Statistical inference with belief functions: A survey

Fabio Cuzzolin

Pith reviewed 2026-05-11 03:15 UTC · model grok-4.3

classification 🧮 math.ST cs.AIcs.LGstat.TH

keywords belief functionsstatistical inferenceuncertaintyDempster-Shafer theoryevidential reasoningimprecise probabilitiesdata scarcity

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The pith

Belief functions allow statistical inference of uncertainty from data too scarce to learn any probability distribution.

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

The survey collects and organizes the principal methods for deriving a belief measure from statistical samples. Belief functions represent uncertainty by assigning degrees of belief to sets of possibilities rather than to single outcomes, which avoids the need for a complete probability law when observations are limited. A reader would care because many practical problems supply too few data points for classical statistical learning yet still require quantified statements about what is known and what remains unknown. The review therefore supplies a practical toolkit for evidential reasoning under ignorance.

Core claim

Belief functions form a mathematical framework for characterising uncertainty that is especially suited to situations in which lack of data prevents learning a probability distribution. The survey reviews the most significant contributions to the problem of inferring a belief measure directly from statistical data, presenting the main constructions, combination rules, and decision procedures developed for this purpose.

What carries the argument

The belief function itself, a monotone set function that quantifies the degree of support for each subset of the frame of discernment, which is learned from data and then used for further inference.

Load-bearing premise

That the reviewed literature contains the essential techniques and that belief functions supply a distinctly more suitable representation than probability when data are scarce.

What would settle it

A head-to-head comparison on multiple real data sets showing that standard probabilistic methods yield equal or better calibrated predictions and decisions than any belief-function procedure in the same low-sample regime.

read the original abstract

Belief functions are a powerful and popular framework for the mathematical characterisation of uncertainty, in particular in situations in which lack of data renders learning a probability distribution for the problem impractical. The first step in a reasoning chain based on belief functions is inference: how to learn a belief measure from the available data. In this survey we focus, in particular, on making inference from statistical data, and review the most significant contributions in the area.

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.

Desk Editor's Note private letter to a colleague

This is a straightforward survey compiling existing work on belief function inference for limited-data settings, with no new results and no stated criteria for selecting the reviewed papers.

read the letter

This paper is a survey reviewing statistical inference methods based on belief functions, especially when data scarcity makes learning probability distributions impractical. It focuses on the inference step in reasoning chains and pulls together prior contributions in the imprecise probabilities area. For someone new to the subfield, the organization could serve as a useful starting point to see how belief functions handle uncertainty without full probabilistic models. The abstract sets a clear scope, and the work stays grounded in cited literature rather than introducing unverified claims. What stands out is the emphasis on practical settings where standard stats fall short, which aligns with real needs in AI and decision-making under ignorance. The main limitation is the lack of any description of how the literature was gathered—no search method, databases, keywords, date ranges, or inclusion rules are mentioned. This leaves the claim of covering the 'most significant contributions' open to questions about completeness or author bias in selection. Without that transparency, it's harder to treat the survey as authoritative. The paper contains no derivations, predictions, or data analysis of its own, so there are no issues with fitting, circularity, or invented entities. It is aimed at readers already interested in belief functions or uncertainty modeling who want a consolidated view rather than novel methods. I would bring it to a reading group only if the group focuses on imprecise probabilities. I would not cite it in my own work in the next year since it adds no unique insights. It deserves peer review because the topic has a dedicated audience and the core organization is coherent, even if revisions should address the selection process.

Referee Report

1 major / 0 minor

Summary. The manuscript is a survey on statistical inference with belief functions. It positions belief functions as a framework for uncertainty characterization that is particularly useful when data scarcity makes learning probability distributions impractical. The paper focuses on the inference step of deriving a belief measure from statistical data and reviews what it describes as the most significant contributions in this area.

Significance. If the review is comprehensive, it could serve as a useful reference consolidating methods for belief-function inference, which offers an alternative to probability theory for handling epistemic uncertainty in low-data regimes. This may aid researchers working at the intersection of statistics, evidence theory, and robust decision-making.

major comments (1)

[Abstract] Abstract: The claim to review 'the most significant contributions in the area' is not supported by any description of the literature search methodology, including databases, keywords, date ranges, citation thresholds, or inclusion/exclusion rules. This omission directly affects the ability to evaluate whether the reviewed works are indeed the most significant or representative.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback on our survey manuscript. We address the major comment below and will revise the paper to improve transparency on the scope and selection of contributions.

read point-by-point responses

Referee: The claim to review 'the most significant contributions in the area' is not supported by any description of the literature search methodology, including databases, keywords, date ranges, citation thresholds, or inclusion/exclusion rules. This omission directly affects the ability to evaluate whether the reviewed works are indeed the most significant or representative.

Authors: We agree that the abstract's phrasing would benefit from greater transparency. This is a narrative survey drawing on the authors' expertise rather than a formal systematic review. In the revised manuscript we will add a short paragraph early in the introduction (and adjust the abstract wording) that describes the main sources consulted (foundational papers by Dempster and Shafer, key articles in the International Journal of Approximate Reasoning, and proceedings of BELIEF and ISIPTA conferences), the time span covered (primarily 1970s to present), and the selection criteria (seminal works plus recent contributions that specifically address statistical inference from data and have demonstrable influence in the field). This will clarify that the review is curated rather than exhaustive, while still justifying the focus on the most significant contributions. revision: yes

Circularity Check

0 steps flagged

No circularity: survey paper with no derivations or predictions

full rationale

This is a literature survey with no internal derivation chain, equations, predictions, or fitted parameters. The abstract and structure consist solely of reviewing external contributions to belief-function inference. No step reduces by construction to the paper's own inputs, self-citations, or ansatzes. The claim of covering 'the most significant contributions' is a curatorial judgment without stated selection criteria, but this is not a mathematical derivation and does not match any enumerated circularity pattern. The paper is self-contained as a review of independent prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a survey paper with no new mathematical content, so it introduces no free parameters, axioms, or invented entities beyond those in the reviewed prior work.

pith-pipeline@v0.9.0 · 5353 in / 950 out tokens · 27254 ms · 2026-05-11T03:15:56.077813+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages

[1]

Connecting Dempster–Shafer belief functions with likelihood-based inference.Synthese, 123(3):347–364,

[Aickin, 2000] Mikel Aickin. Connecting Dempster–Shafer belief functions with likelihood-based inference.Synthese, 123(3):347–364,

work page 2000
[2]

Fiducial inference and be- lief functions

[Almond, 1992] Russell Almond. Fiducial inference and be- lief functions. Technical report, University of Washington,

work page 1992
[3]

Novelty detection in the belief functions frame- work

[Aregui and Denœux, 2006 ] Astride Aregui and Thierry Denœux. Novelty detection in the belief functions frame- work. InProceedings of the International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2006), volume 6, pages 412–419,

work page 2006
[4]

Credal and interval deep evidential classifications.arXiv preprint arXiv:2512.05526,

[Caprioet al., 2025 ] Michele Caprio, Shireen K Manchin- gal, and Fabio Cuzzolin. Credal and interval deep evi- dential classifications.arXiv preprint arXiv:2512.05526,

work page arXiv 2025
[5]

Ambiguity reduction through new statistical data.International Journal of Approximate Reasoning, 24(2-3):283–299,

[Chateauneuf and Vergnaud, 2000] Alain Chateauneuf and Jean-Christophe Vergnaud. Ambiguity reduction through new statistical data.International Journal of Approximate Reasoning, 24(2-3):283–299,

work page 2000
[6]

Geometric analysis of belief space and conditional subspaces

[Cuzzolin and Frezza, 2001] Fabio Cuzzolin and Ruggero Frezza. Geometric analysis of belief space and conditional subspaces. InISIPTA, pages 122–132,

work page 2001
[7]

Belief modeling regression for pose estimation

[Cuzzolin and Gong, 2013] Fabio Cuzzolin and Wenjuan Gong. Belief modeling regression for pose estimation. InProceedings of the 16th International Conference on Information Fusion (FUSION 2013), pages 1398–1405,

work page 2013
[8]

Springer,

[Cuzzolin and Sultana, 2024] Fabio Cuzzolin and Maryam Sultana.Epistemic Uncertainty in Artificial Intelligence. Springer,

work page 2024
[9]

Geometry of dempster’s rule of combination.IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 34(2):961–977,

[Cuzzolin, 2004] Fabio Cuzzolin. Geometry of dempster’s rule of combination.IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 34(2):961–977,

work page 2004
[10]

Three alternative combi- natorial formulations of the theory of evidence.Intelligent Data Analysis, 14(4):439–464,

[Cuzzolin, 2010] Fabio Cuzzolin. Three alternative combi- natorial formulations of the theory of evidence.Intelligent Data Analysis, 14(4):439–464,

work page 2010
[11]

Springer,

[Cuzzolin, 2014] Fabio Cuzzolin.Belief functions: theory and applications. Springer,

work page 2014
[12]

Belief likelihood function for generalised logistic regression.arXiv preprint arXiv:1808.02560, 2018a

[Cuzzolin, 2018a] Fabio Cuzzolin. Belief likelihood func- tion for generalised logistic regression.arXiv preprint arXiv:1808.02560,

work page arXiv
[13]

Uncertainty measures: The big picture.arXiv preprint arXiv:2104.06839,

[Cuzzolin, 2021] Fabio Cuzzolin. Uncertainty measures: The big picture.arXiv preprint arXiv:2104.06839,

work page arXiv 2021
[14]

Reasoning with random sets: An agenda for the future.arXiv preprint arXiv:2401.09435,

[Cuzzolin, 2023] Fabio Cuzzolin. Reasoning with ran- dom sets: An agenda for the future.arXiv preprint arXiv:2401.09435,

work page arXiv 2023
[15]

Generalising realisability in statistical learning theory under epistemic uncertainty

[Cuzzolin, 2024a] Fabio Cuzzolin. Generalising realisability in statistical learning theory under epistemic uncertainty. arXiv preprint arXiv:2402.14759,

work page arXiv
[16]

Lambert Academic Pub- lishing, September

[Cuzzolin, September 2014] Fabio Cuzzolin.Visions of a generalized probability theory. Lambert Academic Pub- lishing, September

work page 2014
[17]

de Campos, Juan F

[de Camposet al., 1994 ] Luis M. de Campos, Juan F. Huete, and Seraf´ın Moral. Probability intervals: a tool for un- certain reasoning.International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2(2):167–196,

work page 1994
[18]

Dempster

[Dempster, 1967] Arthur P. Dempster. Upper and lower probability inferences based on a sample from a finite uni- variate population.Biometrika, 54(3-4):515–528,

work page 1967
[19]

Upper and lower prob- abilities induced by a multivalued mapping

[Dempster, 2008] Arthur P Dempster. Upper and lower prob- abilities induced by a multivalued mapping. InClassic works of the Dempster-Shafer theory of belief functions, pages 57–72. Springer,

work page 2008
[20]

Dempster

[Denœux and Dempster, 2008 ] Thierry Denœux and Arthur P. Dempster. The Dempster–Shafer calculus for statisticians.International Journal of Approximate Reasoning, 48(2):365–377,

work page 2008
[21]

Constructing belief func- tions from sample data using multinomial confidence re- gions.International Journal of Approximate Reasoning, 42(3):228–252,

[Denœux, 2006 ] Thierry Denœux. Constructing belief func- tions from sample data using multinomial confidence re- gions.International Journal of Approximate Reasoning, 42(3):228–252,

work page 2006
[22]

Maximum likelihood from evidential data: An extension of the EM algorithm

[Denœux, 2010 ] Thierry Denœux. Maximum likelihood from evidential data: An extension of the EM algorithm. In Christian Borgelt, Gil Gonz ´alez-Rodr´ıguez, Wolfgang Trutschnig, Mar´ıa Asunci´on Lubiano, Mar´ıa ´Angeles Gil, Przemysław Grzegorzewski, and Olgierd Hryniewicz, ed- itors,Combining Soft Computing and Statistical Methods in Data Analysis, pages ...

work page 2010
[23]

Likelihood-based belief function: Justification and some extensions to low-quality data.International Journal of Approximate Reasoning, 55(7):1535–1547,

[Denœux, 2014 ] Thierry Denœux. Likelihood-based belief function: Justification and some extensions to low-quality data.International Journal of Approximate Reasoning, 55(7):1535–1547,

work page 2014
[24]

Edlefsen, Chuanhai Liu, and Arthur P

[Edlefsenet al., 2009 ] Paul T. Edlefsen, Chuanhai Liu, and Arthur P. Dempster. Estimating limits from Poisson count- ing data using Dempster–Shafer analysis.The Annals of Applied Statistics, 3(2):764–790,

work page 2009
[25]

Myers, and Kari Sentz

[Fersonet al., 2003 ] Scott Ferson, Vladik Kreinovich, Lev Ginzburg, Davis S. Myers, and Kari Sentz. Constructing probability boxes and Dempster–Shafer structures. Tech- nical Report SAND2002-4015, Sandia National Laborato- ries,

work page 2003
[26]

[Fisher, 1922] Ronald A. Fisher. On the mathematical foun- dations of theoretical statistics.Philosophical Transac- tions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 222(594-604):309– 368,

work page 1922
[27]

[Fisher, 1935] Ronald A. Fisher. The fiducial argument in statistical inference.Annals of Human Genetics, 6(4):391– 398,

work page 1935
[28]

Attribute sampling: A belief-function ap- proach to statistical audit evidence.Auditing: A Journal of Practice & Theory, 19(1):145–155,

[Gillett and Srivastava, 2000] Peter R Gillett and Rajendra P Srivastava. Attribute sampling: A belief-function ap- proach to statistical audit evidence.Auditing: A Journal of Practice & Theory, 19(1):145–155,

work page 2000
[29]

Klopotek

[Klopotek, 1996] Mieczyslaw A. Klopotek. Identification of belief structure in Dempster–Shafer theory.Foundations of Computing and Decison Sciences, 21(1):35–54,

work page 1996
[30]

Springer-Verlag,

[Kohlas and Monney, 1995] J¨urg Kohlas and Paul-Andr ´e Monney.A Mathematical Theory of Hints: An Approach to Dempster–Shafer Theory of Evidence, volume 425 of Lecture Notes in Economics and Mathematical Systems. Springer-Verlag,

work page 1995
[31]

[Kong, 1988] C. T. A. Kong. A belief function generaliza- tion of Gibbs ensemble. Technical Report S-122, Harvard University,

work page 1988
[32]

Probabilistic analysis of belief functions

[Kramosil, 2002] Ivan Kramosil. Probabilistic analysis of belief functions. InISFR International Series on Systems Science and Engineering, volume

work page 2002
[33]

[Kyburg, 1987] Henry E. Kyburg. Bayesian and non- Bayesian evidential updating.Artificial Intelligence, 31(3):271–294,

work page 1987
[34]

Inference about constrained parameters using the elastic belief method.International Journal of Approxi- mate Reasoning, 53(5):709–727,

[Leaf and Liu, 2012] Duncan Ermini Leaf and Chuanhai Liu. Inference about constrained parameters using the elastic belief method.International Journal of Approxi- mate Reasoning, 53(5):709–727,

work page 2012
[35]

[Lemmer, 1986] John F. Lemmer. Confidence factors, em- piricism, and the Dempster–Shafer theory of evidence. In L. N. Kanal and J. F. Lemmers, editors,Uncertainty in Ar- tificial Intelligence, pages 167–196. North Holland,

work page 1986
[36]

The MIT Press, Cambridge, Massachusetts,

[Levi, 1980] Isaac Levi.The enterprise of knowledge: An essay on knowledge, credal probability, and chance. The MIT Press, Cambridge, Massachusetts,

work page 1980
[37]

Epistemic artifi- cial intelligence is essential for machine learning models to truly’know when they do not know’.arXiv preprint arXiv:2505.04950,

[Manchingalet al., 2025a ] Shireen Kudukkil Manchingal, Andrew Bradley, Julian FP Kooij, Keivan Shariatmadar, Neil Yorke-Smith, and Fabio Cuzzolin. Epistemic artifi- cial intelligence is essential for machine learning models to truly’know when they do not know’.arXiv preprint arXiv:2505.04950,

work page arXiv
[38]

Dempster-Shafer theory and statistical in- ference with weak beliefs.Statistical Science, 25(1):72– 87,

[Martinet al., 2010 ] Ryan Martin, Jianchun Zhang, and Chuanhai Liu. Dempster-Shafer theory and statistical in- ference with weak beliefs.Statistical Science, 25(1):72– 87,

work page 2010
[39]

[Molchanov, 2005] Ilya S Molchanov.Theory of random sets, volume

work page 2005
[40]

[Moral, 2014] Seraf´ın Moral. Comments on ‘Likelihood- based belief function: Justification and some extensions to low-quality data’ by thierry denœux.International Jour- nal of Approximate Reasoning, 55(7):1591–1593,

work page 2014
[41]

[Nguyen, 1978] Hung T. Nguyen. On random sets and belief functions.Journal of Mathematical Analysis and Applica- tions, 65:531–542,

work page 1978
[42]

Mathematical foundations for a theory of confidence structures.International Jour- nal of Approximate Reasoning, 53(7):1003–1019,

[Perrusselet al., 2012 ] Laurent Perrussel, Luis Enrique Su- car, and Michael Scott Balch. Mathematical foundations for a theory of confidence structures.International Jour- nal of Approximate Reasoning, 53(7):1003–1019,

work page 2012
[43]

Statistical evidence and belief functions

[Seidenfeld, 1978] Teddy Seidenfeld. Statistical evidence and belief functions. InProceedings of the Biennial Meet- ing of the Philosophy of Science Association, pages 478– 489,

work page 1978
[44]

Shafer.A Mathematical Theory of Evi- dence

[Shafer, 1976] G. Shafer.A Mathematical Theory of Evi- dence. Princeton University Press,

work page 1976
[45]

Belief functions and parametric models.Journal of the Royal Statistical Society: Series B (Methodological), 44(3):322–339,

[Shafer, 1982] Glenn Shafer. Belief functions and parametric models.Journal of the Royal Statistical Society: Series B (Methodological), 44(3):322–339,

work page 1982
[46]

Belief functions : the disjunc- tive rule of combination and the generalized Bayesian the- orem.International Journal of Approximate Reasoning, 9(1):1–35,

[Smets, 1993] Philippe Smets. Belief functions : the disjunc- tive rule of combination and the generalized Bayesian the- orem.International Journal of Approximate Reasoning, 9(1):1–35,

work page 1993
[47]

Belief func- tion representation of statistical audit evidence.Inter- national Journal of Intelligent Systems, 15(4):277–290,

[Van den Acker, 2000] Carine Van den Acker. Belief func- tion representation of statistical audit evidence.Inter- national Journal of Intelligent Systems, 15(4):277–290,

work page 2000
[48]

[Walley and Fine, 1982] Peter Walley and Terrence L. Fine. Towards a frequentist theory of upper and lower probabil- ity.The Annals of Statistics, 10(3):741–761,

work page 1982
[49]

Chapman and Hall, New York,

[Walley, 1991] Peter Walley.Statistical Reasoning with Im- precise Probabilities. Chapman and Hall, New York,

work page 1991
[50]

A review of uncertainty representation and quantification in neural networks.IEEE Transactions on Pattern Analysis and Ma- chine Intelligence,

[Wanget al., 2025 ] Kaizheng Wang, Fabio Cuzzolin, Keivan Shariatmadar, David Moens, and Hans Hallez. A review of uncertainty representation and quantification in neural networks.IEEE Transactions on Pattern Analysis and Ma- chine Intelligence,

work page 2025
[51]

Dempster–Shafer inference with weak beliefs.Statistica Sinica, 21:475–494,

[Zhang and Liu, 2011] Jianchun Zhang and Chuanhai Liu. Dempster–Shafer inference with weak beliefs.Statistica Sinica, 21:475–494,

work page 2011
[52]

The total belief theorem

[Zhou and Cuzzolin, 2017] Chunlai Zhou and Fabio Cuz- zolin. The total belief theorem. InUAI,

work page 2017
[53]

Parametric estimation of Dempster– Shafer belief functions

[Zribi and Benjelloun, July 2003] Mourad Zribi and Mo- hammed Benjelloun. Parametric estimation of Dempster– Shafer belief functions. InProceedings of the Sixth In- ternational Conference on Information Fusion (FUSION 2003), volume 1, pages 485–491, July 2003

work page 2003