## 3 Methodology
### Explainability of Uncertain Predictions via SHAP  
CP (Section 3.3) determines where a model is confident in its outputs (\(|\mathcal{C}(\mathbf{x}_{i})|=1\)) or remains uncertain (\(|\mathcal{C}(\mathbf{x}_{i})|=2\)). Here, \(\mathcal{C}(\mathbf{x}_{i})\) denotes the set of plausible labels returned by the conformal predictor for an instance \(\mathbf{x}_{i}\). When \(\mathcal{C}(\mathbf{x}_{i})\) contains only one label, the prediction is deemed certain. Conversely, if it contains two labels, the prediction is uncertain, indicating an elevated level of ambiguity.  
Despite highlighting these uncertain cases, CP alone does not explain why such ambiguity arises. To understand the drivers behind model certainty and uncertainty, we employ an explainable AI (XAI) approach. While several XAI methods exist, we selected SHAP (SHapley Additive exPlanations) [19] due to its distinct advantages for our research questions. First, SHAP provides local, instance-level explanations, which are essential for analyzing why an individual prediction results in a certain (single-label) or uncertain (multi-label) conformal set. Second, its foundation in game theory ensures theoretical guarantees of consistency and accuracy in attributing feature importance, overcoming the potential instability of other methods like LIME.  
_SHAP Formalization_. Consider a binary classifier \(M:\mathcal{X}\rightarrow[0,1]\) with a log-odds representation:  
\[f(\mathbf{x})=(M(\mathbf{x}))=\text{ln}\Big{(}\frac{M(\mathbf{x})}{1-M( \mathbf{x})}\Big{)}. \tag{20}\]  
For a given instance \(\mathbf{x}_{i}\) and feature \(j\), the SHAP value \(\phi_{ij}\) is defined so that  
\[f(\mathbf{x}_{i})=\phi_{i0}+\sum_{k=1}^{d}\phi_{ik}, \tag{21}\]  
where \(\phi_{i0}=\mathbb{E}_{\mathbf{x}\sim\mathcal{D}_{\text{train}}}[f(\mathbf{x})]\) is the baseline expectation. Each \(\phi_{ij}\) quantifies the contribution of feature \(j\) to the instance's deviation from the baseline, computed via  
\[\phi_{ij}=\sum_{S\subseteq\{1,\ldots,d\}\setminus\{j\}}\frac{|S|!\left(d-|S|- 1\right)!}{d!}\left[f(S\cup\{j\})-f(S)\right], \tag{22}\]  
where \(f(S)\) is the conditional expectation of \(f(\mathbf{x})\) given the subset \(S\).  
Cluster Analysis Across Calibration Approaches.We apply this SHAP-based explanation framework to the CP sets \(\mathcal{C}(\mathbf{x}_{i})\) derived from various calibration methods. Specifically, we partition the test set \(\mathcal{D}_{\text{test}}\) into:  
\[\begin{split}\mathcal{C}_{\text{certain}}&=\{ \mathbf{x}_{i}\in\mathcal{D}_{\text{test}}:|\mathcal{C}(\mathbf{x}_{i})|=1\}, \\ \mathcal{C}_{\text{uncertain}}&=\{\mathbf{x}_{i}\in \mathcal{D}_{\text{test}}:|\mathcal{C}(\mathbf{x}_{i})|=2\}.\end{split} \tag{23}\]  
For both certain and uncertain groups, we calculate the mean absolute SHAP values per feature:  
\[\begin{split}\overline{\phi}_{j}^{\text{certain}}& =\frac{1}{|\mathcal{C}_{\text{certain}}|}\sum_{\mathbf{x}_{i}\in \mathcal{C}_{\text{certain}}}|\phi_{ij}|,\\ \overline{\phi}_{j}^{\text{uncertain}}&=\frac{1}{| \mathcal{C}_{\text{uncertain}}|}\sum_{\mathbf{x}_{i}\in\mathcal{C}_{\text{ uncertain}}}|\phi_{ij}|.\end{split} \tag{24}\]  
Conducting this analysis for each calibration approach in tandem with chosen CP methods exposes how feature importance varies under different model-tuning strategies, ultimately revealing the factors that cause a model to remain uncertain.