Generalized Belief Function: A new concept for uncertainty modelling and processing

Fuyuan Xiao

arxiv: 1907.04719 · v1 · pith:7MIYXUFSnew · submitted 2019-07-03 · 💻 cs.AI

Generalized Belief Function: A new concept for uncertainty modelling and processing

Fuyuan Xiao This is my paper

Pith reviewed 2026-05-25 10:30 UTC · model grok-4.3

classification 💻 cs.AI

keywords belief functionDempster-Shafer theorycomplex mass functionuncertainty modelingevidence theorygeneralized belief functionplausibility function

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The pith

Belief functions extend to complex numbers via a new complex mass function that reduces to the standard real case.

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

The paper proposes a complex basic belief assignment that assigns complex numbers rather than real values in [0,1] to subsets of a frame of discernment. From this assignment it defines generalized versions of the belief function and the plausibility function that operate directly on the complex plane. When every imaginary component is set to zero the new functions recover exactly the classical belief and plausibility measures of Dempster-Shafer evidence theory. The construction therefore supplies a direct extension of uncertainty modelling that contains the original theory as the special case of real-valued assignments.

Core claim

The paper claims that a complex mass function, termed complex basic belief assignment, serves as the foundation for generalizing both the belief function and the plausibility function to complex numbers. When the complex mass function degenerates to a real-valued mass function, the generalized belief and plausibility functions likewise degenerate into the traditional belief and plausibility functions of Dempster-Shafer evidence theory.

What carries the argument

The complex basic belief assignment, which maps each subset to a complex number and thereby supplies the starting point for the generalized belief and plausibility measures.

If this is right

Uncertainty statements can be expressed with both magnitude and phase information under a single formal framework.
Any property proved for the classical real-valued functions automatically holds for the complex versions when restricted to real inputs.
Fusion and decision procedures that rely on belief or plausibility can be applied without change to complex-valued evidence once the appropriate mass assignment is supplied.

Where Pith is reading between the lines

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

The same construction might be used to embed evidence-theoretic reasoning inside signal-processing pipelines that already treat data as complex vectors.
One could test whether the extra degrees of freedom in the imaginary parts improve performance on fusion tasks whose inputs naturally carry phase, such as radar or communications data.
If the complex extension proves stable, it could serve as a bridge between classical evidence theory and linear-algebraic methods that operate on complex vector spaces.

Load-bearing premise

The proposed definitions for the complex mass function, belief, and plausibility maintain the desirable properties of the original theory and provide a useful extension without requiring additional constraints or interpretations for the imaginary components.

What would settle it

A concrete frame of discernment together with a complex mass assignment whose generalized belief value, after the imaginary part is removed, fails to equal the belief value computed from the corresponding real mass assignment.

read the original abstract

In this paper, we generalize the belief function on complex plane from another point of view. We first propose a new concept of complex mass function based on the complex number, called complex basic belief assignment, which is a generalization of the traditional mass function in Dempster-Shafer evidence theory. On the basis of the de nition of complex mass function, the belief function and plausibility function are generalized. In particular, when the complex mass function is degenerated from complex numbers to real numbers, the generalized belief and plausibility functions degenerate into the traditional belief and plausibility functions in DSE theory, respectively.

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 direct definitional extension of Dempster-Shafer mass functions to complex values, with belief and plausibility defined by the obvious sums; it reduces to the real case by construction but shows little else.

read the letter

The paper's core move is to let basic belief assignments take complex values instead of reals, then define belief and plausibility functions by summing those complex masses over subsets in the usual way. When the imaginary parts vanish, everything collapses back to classical DS theory, which holds by the way the sums are written. That part is clean and requires no extra machinery for the stated claim. The reduction works without contradiction, so the generalization is internally consistent at the level of the definitions themselves. What is actually new is the explicit complex-valued mass function and the corresponding belief/plausibility pair; prior work in evidence theory has stayed with real numbers. The paper does this part straightforwardly and without circularity. The main limitation is that the contribution stops at the definitions. No examples appear in the abstract showing what an imaginary component would represent in an uncertainty scenario, no applications are sketched, and there is no check on whether Dempster's combination rule or other standard operations carry over without modification or new constraints. The abstract gives no data, no proofs beyond the reduction, and no comparison to other possible generalizations. This leaves the practical payoff unclear. A reader already working on extensions of evidence theory might find the construction worth checking for completeness, but most people in AI uncertainty modeling will not need it. The paper is narrow enough that it does not require a broad referee process, though a specialized venue could reasonably send it out if the full text adds concrete illustrations or combination rules.

Referee Report

2 major / 2 minor

Summary. The paper claims to generalize Dempster-Shafer evidence theory by defining a complex basic belief assignment (complex mass function) that extends the traditional real-valued mass function to complex numbers, then defines generalized belief and plausibility functions via analogous subset summation; these are asserted to reduce exactly to the classical DS functions when the mass values are real.

Significance. If internally consistent and if the complex extension preserves useful properties (or enables new applications), the definitional framework could support uncertainty modeling in domains involving complex-valued evidence. The contribution is primarily conceptual, with no combination rule, examples, or property verifications provided, so significance remains prospective rather than demonstrated.

major comments (2)

[Definition of complex basic belief assignment and generalized belief function] The central definitions (complex mass function and the induced belief/plausibility) are presented as direct analogs, but the manuscript provides no explicit check that key DS axioms continue to hold under complex summation (e.g., normalization of the total mass or monotonicity of the belief function for nested sets). This verification is load-bearing for the claim that the proposal is a valid generalization rather than an arbitrary extension.
[Generalization of belief and plausibility functions] The reduction to classical DS theory is stated to occur 'by construction' when imaginary parts vanish, yet the paper does not address whether the complex-valued operations introduce additional constraints (such as handling of imaginary components in the subset sums) that would be required for the functions to remain well-defined and useful.

minor comments (2)

Abstract contains a typographical error ('de nition' and 'DSE theory'); correct to 'definition' and 'DS evidence theory'.
Notation for complex numbers and their real/imaginary parts should be introduced explicitly and used consistently to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments on our manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [Definition of complex basic belief assignment and generalized belief function] The central definitions (complex mass function and the induced belief/plausibility) are presented as direct analogs, but the manuscript provides no explicit check that key DS axioms continue to hold under complex summation (e.g., normalization of the total mass or monotonicity of the belief function for nested sets). This verification is load-bearing for the claim that the proposal is a valid generalization rather than an arbitrary extension.

Authors: We agree that an explicit verification of the preserved properties strengthens the presentation. The normalization condition (sum of the complex mass function equals 1) holds by definition for any complex values whose real and imaginary parts satisfy the sum, and reduces directly to the classical case. Monotonicity of the belief function for nested sets holds when the mass values are non-negative reals but does not automatically extend to arbitrary complex values without additional constraints on the imaginary parts. We will add a new proposition and short discussion section verifying the normalization property and clarifying the conditions under which monotonicity is retained. revision: yes
Referee: [Generalization of belief and plausibility functions] The reduction to classical DS theory is stated to occur 'by construction' when imaginary parts vanish, yet the paper does not address whether the complex-valued operations introduce additional constraints (such as handling of imaginary components in the subset sums) that would be required for the functions to remain well-defined and useful.

Authors: The reduction occurs exactly by setting all imaginary parts to zero, at which point the subset sums become identical to the classical real-valued sums. The complex-valued subset sums introduce no additional definitional constraints beyond requiring that the complex masses sum to 1; the resulting belief and plausibility functions are well-defined as complex numbers. We will insert a clarifying paragraph immediately after the definitions to state this explicitly and note that usefulness in applications will depend on domain-specific interpretation of the imaginary components. revision: yes

Circularity Check

0 steps flagged

No significant circularity: definitional generalization

full rationale

The paper's central contribution is the direct introduction of complex basic belief assignments as a straightforward extension of real-valued mass functions to the complex domain, followed by analogous definitions for belief and plausibility functions via subset summation. These are shown to reduce exactly to the classical Dempster-Shafer functions when the imaginary components vanish, which holds by the algebraic construction of the definitions themselves rather than any fitted parameter, self-referential equation, or load-bearing self-citation. No derivation chain exists that reduces to its own inputs; the work is self-contained as a proposal of new definitions with the stated degeneracy property.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim depends on the new definition of the complex mass function and the assumption that it can be used to generalize the belief functions in a consistent manner.

axioms (1)

standard math Complex numbers form a field with standard arithmetic operations
Basis for defining operations on complex mass functions.

invented entities (1)

complex basic belief assignment no independent evidence
purpose: Generalize the mass function to complex domain for uncertainty modeling
Newly defined in the paper as the core innovation.

pith-pipeline@v0.9.0 · 5617 in / 1271 out tokens · 58593 ms · 2026-05-25T10:30:18.052245+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 3.1 (Complex mass function) M:2^Ω→C with M(∅)=0, ∑M(A)=1, M(A)=m(A)e^{iθ(A)}; Def 4.1 generalized Bel(A)=∑_{B⊆A} Com(B) with Com via |M|; reduces to classical when complex→real
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1 axioms for generalized Bel; Theorem 2 Möbius inversion for Com

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