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arxiv: 1906.11409 · v1 · pith:KCA3TYYRnew · submitted 2019-06-27 · 💻 cs.AI

Generalization of Dempster-Shafer theory: A complex belief function

Fuyuan Xiao This is my paper

Pith reviewed 2026-05-25 15:12 UTC · model grok-4.3

classification 💻 cs.AI
keywords complex belief functiongeneralized Dempster-Shafer theorycomplex basic belief assignmentevidence combination ruleuncertainty modelingphase informationDempster combination
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The pith

Modeling belief masses as complex numbers removes the conflict restriction from Dempster's combination rule.

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

The paper extends Dempster-Shafer evidence theory by representing basic belief assignments as complex numbers rather than real values. This lets the model capture uncertainty that occurs together with phase or periodicity shifts in data. A new combination rule is derived that works without requiring the conflict coefficient between sources to stay below one. The classical theory is recovered exactly when the complex values reduce to real numbers and conflict remains low. The change matters for fusing evidence from sources whose timing or oscillation properties carry information alongside their uncertainty.

Core claim

A mass function in the generalized Dempster-Shafer evidence theory is modeled by a complex number, called a complex basic belief assignment, which has more powerful ability to express uncertain information. Based on that, a generalized Dempster's combination rule is exploited. In contrast to the classical Dempster's combination rule, the condition in terms of the conflict coefficient between the evidences K<1 is released in the generalized Dempster's combination rule. Hence, it is more general and applicable than the classical Dempster's combination rule. When the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster's combination rule degeneresr

What carries the argument

The complex basic belief assignment, a mass function whose values are complex numbers, which carries both belief magnitude and phase information into the generalized combination rule.

If this is right

  • Evidence sources can be combined even when their conflict coefficient K meets or exceeds one.
  • Uncertainty and concurrent phase or periodicity changes are expressed in a single mass function.
  • The classical theory is recovered precisely when complex values reduce to real numbers and K remains below one.
  • The approach applies directly to data whose timing or oscillation properties matter for uncertainty modeling.

Where Pith is reading between the lines

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

  • The complex representation could integrate with existing complex-valued signal processing or neural network pipelines that already track phase.
  • One could check whether the new rule preserves algebraic properties such as associativity or commutativity under complex arithmetic.
  • Practical tests on time-series sensor data would reveal whether the added phase dimension changes fusion accuracy compared with real-valued baselines.
  • Interpretation rules for the imaginary part of combined beliefs would need to be developed before routine use.

Load-bearing premise

Representing belief masses as complex numbers faithfully encodes both the degree of belief and any phase-related fluctuations in a way that produces coherent combined results under the new rule.

What would settle it

Apply the generalized rule to two complex basic belief assignments whose real parts form a classical pair with conflict coefficient at least one; if the output masses fail to sum to one or yield uninterpretable negative components, the generalization does not hold.

Figures

Figures reproduced from arXiv: 1906.11409 by Fuyuan Xiao.

Figure 1
Figure 1. Figure 1: The variations of parameters x and y [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An example of the variation of |K| between two CBBAs from the front and the side angles. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: An example of the variation of |K| between two CBBAs from the oblique angles. Example 2: Supposing that there are two CBBAs M1 and M2 in the frame of discernment Ω = {A, B}, and the two CBBAs are given as follows: M1 :M1(A) = 0.2031e i arctan(−1.7678) , M1(B) = 0.7842e i arctan(0.5051) , M1(A, B) = 0.2669e i arctan(−0.8839); M2 :M2(A) = 0.3606e i arctan(3.4641) , M2(B) = 0.6245e i arctan(0.2887) , M2(A, B)… view at source ↗
read the original abstract

Dempster-Shafer evidence theory has been widely used in various fields of applications, because of the flexibility and effectiveness in modeling uncertainties without prior information. However, the existing evidence theory is insufficient to consider the situations where it has no capability to express the fluctuations of data at a given phase of time during their execution, and the uncertainty and imprecision which are inevitably involved in the data occur concurrently with changes to the phase or periodicity of the data. In this paper, therefore, a generalized Dempster-Shafer evidence theory is proposed. To be specific, a mass function in the generalized Dempster-Shafer evidence theory is modeled by a complex number, called as a complex basic belief assignment, which has more powerful ability to express uncertain information. Based on that, a generalized Dempster's combination rule is exploited. In contrast to the classical Dempster's combination rule, the condition in terms of the conflict coefficient between the evidences K<1 is released in the generalized Dempster's combination rule. Hence, it is more general and applicable than the classical Dempster's combination rule. When the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster's combination rule degenerates to the classical evidence theory under the condition that the conflict coefficient between the evidences K is less than 1. In a word, this generalized Dempster-Shafer evidence theory provides a promising way to model and handle more uncertain 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

4 major / 1 minor

Summary. The manuscript proposes a generalization of Dempster-Shafer evidence theory in which basic belief assignments are represented as complex numbers (complex BBAs) to capture fluctuations of data at given phases and concurrent uncertainties with phase or periodicity changes. It introduces a generalized Dempster combination rule that removes the classical requirement that the conflict coefficient K be less than 1. The paper states that the new rule degenerates to the classical Dempster rule when the complex masses reduce to real numbers and K<1.

Significance. If the complex representation and rule were shown to be coherent and to preserve the core properties of belief functions while meaningfully encoding phase information, the work could extend evidence theory to domains involving periodic or phased data. The removal of the K<1 restriction would address a known limitation of classical DS theory in high-conflict settings. No such demonstration, derivation, or validation is supplied.

major comments (4)
  1. [Abstract] Abstract: the central claims—that complex BBAs have 'more powerful ability to express uncertain information' and that the generalized rule is valid without the K<1 condition—are introduced by definition with no derivation, semantic mapping from phase/periodicity to the imaginary part, or proof that the resulting quantities remain valid belief functions (non-negative real parts, normalization).
  2. [Abstract] Abstract: no constraints are stated on the complex masses (e.g., Re(m(A)) ≥ 0, ∑ Re(m(A)) = 1, or bounds on |m(A)|), so the generalized rule can produce masses with negative real parts or fail to normalize, undermining the claim that it is a coherent generalization.
  3. [Abstract] Abstract: the degeneration statement only recovers the classical case under K<1; it does not establish that the complex rule itself is well-defined or that the complex masses correspond to a valid extension of belief functions when K≥1.
  4. [Abstract] Abstract: the manuscript contains no example, numerical check, or formal property (e.g., associativity, commutativity, or reduction to Bayesian updating) to support the new rule, leaving the central claim without any empirical or theoretical grounding.
minor comments (1)
  1. [Abstract] Abstract contains minor grammatical issues ('called as a complex basic belief assignment', 'in a word') that should be corrected for clarity.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for the constructive critique of our manuscript. The comments correctly identify that the current version is primarily a definitional proposal without supporting derivations, constraints, or examples. We address each point below and commit to revisions that add the requested material.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims—that complex BBAs have 'more powerful ability to express uncertain information' and that the generalized rule is valid without the K<1 condition—are introduced by definition with no derivation, semantic mapping from phase/periodicity to the imaginary part, or proof that the resulting quantities remain valid belief functions (non-negative real parts, normalization).

    Authors: The motivation for complex BBAs is to encode phase/periodicity via the imaginary component, but we agree the abstract and manuscript lack an explicit semantic mapping and formal proof that real parts stay non-negative with proper normalization. We will revise by adding a new section that derives these properties from the definition and provides the phase-to-imaginary mapping. revision: yes

  2. Referee: [Abstract] Abstract: no constraints are stated on the complex masses (e.g., Re(m(A)) ≥ 0, ∑ Re(m(A)) = 1, or bounds on |m(A)|), so the generalized rule can produce masses with negative real parts or fail to normalize, undermining the claim that it is a coherent generalization.

    Authors: We acknowledge that explicit constraints are missing from the presented text. The revision will state the required conditions (Re(m(A)) ≥ 0, sum of real parts equals 1, and appropriate bounds) and prove that the generalized combination rule preserves them. revision: yes

  3. Referee: [Abstract] Abstract: the degeneration statement only recovers the classical case under K<1; it does not establish that the complex rule itself is well-defined or that the complex masses correspond to a valid extension of belief functions when K≥1.

    Authors: The degeneration claim is limited to the K<1 case as stated. For K≥1 the rule is defined by construction, but we agree a formal argument for well-definedness is absent. The revision will include a derivation showing the rule remains well-defined for K≥1 while recovering the classical rule when masses are real and K<1. revision: yes

  4. Referee: [Abstract] Abstract: the manuscript contains no example, numerical check, or formal property (e.g., associativity, commutativity, or reduction to Bayesian updating) to support the new rule, leaving the central claim without any empirical or theoretical grounding.

    Authors: The current manuscript is introductory and supplies none of the requested checks. We will add a numerical example, verify commutativity (and associativity where it holds), and demonstrate reduction to Bayesian updating in the appropriate limit. revision: yes

Circularity Check

2 steps flagged

Complex BBA and generalized combination rule introduced by direct definition without independent derivation

specific steps
  1. self definitional [Abstract]
    "a mass function in the generalized Dempster-Shafer evidence theory is modeled by a complex number, called as a complex basic belief assignment, which has more powerful ability to express uncertain information. Based on that, a generalized Dempster's combination rule is exploited. In contrast to the classical Dempster's combination rule, the condition in terms of the conflict coefficient between the evidences K<1 is released in the generalized Dempster's combination rule."

    The generalized theory is introduced by defining the mass function as a complex number (CBBA) and then exploiting the combination rule as its direct analogue; the release of K<1 and the asserted greater expressive power are therefore true by the act of definition rather than derived from first principles or shown to preserve belief-function semantics (e.g., non-negative real parts).

  2. self definitional [Abstract]
    "When the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster's combination rule degenerates to the classical evidence theory under the condition that the conflict coefficient between the evidences K is less than 1."

    The degeneration statement is tautological: the generalized rule is constructed so that it recovers the classical rule precisely when the inputs are real and K<1, confirming that the extension itself is the definitional move rather than an independently justified generalization.

full rationale

The paper defines the generalization by modeling mass functions as complex numbers and adapting the combination rule as the direct complex analogue, releasing the K<1 condition by construction. The claimed ability to capture phase/periodicity fluctuations and the validity of the rule without real-valued constraints reduce to this definitional choice rather than a derived result. Degeneration to classical DS theory is explicitly built into the definition. No external principle or validation is invoked to establish the complex case as a coherent belief function.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Review performed on abstract only; full definitions, proofs, and any additional assumptions are unavailable.

axioms (2)
  • standard math Standard axioms of classical Dempster-Shafer theory
    The generalization is stated to reduce to classical DS theory under real numbers and K<1.
  • ad hoc to paper Complex numbers can represent phase and periodicity fluctuations in uncertain data
    Invoked to justify the complex basic belief assignment for handling concurrent uncertainty and phase changes.
invented entities (1)
  • complex basic belief assignment no independent evidence
    purpose: To model uncertain information with phase or periodicity components
    New entity introduced in the abstract as the core modeling device.

pith-pipeline@v0.9.0 · 5785 in / 1412 out tokens · 32895 ms · 2026-05-25T15:12:20.493685+00:00 · methodology

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