pith. sign in

arxiv: 2604.09272 · v1 · submitted 2026-04-10 · 💻 cs.LO

A Domain-Theoretic Foundation for Imprecise Probability and Credal Sets

Abbas Edalat , Pietro Di Gianantonio , Amin Farjudian This is my paper

Pith reviewed 2026-05-10 16:18 UTC · model grok-4.3

classification 💻 cs.LO
keywords imprecise probabilitycredal setsdomain theoryconditional probabilityBayesian updatingChoquet integrationcapacity theoryiterated function systems
0
0 comments X

The pith

Domain theory supplies a foundation for imprecise probability and credal sets on general topological spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a domain-theoretic framework to reason about imprecise probabilities arising from incomplete event descriptions or from sets of probability distributions known as credal sets. It develops theories of conditional probability, Bayesian updating, and conditional independence, together with logical predicates that satisfy soundness and completeness results. A central technical step is the construction of a Scott-continuous mapping from any credal set to the domain of intervals that recovers classical results from capacity theory and Choquet integration. The work also introduces new computable families of credal sets generated by iterated function systems with imprecise weights, with the overall aim of unifying logical, topological, and measure-theoretic perspectives on uncertainty.

Core claim

The paper establishes a domain-theoretic framework for imprecise probability reasoning on general topological spaces whose open sets form a countably based continuous lattice. Within this framework it constructs conditional probability, derives inference rules for Bayesian updating under both partial event information and credal-set uncertainty, and extends the constructions to conditional independence. Logical predicates for these notions are shown to be sound and complete. The key contribution is a Scott-continuous mapping from arbitrary credal sets to the interval domain that realises capacity theory and Choquet integration in domain-theoretic terms. A new class of credal sets is defined,

What carries the argument

The Scott-continuous mapping from any credal set to the domain of intervals, which realises classical capacity theory and Choquet integration results.

If this is right

  • Novel inference rules become available for Bayesian updating when both event descriptions and probability measures are imprecise.
  • The framework yields a theory of conditional independence for imprecise probabilistic events.
  • Logical predicates for conditional probability, Bayesian updating, and conditional independence admit soundness and completeness theorems.
  • New families of computationally tractable credal sets are obtained from iterated function systems with imprecise probability weights.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The Scott-continuous mapping may support algorithmic implementations of robust inference in systems that must handle set-valued probabilities.
  • The unification of logical predicates with topological domains could be tested on concrete spaces such as the real line to verify preservation of independence relations.
  • The iterated-function-system construction suggests a route to generating tractable models for high-dimensional imprecise probability problems.

Load-bearing premise

The topological space must have a countably based continuous lattice of open sets and the domain-theoretic constructions for conditional probability and credal sets must extend without further restrictions.

What would settle it

An explicit credal set on a topological space lacking a countably based continuous lattice of open sets for which no Scott-continuous map to the interval domain preserves the Choquet integral.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a domain-theoretic framework for imprecise probability reasoning on general topological spaces whose open-set lattice is countably based and continuous. It constructs theories of conditional probability, Bayesian updating, and conditional independence that accommodate both partial event descriptions and credal sets; formulates logical predicates for these notions together with soundness and completeness theorems; constructs a Scott-continuous mapping from arbitrary credal sets to the interval domain that realises classical results from capacity theory and Choquet integration; and introduces a new family of credal sets generated by iterated function systems with imprecise probability weights. The resulting framework is claimed to be computable and to unify logical, topological, and measure-theoretic perspectives.

Significance. If the central constructions and theorems hold, the work supplies a computable, domain-theoretic foundation that integrates imprecise probability with logic and topology on a broad class of spaces. The soundness and completeness results for the logical predicates, together with the explicit Scott-continuous realisation of capacity-theoretic results, constitute concrete technical strengths. The new IFS-generated credal sets enlarge the class of tractable imprecise models. These contributions could support robust inference under combined partial and set-valued uncertainty.

minor comments (3)
  1. The introduction should explicitly state the precise topological assumptions (countably based continuous lattice of opens) at the outset rather than deferring them to the technical sections, to help readers assess the scope immediately.
  2. Notation for the interval domain and the Scott-continuous mapping should be introduced with a short preliminary subsection before the main construction, to improve readability for readers less familiar with domain theory.
  3. A brief comparison paragraph with existing domain-theoretic treatments of probability (e.g., those based on the probabilistic powerdomain) would clarify the novelty of the credal-set mapping.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the detailed and positive summary of our manuscript, for highlighting its potential to provide a computable domain-theoretic foundation integrating imprecise probability with logic and topology, and for recommending minor revision. The recognition of the soundness/completeness theorems, the Scott-continuous realisation of capacity-theoretic results, and the new IFS-generated credal sets is encouraging. No specific major comments or points of criticism appear in the report.

Circularity Check

0 steps flagged

No significant circularity; minor self-citations not load-bearing

full rationale

The paper develops a new domain-theoretic framework for imprecise probabilities on general topological spaces. The key constructions, such as the Scott-continuous mapping from credal sets to the interval domain realizing Choquet integration, are presented as original contributions building on standard domain theory and capacity theory. Logical predicates for conditional probability and independence lead to derived soundness and completeness results. While the authors likely cite their prior domain-theoretic work, these citations support the foundational tools rather than defining the new results by construction. No fitted parameters renamed as predictions or self-definitional loops are evident in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on standard domain-theoretic assumptions about continuous lattices and Scott-continuity, plus the domain assumption that open sets form a countably based continuous lattice; it introduces one new entity (the IFS credal sets) without independent evidence.

axioms (2)
  • domain assumption The topological space has a countably based continuous lattice of open sets
    Explicitly stated as the setting for the entire framework in the abstract.
  • standard math Domain theory supplies the continuity and computability structure needed for probability and inference
    Invoked via Scott-continuous mappings and continuous lattices throughout the described constructions.
invented entities (1)
  • New family of credal sets generated by iterated function systems with imprecise probability weights no independent evidence
    purpose: Broaden the scope of computationally tractable imprecise probabilistic models
    Introduced in the abstract as a new family without external validation or falsifiable handle provided.

pith-pipeline@v0.9.0 · 5510 in / 1509 out tokens · 45689 ms · 2026-05-10T16:18:21.797028+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    Domain theory in logical form

    Samson Abramsky. Domain theory in logical form. Annals of pure and applied logic , 51(1-2):1--77, 1991

    work page 1991

  2. [2]

    Thomas Augustin, Frank P. A. Coolen, Gert de Cooman, and Matthias C. M. Troffaes. Introduction to imprecise probabilities. Wiley, 2014

    work page 2014

  3. [3]

    Domain theory, handbook of logic in computer science (vol

    Samson Abramsky and Achim Jung. Domain theory, handbook of logic in computer science (vol. 3): semantic structures, 1995

    work page 1995

  4. [4]

    An extension result for continuous valuations

    Mauricio Alvarez-Manilla, Abbas Edalat, and Nasser Saheb-Djahromi. An extension result for continuous valuations. Journal of the London Mathematical Society , 61(2):629--640, 2000

    work page 2000

  5. [5]

    James O. Berger. Statistical Decision Theory and Bayesian Analysis . Springer-Verlag, New York, 2nd edition, 1985

    work page 1985

  6. [6]

    Theory of capacities

    Gustave Choquet. Theory of capacities. Annales de l'Institut Fourier , 5:131--295, 1954

    work page 1954

  7. [7]

    Continuity of attractors and invariant measures for iterated function systems

    PM Centore and ER Vrscay. Continuity of attractors and invariant measures for iterated function systems. Canadian Mathematical Bulletin , 37(3):315--329, 1994

    work page 1994

  8. [8]

    Dynamical systems, measures, and fractals via domain theory

    Abbas Edalat. Dynamical systems, measures, and fractals via domain theory. Information and Computation , 120(1):32--48, 1995

    work page 1995

  9. [9]

    A computational model for metric spaces

    Abbas Edalat and Reinhold Heckmann. A computational model for metric spaces. Theoretical computer science , 193(1-2):53--73, 1998

    work page 1998

  10. [10]

    Foundation of a computable solid modelling

    Abbas Edalat and Andr \'e Lieutier. Foundation of a computable solid modelling. Theoretical Computer Science , 284(2):319--345, 2002

    work page 2002

  11. [11]

    Myers, and Kari Sentz

    Scott Ferson, Vladik Kreinovich, Lev Ginzburg, Davis S. Myers, and Kari Sentz. Experimental Uncertainty Estimation and Statistics for Data with Interval Uncertainty . Sandia National Laboratories, 2003. SAND2003-4015

    work page 2003

  12. [12]

    Iterated function systems: A comprehensive survey

    Henning Fernau and Ludwig Staiger. Iterated function systems: A comprehensive survey. ACM Computing Surveys , 55(2):1--42, 2022

    work page 2022

  13. [13]

    Continuous lattices and domains , volume 93

    Gerhard Gierz, Karl Heinrich Hofmann, Klaus Keimel, Jimmie D Lawson, Michael Mislove, and Dana S Scott. Continuous lattices and domains , volume 93. Cambridge university press, 2003

    work page 2003

  14. [14]

    Non-Hausdorff Topology and Domain Theory

    Jean Goubault-Larrecq. Non-Hausdorff Topology and Domain Theory . Cambridge University Press, 2013

    work page 2013

  15. [15]

    A domain-theoretic approach to statistical programming languages

    Jean Goubault-Larrecq, Xiaodong Jia, and Clément Théron. A domain-theoretic approach to statistical programming languages. Journal of the ACM , 1:35, 2023

    work page 2023

  16. [16]

    Set Functions, Games and Capacities in Decision Making , volume 46 of Theory and Decision Library C

    Michel Grabisch. Set Functions, Games and Capacities in Decision Making , volume 46 of Theory and Decision Library C . Springer, 2016

    work page 2016

  17. [17]

    Klaus Keimel and Jimmie D. Lawson. Measure extension theorems for T_0 -spaces. Topology and its Applications , 149(1):57--83, 2005

    work page 2005

  18. [18]

    Andrew Moshier and Achim Jung

    M. Andrew Moshier and Achim Jung. A logic for probabilities in semantics. In Computer Science Logic , volume 2471 of Lecture Notes in Computer Science , pages 216--231. Springer Berlin Heidelberg, 2002

    work page 2002

  19. [19]

    Post-graduate lecture notes in advanced domain theory (incorporating the Pisa Notes )

    Gordon D Plotkin. Post-graduate lecture notes in advanced domain theory (incorporating the Pisa Notes ). Dept. of Computer Science, Univ. of Edinburgh , 1981

    work page 1981

  20. [20]

    Outline of a mathematical theory of computation

    Dana Scott. Outline of a mathematical theory of computation. In Proc. 4 th Princet. Conf. Inf. Sci. , pages 169--176, 1970

    work page 1970

  21. [21]

    Michael B. Smyth. Effectively given domains. Theoretical Computer Science , 5(3):257--274, 1977

    work page 1977

  22. [22]

    Topology via logic

    Steven Vickers. Topology via logic . Cambridge University Press, 1989

    work page 1989

  23. [23]

    Statistical Reasoning with Imprecise Probabilities

    Peter Walley. Statistical Reasoning with Imprecise Probabilities . Chapman and Hall, London, 1991

    work page 1991