{VCP, Probability}
        \label{fig:ablation-vcp-prob_star}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_star_1.pdf}
        \caption{VCP, Misspecification}
        \label{fig:ablation-vcp-mis_star}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_star_2.pdf}
        \caption{CQR, Probability}
        \label{fig:ablation-cqr-prob_star}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.24\linewidth}
        \centering
        \includegraphics[width=\linewidth]{figure/ablation_study_star_3.pdf}
        \caption{CQR, Misspecification}
        \label{fig:ablation-cqr-mis_star}
    \end{subfigure}
    
    \caption{Ablation studies of dataset STAR on different misspecification level (a, c) and probability hyperparameter (b, d), including the comparison with VCP (a-b) and CQR (c-d).}
    \label{fig:ablation_star}
    
\end{figure*}


\input{text/experiment_details}




% \section{The Use of Large Language Models (LLMs)}
% \label{appendix:use of llm}
% In preparing this paper, a large language model (LLM) was employed solely for language refinement purposes, such as improving the clarity and fluency of expressions. The LLM did not contribute to research ideation, methodology, data analysis, or substantive content generation. The authors fully acknowledge responsibility for all contents of the paper, including any text polished with the assistance of the LLM.