Evidential distance measure in complex belief function theory

Fuyuan Xiao

arxiv: 1907.00716 · v1 · pith:LG62MZEDnew · submitted 2019-06-27 · 💻 cs.AI

Evidential distance measure in complex belief function theory

Fuyuan Xiao This is my paper

Pith reviewed 2026-05-25 15:09 UTC · model grok-4.3

classification 💻 cs.AI

keywords evidential distancecomplex belief functionJousselme distancebasic belief assignmentdissimilarity measureevidence theorycomplex numbersCBBAs

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The pith

A distance measure on complex basic belief assignments reduces exactly to Jousselme distance when the values become real numbers.

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

The paper introduces a distance that quantifies dissimilarity between complex basic belief assignments, representations of evidence that use complex numbers instead of real probabilities. The construction is chosen so the new distance matches Jousselme et al.'s distance exactly once the complex numbers are replaced by their real parts. This supplies a dissimilarity on the larger domain of the complex plane while preserving backward compatibility with the standard real-valued case. A reader would care because the extension keeps existing tools intact yet permits evidence to be handled in a broader algebraic setting.

Core claim

The paper proposes an evidential distance measure defined on complex basic belief assignments (CBBAs) composed of complex numbers; when the CBBAs degenerate from complex numbers to real numbers the proposed distance reduces to Jousselme et al.'s distance, thereby extending dissimilarity measurement to the complex plane space.

What carries the argument

The proposed evidential distance measure on CBBAs that is required to recover Jousselme distance under real degeneration.

If this is right

Differences between evidences can be measured inside the complex plane rather than only on the real line.
Any application that already uses Jousselme distance on real BBAs can substitute the new distance without changing results on the real subset.
Evidence theory gains a dissimilarity that is native to complex-valued representations.
The measure supplies a concrete bridge between classical Dempster-Shafer theory and any future complex-valued extensions.

Where Pith is reading between the lines

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

The same reduction property could be checked on other existing distances in evidence theory to see which ones generalize most cleanly.
Numerical experiments on small complex assignments would immediately reveal whether the distance behaves as a metric or satisfies triangle inequality.
If the distance turns out to be computationally tractable, it could be inserted into existing clustering or fusion algorithms that currently rely on real-valued distances.

Load-bearing premise

That a distance can be defined on complex basic belief assignments which exactly recovers Jousselme distance when the assignments become real while remaining a valid dissimilarity on the complex domain.

What would settle it

An explicit pair of real-valued basic belief assignments for which the proposed distance yields a numerical value different from the Jousselme distance.

read the original abstract

In this paper, an evidential distance measure is proposed which can measure the difference or dissimilarity between complex basic belief assignments (CBBAs), in which the CBBAs are composed of complex numbers. When the CBBAs are degenerated from complex numbers to real numbers, i.e., BBAs, the proposed distance will degrade into the Jousselme et al.'s distance. Therefore, the proposed distance provides a promising way to measure the differences between evidences in a more general framework of complex plane space.

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

The paper asserts a distance on complex BBAs that reduces to Jousselme's but supplies no formula or degeneration steps to verify the claim.

read the letter

The main point is that this paper defines an evidential distance for complex basic belief assignments where the masses are complex numbers, with the key property that it reduces to the Jousselme distance when the imaginary parts vanish. That is the entire contribution as described in the abstract. The work does a reasonable job of identifying the need for a dissimilarity measure in the complex setting and framing the new distance as a direct extension of an existing one. This keeps the proposal simple and potentially compatible with prior results in the belief function literature. The reduction claim, if it holds, would let users move between real and complex cases without changing their distance calculations. The soft spot is exactly the one flagged in the stress test: the abstract states the reduction property but gives neither the explicit definition of the new distance nor any algebraic steps showing how the limit recovers Jousselme's formula, which depends on the specific 2^|Θ| by 2^|Θ| intersection matrix. Without that construction it is impossible to confirm the measure remains a valid dissimilarity on the complex domain or to check basic properties such as non-negativity and symmetry under complex arithmetic. The citation pattern is standard and points back to Jousselme et al. without circularity or invented entities. This paper is aimed at the narrow group already working on complex-valued extensions of Dempster-Shafer theory. Most readers outside that subfield will see little to use or cite. If the full manuscript contains the missing definition, the degeneration proof, and some verification examples, then it deserves peer review so that specialists can check the math. If those elements are absent, the paper is too preliminary for review.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes an evidential distance measure defined on complex basic belief assignments (CBBAs). The central claim is that this distance reduces exactly to Jousselme et al.'s distance when the imaginary parts of the CBBAs are set to zero (i.e., when CBBAs degenerate to ordinary BBAs), thereby extending dissimilarity measurement to the complex plane.

Significance. If the explicit construction and the exact reduction property can be verified, the work would supply a generalization of an established evidential distance to a broader algebraic setting; this could be useful for applications that already employ complex-valued belief functions.

major comments (1)

[Abstract] Abstract (and presumably the definition section): the reduction claim is stated but neither the explicit formula for the proposed distance d_C nor the algebraic steps demonstrating that lim_{Im→0} d_C(CBBA) recovers Jousselme et al.'s formula (which depends on the specific 2^|Θ|×2^|Θ| intersection matrix) are supplied. Without this construction it is impossible to confirm that d_C remains a valid dissimilarity on the complex domain.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and constructive comments on our manuscript. We address the major comment below and will revise the paper accordingly.

read point-by-point responses

Referee: [Abstract] Abstract (and presumably the definition section): the reduction claim is stated but neither the explicit formula for the proposed distance d_C nor the algebraic steps demonstrating that lim_{Im→0} d_C(CBBA) recovers Jousselme et al.'s formula (which depends on the specific 2^|Θ|×2^|Θ| intersection matrix) are supplied. Without this construction it is impossible to confirm that d_C remains a valid dissimilarity on the complex domain.

Authors: We agree that providing the explicit formula for the proposed evidential distance d_C and the detailed algebraic steps for the reduction to Jousselme's distance is necessary to substantiate the central claim. In the revised manuscript, we will add the complete definition of d_C in the relevant section and include a proof or derivation showing that as the imaginary parts of the CBBAs approach zero, d_C reduces exactly to the Jousselme distance, accounting for the intersection matrix. This will also allow verification of its properties as a dissimilarity measure in the complex domain. revision: yes

Circularity Check

0 steps flagged

No circularity: proposed distance is a new definition whose degeneration property is a verifiable feature, not a reduction to inputs

full rationale

The paper introduces a novel distance on complex basic belief assignments and states that it recovers Jousselme et al.'s distance under real degeneration. This is a property of the explicit construction rather than a prediction derived from fitted parameters, self-citations, or ansatzes. No load-bearing self-citation chains, uniqueness theorems imported from the authors' prior work, or renamings of known results appear in the provided claims. The central result is the definition itself, which is self-contained and independent of the target reduction; verification of the limit would be algebraic but does not render the proposal circular by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated assumption that complex BBAs form a coherent mathematical structure on which a distance can be defined.

pith-pipeline@v0.9.0 · 5595 in / 1099 out tokens · 19901 ms · 2026-05-25T15:09:45.832713+00:00 · methodology

Evidential distance measure in complex belief function theory

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)