A Domain-Theoretic Foundation for Imprecise Probability and Credal Sets

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.

• The paper introduces a domain-theoretic approach that formalizes imprecise probabilities and credal sets through interval representations.
• It employs Scott-continuous mappings to ensure computable approximations and robust Bayesian updates for handling partial event observations.
• The work unifies logical, topological, and measure-theoretic methods, paving the way for safety-critical, risk-aware inference in uncertain environments.

A Domain-Theoretic Foundation for Imprecise Probability and Credal Sets

Introduction and Motivation

This work develops a domain-theoretic foundation for imprecise probability, specifically targeting inference under both partial event descriptions and distributional ambiguity modeled via credal sets. The framework is constructed on countably based continuous lattices of open sets for general second-countable, locally compact, sober topological spaces. This setting subsumes standard Polish, metric, and domain-theoretic spaces underpinning mainstream probability theory.

Contrary to classical probability's reliance on precise events and single-valued distributions, the presented framework robustly incorporates partial observations and set-valued models (credal sets), supporting robust inference suitable for safety-critical and epistemically uncertain environments.

Event Domains and Interval Probability

Events are formalized as disjoint pairs of open sets $(O_1, O_2)$ in a space $D$, leading to the bounded-complete event domain $\mathcal{E}(D)$. This structure naturally supports partial event specification: incomplete evidence is represented by a range of admissible sets between $O_1$ (definite inclusion) and $D \setminus O_2$ (definite exclusion). The critical technical device is the Scott-continuous interval probability map $\Pi_\sigma: \mathcal{E}(D) \rightarrow I[0,1]$, assigning each event an interval $[\sigma(O_1), 1-\sigma(O_2)]$, where $\sigma$ is a probability valuation.

A key theoretical contribution is the full domain-theoretic formalization of these interval probabilities, ensuring support for computable approximation and continuity properties essential for algorithmic applications.

Credal Sets and Capacity Theory

On the distributional side, imprecision is modeled by convex, compact sets of probability valuations—credal sets—which are handled as elements of the upper space $U(\mathcal{P}(X))$ endowed with the Scott topology. Singletons in this space recover standard (precise) probabilities, but the key insight is the construction of Scott-continuous envelope maps from credal sets to intervals, which generalize classical lower and upper probabilities (belief/plausibility) from capacity theory and are representable as Choquet integrals.

For convex polytopal credal sets, endpoint calculations reduce to minima and maxima over vertices, enforcing computational tractability.

Conditional Probability, Independence, and Inference

Conditional probability is extended to imprecise events and credal sets via interval-valued mappings. The framework provides predicates and corresponding logical inference rules for lower and upper conditional probabilities, with full soundness and completeness theorems. The conditional probability interval for events $(V_1,V_2)$ given $D$0 is:

$D$1

This result extends through the Scott-continuous lifting to credal sets.

The theory incorporates robust versions of conditional independence for imprecise events, giving both a general (Fréchet) rule and a strong independence rule. The Fréchet rule provides conservative bounds (with interval width reflecting dependence uncertainty), while strong independence assumptions yield sharper intervals (products of marginal endpoints).

Bayesian Updating under Imprecision

A monotonicity-based analysis of Bayes’ rule yields new, exact endpoint formulas for posterior probabilities under joint imprecision of priors, likelihoods, and evidence. The sharp interval extension of Bayesian updating is proven, with explicit inference rules (B1–B4) and completeness established. For sets of priors (credal sets), the posterior is given by the range:

$D$2

Numerical exemplars document that classical point-estimates are always contained within the resulting intervals, but often significantly underrepresent true uncertainty and risk.

Iterated Function Systems and Computational Models

The authors introduce a new class of credal sets, generated as invariant measures of Iterated Function Systems (IFS) with imprecise probability weights. Extending Hutchinson’s classical fixed-point theory, they establish Scott-continuity of the mapping from admissible probability weights to the space of invariant measures, both at the level of measures and credal sets. This extends the computational tractability of imprecise probabilistic models to broad classes of fractal and dynamical systems.

Additionally, a foundation is laid for interval-based Markov chains, showing that tight interval bounds on long-run stationary distributions can be obtained via direct bilinear optimization on the vertices of the credal polytope.

Logical and Topological Integration

Throughout, predicates for probability, conditional probability, and independence are specified via the theory of approximable relations on countable bases, underpinned by domain-theoretic duality and computation topology. All key maps (interval probability, conditioning, Bayesian update, independence) are proven Scott-continuous and thus compatible with the computational approximation paradigm.

Implications and Future Directions

The establishment of a robust, computable, and logically-sound framework for imprecise probability substantially strengthens the theoretical foundations for credal (imprecise Bayesian) networks, robust inference, and risk-aware sequential decision making. By guaranteeing Scott continuity, all inferences are computably approximable—a critical property for any real-world deployment under computational constraint.

This approach unifies logical, topological, and measure-theoretic treatments of uncertainty and paves the way for:

• Fully domain-theoretic credal networks and Bayesian networks with imprecise parameters/observations
• Exact and approximate algorithms for robust probabilistic inference
• Extension to other types of set-valued uncertainty models (e.g., capacities, possibility measures)
• Safety verification and robust learning in settings with epistemic/model uncertainty

Conclusion

This work rigorously advances the mathematical and computational apparatus for imprecise probability. It systematically integrates domain theory, topology, and measure theory, constructs interval representations and inference for both event imprecision and credal sets, and extends the machinery to complex systems such as IFSs and Markov chains with interval parameters. The resulting sound, complete, and computable rules position this framework as a foundational tool for future research and applications in robust AI, risk analysis, and formal methods for uncertain reasoning (2604.09272).