Credal Two-Sample Tests of Epistemic Uncertainty (2410.12921v2)

Abstract: We introduce credal two-sample testing, a new hypothesis testing framework for comparing credal sets -- convex sets of probability measures where each element captures aleatoric uncertainty and the set itself represents epistemic uncertainty that arises from the modeller's partial ignorance. Compared to classical two-sample tests, which focus on comparing precise distributions, the proposed framework provides a broader and more versatile set of hypotheses. This approach enables the direct integration of epistemic uncertainty, effectively addressing the challenges arising from partial ignorance in hypothesis testing. By generalising two-sample test to compare credal sets, our framework enables reasoning for equality, inclusion, intersection, and mutual exclusivity, each offering unique insights into the modeller's epistemic beliefs. As the first work on nonparametric hypothesis testing for comparing credal sets, we focus on finitely generated credal sets derived from i.i.d. samples from multiple distributions -- referred to as credal samples. We formalise these tests as two-sample tests with nuisance parameters and introduce the first permutation-based solution for this class of problems, significantly improving existing methods. Our approach properly incorporates the modeller's epistemic uncertainty into hypothesis testing, leading to more robust and credible conclusions, with kernel-based implementations for real-world applications.

• The paper introduces a novel permutation-based credal testing framework that robustly models epistemic uncertainty.
• It employs kernel-based methods to compare credal sets, achieving superior power and consistent Type I error control.
• The framework offers practical applications in machine learning, including domain generalization and distributionally robust optimization.

Overview of "Credal Two-Sample Tests of Epistemic Ignorance"

The paper introduces a novel framework for hypothesis testing in the context of credal sets, which offer a mechanism to model epistemic uncertainty. Unlike classical two-sample tests that solely compare precise probability distributions, this approach accounts for uncertainties due to partial ignorance by comparing credal sets—collections of probability measures representing diverse epistemic beliefs.

Framework Considerations

The researchers propose "credal two-sample testing," where the epistemic uncertainty is inherently modeled via credal sets. These sets are convex hulls of discrete sets of probability distributions, effectively capturing a modeler's partial ignorance without reliance on a single prior. This offers a robust Bayesian interpretation by encompassing all viable priors. The research introduces several null hypotheses centered on credal sets: equality, inclusion, intersection, and mutual exclusivity, which provide a structured examination of epistemic beliefs.

Technical Contributions

1. Permutation-Based Solution: The paper presents the first permutation-based methodology for credal set hypothesis testing, enhancing test robustness by accommodating modeler uncertainty. This approach achieves non-parametric independence, controlling Type I error asymptotically while ensuring test consistency.
2. Kernel-Based Implementation: The researchers develop kernel methods to implement these tests, conferring flexibility to handle a broad spectrum of data types, including continuous data, sets, and images.
3. Estimation and Testing: The framework rigorously treats credal testing as precise testing with nuisance parameters, employing a two-stage approach—epistemic alignment for parameter estimation followed by hypothesis testing. An adaptive sample splitting strategy ensures accurate Type I error control, preventing estimation error from skewing test results.

Results and Implications

The proposed methods empirically outperform existing approaches, such as MMDQ, demonstrating superior power and robustness. The adaptive splitting ensures valid Type I error control while maintaining sensitivity to detect true differences under alternatives.

Potential Applications

The credal testing framework can be transformative in areas like domain generalization and distributionally robust optimization. For example, in machine learning, it provides a method to validate models trained under the assumption that deployment distributions are credal sets, thus mitigating potential model risks due to distributional assumptions.

Future Directions

Exploration of alternative credal set generation methods, enhanced kernel selection for improved test power, and advanced multi-sample techniques are promising avenues for increasing the efficacy of credal hypothesis tests. Moreover, adaptive strategies for determining sample splits could offer further improvements in test power and validity.

Limitations

While credal sets offer a structured approach to modeling uncertainty, they paradoxically suggest increased epistemic uncertainty with more information. This counterintuitive result poses practical challenges. Nevertheless, with subjective judgment on data quality and distribution selection, these concerns can be alleviated.

Conclusion

This work significantly advances the statistical toolbox available for tackling epistemic uncertainties in hypothesis testing. By addressing the nuances of credal sets, the proposed framework offers a robust avenue for making credible conclusions in the presence of partial ignorance, positioning itself as a pivotal development in statistical reasoning.