# Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss, and Simpson's Paradox

arXiv: https://arxiv.org/abs/1712.08946 · HTML: https://mlsanity.com/paper/1712.08946/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Dec 2017.

## Built on by (most cited first, 8 of 8)

- [Credal Bayesian Deep Learning](https://mlsanity.com/paper/2302.09656/) (2023)
- [Credal Learning Theory](https://mlsanity.com/paper/2402.00957/) (2024)
- [A Novel Bayes' Theorem for Upper Probabilities](https://mlsanity.com/paper/2307.06831/) (2023)
- [Distributionally Robust Statistical Verification with Imprecise Neural Networks](https://mlsanity.com/paper/2308.14815/) (2023)
- [Conformal Prediction Regions are Imprecise Highest Density Regions](https://mlsanity.com/paper/2502.06331/) (2025)
- [Optimal Transport for ε-Contaminated Credal Sets: To the Memory of Sayan Mukherjee](https://mlsanity.com/paper/2410.03267/) (2024)
- [Robust Predictive Uncertainty and Double Descent in Contaminated Bayesian Random Features](https://mlsanity.com/paper/2602.19126/) (2026)
- [Binary Expansion Group Intersection Network](https://mlsanity.com/paper/2603.24763/) (2026)

## Abstract (arXiv)

Statistical learning using imprecise probabilities is gaining more attention because it presents an alternative strategy for reducing irreplicable findings by freeing the user from the task of making up unwarranted high-resolution assumptions. However, model updating as a mathematical operation is inherently exact, hence updating imprecise models requires the user's judgment in choosing among competing updating rules. These rules often lead to incompatible inferences, and can exhibit unsettling phenomena like dilation, contraction and sure loss, which cannot occur with the Bayes rule and precise probabilities. We revisit a number of famous "paradoxes", including the three prisoners/Monty Hall problem, revealing that a logical fallacy arises from a set of marginally plausible yet jointly incommensurable assumptions when updating the underlying imprecise model. We establish an equivalence between Simpson's paradox and an implicit adoption of a pair of aggregation rules that induce sure loss. We also explore behavioral discrepancies between the generalized Bayes rule, Dempster's rule and the Geometric rule as alternative posterior updating rules for Choquet capacities of order 2. We show that both the generalized Bayes rule and Geometric rule are incapable of updating without prior information regardless of how strong the information in our data is, and that Dempster's rule and the Geometric rule can mathematically contradict each other with respect to dilation and contraction. Our findings show that unsettling updates reflect a collision between the rules' assumptions and the inexactness allowed by the model itself, highlighting the invaluable role of judicious judgment in handling low-resolution information, and the care we must take when applying learning rules to update imprecise probabilities.