## 3 Methodology
### Conformal Prediction  
CP extends UQ to generate provably valid prediction sets for binary outcomes, ensuring coverage guarantees without distributional assumptions. The framework's validity rests on the minimal assumption of exchangeability, which posits that the calibration data and new test instances are drawn from the same underlying data-generating process, regardless of its form. This is a significant advantage in healthcare, where data is often too complex to fit traditional parametric distributions. By relying only on exchangeability, CP remains valid even when used with complex \"black-box\" models whose outputs have unknown distributions. Given a binary classifier \(M:\mathcal{X}\rightarrow[0,1]\) producing estimates \(\hat{p}_{i}=\Pr(y_{i}=1\mid\mathbf{x}_{i})\), a non-conformity score \(S(\mathbf{x},y)\) quantifies model uncertainty for labels \(y\in\{0,1\}\):  
\[S(\mathbf{x},y)= \begin{cases}1-\hat{p}_{i}&\text{if $y=1$,}\\ \hat{p}_{i}&\text{if $y=0$,}\end{cases}  \tag{14}\]  
where lower scores indicate stronger agreement between \(\mathbf{x}_{i}\) and \(y\). Using the calibration set \(\mathcal{D}_{\text{cal}}=\{(\mathbf{x}_{j},y_{j})\}_{j=1}^{m}\), scores \(s_{j}=S(\mathbf{x}_{j},y_{j})\) are computed, and the \((1-\alpha)\)-quantile \(\hat{q}\) is derived as:  
\[\hat{q}=\inf\biggl{\{}q\in\mathbb{R}\,:\,\frac{\left|\{j\in\mathcal{D}_{\text{ cal}}\cup\{(\mathbf{x}_{n+1},y_{n+1})\}:s_{j}\leq q\}\right|}{m+1}\geq 1-\alpha \biggr{\}} \tag{15}\]  
For a new instance \(\mathbf{x}_{n+1}\), the prediction set \(\mathcal{C}(\mathbf{x}_{n+1})\subseteq\{0,1\}\) is:  
\[\mathcal{C}(\mathbf{x}_{n+1})=\{y\in\{0,1\}\,:\,S(\mathbf{x}_{n+1},y)\leq\hat {q}\} \tag{16}\]  
Under exchangeability of \(\mathcal{D}_{\text{cal}}\) and test data, CP guarantees:  
\[\Pr(y_{n+1}\in\mathcal{C}(\mathbf{x}_{n+1}))\geq 1-\alpha \tag{17}\]  
irrespective of \(M\)'s accuracy. Prediction sets may yield confident predictions (\(\mathcal{C}=\{0\}\) or \(\{1\}\)) or abstain (\(\mathcal{C}=\{0,1\}\)) when uncertainty exceeds \(\alpha\). In PPM, this enables risk-aware decision-making by flagging uncertain cases for human review while ensuring auditability through provable coverage rates. The framework's validity depends critically on exchangeability--a challenge in temporal processes with concept drift, addressed in subsequent subsections via adaptive methods. In this study, we adopt the SCP approach, a computationally efficient and widely used variant of the original transductive framework. Also known as Inductive Conformal Prediction, SCP decouples the calibration from the prediction phase, making it highly scalable and practical for real-world applications. The procedure involves the following steps: First, the available data is partitioned into two disjoint sets: a proper training set, \(\mathcal{D}_{\text{train}}\), and a calibration set, \(\mathcal{D}_{\text{cal}}\). The model \(M\) is trained exclusively on \(\mathcal{D}_{\text{train}}\). Second, the trained model is used to compute a non-conformity score, \(s_{j}=S(\mathbf{x}_{j},y_{j})\), for every data point \((\mathbf{x}_{j},y_{j})\) in the held-out calibration set. This yields a set of \(m=|\mathcal{D}_{\text{cal}}|\) calibration scores, \(\{s_{1},s_{2},\ldots,s_{m}\}\), which provides an empirical measure of the errors the trained model makes on data it has not seen before. Third, this set of calibration scores is used to determine a single critical threshold, \(\hat{q}\), that will guarantee the desired coverage rate, \(1-\alpha\). To account for finite sample effects, this threshold is calculated as the appropriate empirical quantile of the calibration scores. Let \(s_{(1)}\leq s_{(2)}\leq\ldots\leq s_{(m)}\) be the scores sorted in non-decreasing order. The threshold \(\hat{q}\) is set to the \(k\)-th smallest score:  
\[\hat{q}=s_{(k)},\quad\text{where}\quad k=\lceil(m+1)(1-\alpha)\rceil \tag{18}\]  
If \(k>m\), we can consider \(\hat{q}=\infty\), ensuring all prediction sets are valid. This value of \(\hat{q}\) is computed only once and is then fixed.