# When Do Credal Sets Stabilize? Fixed-Point Theorems for Credal Set Updates

arXiv: https://arxiv.org/abs/2510.04769 · HTML: https://mlsanity.com/paper/2510.04769/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 25 of its 64 references that are in the map.
- First posted on arXiv 6 Oct 2025.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 25 of 25)

- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Continual Lifelong Learning with Neural Networks: A Review](https://mlsanity.com/paper/1802.07569/) (2018)
- [Performative Prediction](https://mlsanity.com/paper/2002.06673/) (2020)
- [Performative Prediction: Past and Future](https://mlsanity.com/paper/2310.16608/) (2023)
- [ROAD-R: The Autonomous Driving Dataset with Logical Requirements](https://mlsanity.com/paper/2210.01597/) (2022)
- [Credal Learning Theory](https://mlsanity.com/paper/2402.00957/) (2024)
- [CreINNs: Credal-Set Interval Neural Networks for Uncertainty Estimation in Classification Tasks](https://mlsanity.com/paper/2401.05043/) (2024)
- [Explaining the Uncertain: Stochastic Shapley Values for Gaussian Process Models](https://mlsanity.com/paper/2305.15167/) (2023)
- [Conformalized Credal Regions for Classification with Ambiguous Ground Truth](https://mlsanity.com/paper/2411.04852/) (2024)
- [Credal Two-Sample Tests of Epistemic Uncertainty](https://mlsanity.com/paper/2410.12921/) (2024)
- [Domain Generalisation via Imprecise Learning](https://mlsanity.com/paper/2404.04669/) (2024)
- [Distributionally Robust Statistical Verification with Imprecise Neural Networks](https://mlsanity.com/paper/2308.14815/) (2023)
- [Integral Imprecise Probability Metrics](https://mlsanity.com/paper/2505.16156/) (2025)
- [Scoring Rules and Calibration for Imprecise Probabilities](https://mlsanity.com/paper/2410.23001/) (2024)
- [Approximating Counterfactual Bounds while Fusing Observational, Biased and Randomised Data Sources](https://mlsanity.com/paper/2307.16577/) (2023)
- [Conformal Prediction Regions are Imprecise Highest Density Regions](https://mlsanity.com/paper/2502.06331/) (2025)
- [A calibration test for evaluating set-based epistemic uncertainty representations](https://mlsanity.com/paper/2502.16299/) (2025)
- [An Imprecise SHAP as a Tool for Explaining the Class Probability Distributions under Limited Training Data](https://mlsanity.com/paper/2106.09111/) (2021)
- [Truthful Elicitation of Imprecise Forecasts](https://mlsanity.com/paper/2503.16395/) (2025)
- [Exact Shapley Attributions in Quadratic-time for FANOVA Gaussian Processes](https://mlsanity.com/paper/2508.14499/) (2025)
- [IBCL: Zero-shot Model Generation under Stability-Plasticity Trade-offs](https://mlsanity.com/paper/2305.14782/) (2023)
- [Generalization Bounds and Stopping Rules for Learning with Self-Selected Data](https://mlsanity.com/paper/2505.07367/) (2025)
- [A Category-Theoretic Analysis of Conformal Prediction](https://mlsanity.com/paper/2507.04441/) (2025)
- [Sensitivity analysis for finite Markov chains in discrete time](https://mlsanity.com/paper/1408.2029/) (2014)
- [Compositional imprecise probability](https://mlsanity.com/paper/2405.09391/) (2024)

## Built on by (most cited first, 1 of 1)

- [Beyond Prediction-Set Size: Quantifying Conformal Uncertainty through Credal Geometry](https://mlsanity.com/paper/2602.01667/) (2026)

## Abstract (arXiv)

Many machine learning algorithms rely on iterative updates of uncertainty representations, ranging from variational inference and expectation-maximization, to reinforcement learning, continual learning, and multi-agent learning. In the presence of imprecision and ambiguity, credal sets -- closed, convex sets of probability distributions -- have emerged as a popular framework for representing imprecise probabilistic beliefs. Under such imprecision, many learning problems in imprecise probabilistic machine learning (IPML) may be viewed as processes involving successive applications of update rules on credal sets. This naturally raises the question of whether this iterative process converges to stable fixed points -- or, more generally, under what conditions on the updating mechanism such fixed points exist, and whether they can be attained. We provide the first analysis of this problem, and illustrate our findings using Credal Bayesian Deep Learning as a concrete example. Our work demonstrates that incorporating imprecision into the learning process not only enriches the representation of uncertainty, but also reveals structural conditions under which stability emerges, thereby offering new insights into the dynamics of iterative learning under imprecision.