Abstract

Conformal Prediction (CP) allows to perform rigorous uncertainty quantification by constructing a prediction set $C(X)$ satisfying $\mathbb{P}_{agg}(Y \in C(X))\geq 1-\alpha$ for a user-chosen $\alpha \in [0,1]$ by relying on calibration data $(X_1,Y_1),...,(X_n,Y_n)$ from $\mathbb{P}=\mathbb{P}_{agg}^{X} \otimes \mathbb{P}_{agg}^{Y|X}$. It is typically implicitly assumed that $\mathbb{P}_{agg}^{Y|X}$ is the ``true'' posterior label distribution. However, in many real-world scenarios, the labels $Y_1,...,Y_n$ are obtained by aggregating expert opinions using a voting procedure, resulting in a one-hot distribution $\mathbb{P}_{vote}^{Y|X}$. This is the case for most datasets, even well-known ones like ImageNet. For such ``voted'' labels, CP guarantees are thus w.r.t. $\mathbb{P}_{vote}=\mathbb{P}_{agg}^X \otimes \mathbb{P}_{vote}^{Y|X}$ rather than the true distribution $\mathbb{P}_{agg}$. In cases with unambiguous ground truth labels, the distinction between $\mathbb{P}_{vote}$ and $\mathbb{P}_{agg}$ is irrelevant. However, when experts do not agree because of ambiguous labels, approximating $\mathbb{P}_{agg}^{Y|X}$ with a one-hot distribution $\mathbb{P}_{vote}^{Y|X}$ ignores this uncertainty. In this paper, we propose to leverage expert opinions to approximate $\mathbb{P}_{agg}^{Y|X}$ using a non-degenerate distribution $\mathbb{P}_{agg}^{Y|X}$. We then develop Monte Carlo CP procedures which provide guarantees w.r.t. $\mathbb{P}_{agg}=\mathbb{P}_{agg}^X \otimes \mathbb{P}_{agg}^{Y|X}$ by sampling multiple synthetic pseudo-labels from $\mathbb{P}_{agg}^{Y|X}$ for each calibration example $X_1,...,X_n$. In a case study of skin condition classification with significant disagreement among expert annotators, we show that applying CP w.r.t. $\mathbb{P}_{vote}$ under-covers expert annotations: calibrated for $72\%$ coverage, it falls short by on average $10\%$; our Monte Carlo CP closes this gap both empirically and theoretically. We also extend Monte Carlo CP to multi-label classification and CP with calibration examples enriched through data augmentation.

David Stutz and Abhijit Guha Roy and Tatiana Matejovicova and Patricia Strachan and Ali Taylan Cemgil and Arnaud Doucet.
Conformal prediction under ambiguous ground truth. TMLR, 2023.