A Domain-Theoretic Foundation for Imprecise Probability and Credal Sets

2026-04-10

AI Summary

This work addresses the dual sources of uncertainty arising from incomplete event descriptions in topological spaces and credal sets—i.e., sets of probability distributions—as well as their interaction, by proposing a unified framework grounded in domain theory that integrates logical, topological, and measure-theoretic perspectives. The authors construct a Scott-continuous mapping from arbitrary credal sets into the interval domain and introduce an iterated function system with uncertain weights to generate novel credal sets, thereby extending the scope of computable models. Building on this foundation, they formulate new definitions of conditional probability and Bayesian updating, establish the soundness and completeness of an associated logical system, and achieve a unified approach to modeling, reasoning, and robust computation under both forms of uncertainty.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Reasoning with Beliefs
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We develop a domain-theoretic framework for imprecise probability reasoning and inference on general topological spaces with a countably based continuous lattice of open sets. We address two distinct forms of uncertainty: partial or incomplete event descriptions, and sets of probability distributions as represented by credal sets -- as well as their combination. Within this framework, we construct a theory of conditional probability and derive novel inference rules for performing Bayesian updating in the presence of these two complementary types of imprecision. These results are extended to a theory of conditional independence for imprecise probabilistic events. We also formulate logical predicates for conditional probability, Bayesian updating, and conditional independence, and we obtain the relevant soundness and completeness results. A key contribution is the construction of a Scott-continuous mapping from any credal set to the domain of intervals, providing a domain-theoretic realisation of classical results from capacity theory and Choquet integration. Finally, we introduce and study a new family of credal sets generated by iterated function systems with imprecise probability weights, broadening the scope of computationally tractable imprecise probabilistic models. The resulting computable framework unifies logical, topological, and measure-theoretic perspectives on uncertainty, supporting robust probabilistic inference under partial and set-valued information.

Problem

• imprecise probability
• credal sets
• conditional probability
• Bayesian updating
• uncertainty

Innovation

• domain theory
• imprecise probability
• credal sets
• Scott-continuous mapping
• Bayesian updating