Conformalized Credal Regions for Classification with Ambiguous Ground Truth

Michele Caprio; David Stutz; Shuo Li; Arnaud Doucet

stat.MLNov 7, 2024

Conformalized Credal Regions for Classification with Ambiguous Ground Truth

Authors: Michele Caprio, David Stutz, Shuo Li, Arnaud Doucet

Organizations: Department of Computer Science University of Manchester · DeepMind · Department of Computer and Information Science University of Pennsylvania · Department of Statistics, University of Oxford

Abstract

An open question in 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 & tables

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 $\delta$ .

Panel of Table 1 — Table 1: Qualitative Examples from the DDx Dataset: The left column presents the original medical images, the middle column shows the label distributions annotated by experts, and the right column displays the credal regions constructed using our method.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Conformal Prediction

Conformal prediction wraps predictive models with calibrated sets or intervals that provide statistical coverage guarantees.

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Conformal prediction (CP) is a widely used frequentist framework to quantify uncertainty by constructing prediction sets with user-specified marginal coverage guarantees. In practice, CP is typically applied on top of probabilistic classifiers, which are able to express aleatoric but not epistemic uncertainty. In this paper, we consider the question of how to optimally employ CP on top of a more expressive formalism, namely credal sets, which can express both aleatoric and epistemic uncertainty. More specifically, we propose probabilistic Bernoulli prediction sets (BPS) and derive a variant that achieves conditional coverage for valid credal sets while remaining minimal in expected size. We then address the more realistic scenario in which the validity of the credal sets is not guaranteed. Assuming access to calibration data with ground-truth distributions over labels, we apply conformal risk control to BPS and derive a PAC-style guarantee: with high probability over the data, the achieved conditional coverage is at least the desired level. We validate our theoretical findings empirically over various datasets.

Abstract

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