Conformalized Credal Regions for Classification with Ambiguous Ground Truth

TL;DR

This work extends conformal prediction to probability-space credal regions, enabling empirical construction of convex, closed predictive regions for classification with ambiguous ground truth. By representing ground-truth uncertainty with plausibility regions $c(X_{n+1})$ and linking them to credal regions $\mathcal{P} = \{\text{Cat}(\lambda): \lambda \in c(X_{n+1})\}$, the authors achieve coverage guarantees at level $1-\alpha$ and introduce Impprecise Highest Density Sets (IHDS) to yield narrower predictive label sets with a $(1-\delta)(1-\alpha)$-type bound. The approach disentangles epistemic and aleatoric uncertainty and provides a practical, more efficient alternative to standard conformal prediction by producing smaller prediction sets while maintaining calibration, demonstrated on synthetic and real datasets with ambiguous ground truth. The results support robust uncertainty quantification in IPML scenarios and offer a principled framework for handling annotation-imprecision in supervised learning.

Abstract

An open question in \emph{Imprecise Probabilistic Machine Learning} is how to empirically derive a credal region (i.e., a closed and convex family of probabilities on the output space) from the available data, without any prior knowledge or assumption. In classification problems, credal regions are a tool that is able to provide provable guarantees under realistic assumptions by characterizing the uncertainty about the distribution of the labels. Building on previous work, we show that credal regions can be directly constructed using conformal methods. This allows us to provide a novel extension of classical conformal prediction to problems with ambiguous ground truth, that is, when the exact labels for given inputs are not exactly known. The resulting construction enjoys desirable practical and theoretical properties: (i) conformal coverage guarantees, (ii) smaller prediction sets (compared to classical conformal prediction regions) and (iii) disentanglement of uncertainty sources (epistemic, aleatoric). We empirically verify our findings on both synthetic and real datasets.

Figures (6)

• Figure 1: Suppose we are in a $3$-class classification setting, so $\mathcal{Y}=\{1,2,3\}$. Then, any probability measure $P$ on $\mathcal{Y}$ can be seen as a probability vector. For example, suppose $P(\{1\})=0.6$, $P(\{2\})=0.3$, and $P(\{3\})=0.1$. We have that $P\equiv (0.6,0.3,0.1)^\top$. Since its elements are positive and sum up to $1$, probability vector $P$ belongs to the unit simplex $\Delta^2$ in $\mathbb{R}^3$, the purple (2D) triangle in the figure. Then, plausibility region $c(X_{n+1})$ is a closed and convex body in $\Delta^2$, such as the depicted orange (2D) heptagon (we depict it with a solid black border to highlight the fact that it is indeed closed).
• Figure 2: Empirical distribution coverage levels. The left plot is on toy dataset; the middle plot is on Dermatology DDx dataset; the right plot is on CIFAR-10H.
• Figure 3: True label coverage levels. The left plot is on the toy dataset; the middle plot is on Dermatology DDx; the third is on CIFAR-10H.
• Figure 4: Average inefficiencies. The left plot is on the toy dataset; the middle plot is on Dermatology DDx; the third is on CIFAR-10H.
• Figure 5: Inefficiency using different combinations of $\alpha$ and $\delta$.

Theorems & Definitions (13)

• Definition 1: Imprecise Highest Density Set, coolen
• Proposition 2: Computing $\underline{P}(A)$
• Lemma 3: Lower Envelopes Avoid Sure Loss
• Proposition 4: Properties of the Plausibility Region
• proof
• Corollary 4.1: Plausibility Region as a Credal Region
• Proposition 5: Probabilistic Correctness
• proof
• Proposition 6: A Version of Type-2 Validity for Our Credal Region
• proof