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Reasoning with random sets: An agenda for the future

Fabio Cuzzolin

TL;DR

This work articulates a forward-looking research agenda for random-set and belief-function theory, advocating a shift toward statistical inference with lower/upper likelihoods, generalized logistic regression, and a true law of total belief. It advances the geometric view of uncertainty and outlines limit theorems, frequentist-inference avenues, and random-set extensions to random variables, all while connecting to high-impact domains like climate modelling and machine learning. By developing a coherent framework that unifies belief measures, capacities, and random sets, the paper lays groundwork for robust, interval-valued inference and decision-making under epistemic uncertainty. The proposed program aims to deepen theoretical foundations (e.g., Radon–Nikodym-type results for capacities), extend the geometry of uncertainty, and enable practically meaningful applications with transparent uncertainty quantification.

Abstract

In this paper, we discuss a potential agenda for future work in the theory of random sets and belief functions, touching upon a number of focal issues: the development of a fully-fledged theory of statistical reasoning with random sets, including the generalisation of logistic regression and of the classical laws of probability; the further development of the geometric approach to uncertainty, to include general random sets, a wider range of uncertainty measures and alternative geometric representations; the application of this new theory to high-impact areas such as climate change, machine learning and statistical learning theory.

Reasoning with random sets: An agenda for the future

TL;DR

This work articulates a forward-looking research agenda for random-set and belief-function theory, advocating a shift toward statistical inference with lower/upper likelihoods, generalized logistic regression, and a true law of total belief. It advances the geometric view of uncertainty and outlines limit theorems, frequentist-inference avenues, and random-set extensions to random variables, all while connecting to high-impact domains like climate modelling and machine learning. By developing a coherent framework that unifies belief measures, capacities, and random sets, the paper lays groundwork for robust, interval-valued inference and decision-making under epistemic uncertainty. The proposed program aims to deepen theoretical foundations (e.g., Radon–Nikodym-type results for capacities), extend the geometry of uncertainty, and enable practically meaningful applications with transparent uncertainty quantification.

Abstract

In this paper, we discuss a potential agenda for future work in the theory of random sets and belief functions, touching upon a number of focal issues: the development of a fully-fledged theory of statistical reasoning with random sets, including the generalisation of logistic regression and of the classical laws of probability; the further development of the geometric approach to uncertainty, to include general random sets, a wider range of uncertainty measures and alternative geometric representations; the application of this new theory to high-impact areas such as climate change, machine learning and statistical learning theory.
Paper Structure (108 sections, 33 theorems, 222 equations, 17 figures)

This paper contains 108 sections, 33 theorems, 222 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Open issues
  3. A research programme
  4. Belief functions
  5. Belief and plausibility measures
  6. Evidence combination
  7. Conditioning
  8. Multivariate analysis
  9. Refinings
  10. A statistical random set theory
  11. Lower and upper likelihoods
  12. Belief likelihood function of repeated trials
  13. Binary trials: The conjunctive case
  14. Focal elements of the belief likelihood
  15. Factorisation for 'sharp' tuples
  16. ...and 93 more sections

Key Result

Lemma 1

For any $n \in \mathbb{Z}$, the belief function $Bel_{{\mathbb X}_1} \oplus \cdots \oplus Bel_{{\mathbb X}_n}$, where ${\mathbb X}_i = {\mathbb X} = \{T,F\}$, has $3^n$ focal elements, namely all possible Cartesian products $A_1\times \cdots \times A_n$ of $n$ non-empty subsets $A_i$ of ${\mathbb X}

Figures (17)

  • Figure 1: (a) Graphical representation of the Dempster combination $Bel_{{\mathbb X}_1} \oplus Bel_{{\mathbb X}_2}$ on ${\mathbb X}_1 \times {\mathbb X}_2$, in the binary case in which ${\mathbb X}_1 = {\mathbb X}_2 = \{T,F\}$. (b) Graphical representation of the disjunctive combination $Bel_{{\mathbb X}_1}$${\cup}$$Bel_{{\mathbb X}_2}$ there.
  • Figure 2: Lower (a) and upper (b) likelihood functions plotted in the space of belief functions defined on the frame ${\mathbb X} = \{ T,F \}$, parameterised by $p = m(T)$ ($X$ axis) and $q = m(F)$ ($Y$ axis), for the case of $k=6$ successes in $n=10$ trials.
  • Figure 3: Pictorial representation of the hypotheses of the total belief theorem (Theorem \ref{['the:total-belief']}).
  • Figure 4: The conditional belief functions considered in our case study. The set-theoretical relations between their focal elements are immaterial to the solution.
  • Figure 5: Graphical representation of the four possible focal elements (\ref{['eq:elastic-bands']}) of the total belief function (\ref{['eq:total-belief-function']}) in our case study.
  • ...and 12 more figures

Theorems & Definitions (70)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • ...and 60 more