Backward Conformal Prediction (2505.13732v1)

Abstract: We introduce $\textit{Backward Conformal Prediction}$, a method that guarantees conformal coverage while providing flexible control over the size of prediction sets. Unlike standard conformal prediction, which fixes the coverage level and allows the conformal set size to vary, our approach defines a rule that constrains how prediction set sizes behave based on the observed data, and adapts the coverage level accordingly. Our method builds on two key foundations: (i) recent results by Gauthier et al. [2025] on post-hoc validity using e-values, which ensure marginal coverage of the form $\mathbb{P}(Y_{\rm test} \in \hat C_n^{\tilde{\alpha}}(X_{\rm test})) \ge 1 - \mathbb{E}[\tilde{\alpha}]$ up to a first-order Taylor approximation for any data-dependent miscoverage $\tilde{\alpha}$, and (ii) a novel leave-one-out estimator $\hat{\alpha}^{\rm LOO}$ of the marginal miscoverage $\mathbb{E}[\tilde{\alpha}]$ based on the calibration set, ensuring that the theoretical guarantees remain computable in practice. This approach is particularly useful in applications where large prediction sets are impractical such as medical diagnosis. We provide theoretical results and empirical evidence supporting the validity of our method, demonstrating that it maintains computable coverage guarantees while ensuring interpretable, well-controlled prediction set sizes.

• The paper introduces backward conformal prediction, a novel method controlling prediction set sizes by adapting coverage, applicable in classification settings.
• The framework is based on e-values for post-hoc validity and uses a leave-one-out estimator to assess miscoverage on calibration data.
• Empirical validation shows the method's robustness on standard datasets, offering practical use cases in healthcare and inventory forecasting.

An Overview of "Backward Conformal Prediction"

The paper "Backward Conformal Prediction" introduces a novel method for conformal prediction that emphasizes controlling the size of prediction sets while maintaining conformal coverage guarantees. Traditional conformal prediction maintains a fixed coverage level, allowing the size of prediction sets to vary based on the data. In contrast, the backward conformal prediction approach adapts the coverage level based on a predefined rule for prediction set sizes. This paper focuses on binary and multi-class classification scenarios, where the outcome space is finite.

Key Contributions

The primary contribution of the paper is the development of a method that offers flexible control over the size of prediction sets, which is achieved by adapting the coverage level in response to observed data patterns. This approach is particularly beneficial in contexts where large prediction sets are impractical, such as in medical diagnostics, where decision interpretability is crucial.

1. Method Development: The proposed backward conformal prediction framework is based on two foundational elements: (i) the use of e-values for post-hoc validity as established in prior research by Gauthier et al., allowing for flexible manipulation of coverage guarantees, and (ii) a novel leave-one-out estimator that assesses the marginal miscoverage level using the calibration set.
2. Theoretical Validation: The paper presents both theoretical justification and empirical evidence to support the effectiveness of backward conformal prediction. The authors demonstrate that this method maintains valid, computable coverage guarantees while offering interpretable control over prediction set sizes.
3. Practical Implications: By prioritizing control over prediction set size, backward conformal prediction provides a flexible alternative for applications that demand both precision and interpretability, such as healthcare and inventory demand forecasting.

Theoretical Foundations

The paper bases its theoretical underpinnings on the properties of e-values, a type of statistical measure that allows for more adaptive inference than traditional p-values. It explores how these e-values contribute to valid post-hoc adjustments of miscoverage levels, enabling controlled-size prediction sets while maintaining marginal coverage.

Furthermore, the leave-one-out estimator provides practitioners with a practical tool to estimate the coverage guarantee on available calibration data. This estimator assists in making informed decisions regarding the reliability of prediction sets in real-world applications.

Experimental Evaluation

The experimental analysis conducted on standard datasets showcases the method's applicability and robustness. The empirical results confirm that backward conformal prediction indeed offers a viable solution for applications demanding both high accuracy and manageable prediction set sizes, with empirical coverage effectively bounding the computed coverage probabilities.

Future Directions

This work opens several avenues for future research. The backward conformal prediction framework could be extended to regression settings or continuous outcome spaces. Additionally, exploring alternative miscoverage definitions and investigating more complex data-dependent rules for set sizes might further enhance the flexibility and applicability of the method.

Overall, this paper contributes significantly to the conformal prediction literature by offering a new perspective on controlling prediction set sizes. It enriches the toolkit available to researchers and practitioners confronting challenges in uncertainty quantification and decision-making under uncertainty.