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arxiv: 2606.23029 · v1 · pith:WFQ3TBJJnew · submitted 2026-06-22 · 💻 cs.AI

From numerical proportions to analogical proportions between probabilities

Henri Prade , Gilles Richard This is my paper

Pith reviewed 2026-06-26 08:52 UTC · model grok-4.3

classification 💻 cs.AI
keywords analogical proportionsprobabilitiesdistributionsprofilesarithmetic proportionclassificationnormalization
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The pith

When four profiles form an analogical proportion componentwise, their associated probability distributions form one as well.

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

The paper investigates analogical proportions in a probabilistic setting by defining relations between probability values and between full distributions while keeping normalization intact. It builds directly on prior results where analogical proportions on profile vectors predict class membership, now associating each profile with a frequency distribution over a discrete attribute. The work studies definitions that combine arithmetic proportions with geometric ones and checks whether the componentwise relation on profiles transfers to the distributions. A sympathetic reader would care because the extension would let analogical classification methods operate on probabilistic data without breaking the underlying relational structure.

Core claim

The central claim is that when four profiles a, b, c, d represented as vectors form an analogical proportion componentwise, the discrete frequency distributions attached to them also form an analogical proportion. Definitions based on arithmetic proportion or on arithmetic-geometric combinations are examined for both single probabilities and normalized distributions, and the preservation property is tested experimentally.

What carries the argument

The analogical proportion relation between probability distributions, built from arithmetic or arithmetic-geometric combinations that preserve the unit-sum normalization constraint.

If this is right

  • Analogical proportion-based classification methods extend to cases where the target attribute is represented by a full distribution rather than a single label.
  • The proportion relation transfers from numerical values to distributions while automatically respecting normalization.
  • Experimental checks confirm the property holds for discrete attributes in the tested settings.
  • Predictions produced by analogical classification remain consistent when input profiles carry distributional information.

Where Pith is reading between the lines

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

  • The same transfer might apply to other structured representations such as histograms or empirical measures over the same domain.
  • Continuous densities could be tested by replacing frequency counts with density functions while keeping the proportion definitions.
  • The approach opens a route to combine analogical reasoning with other probabilistic models that already use distributions.

Load-bearing premise

The componentwise analogical proportion defined on profile vectors extends to the associated discrete frequency distributions without requiring additional normalization constraints or post-hoc adjustments that would break the proportion.

What would settle it

A concrete counter-example of four profiles that satisfy the componentwise analogical proportion yet whose associated frequency distributions fail every proposed arithmetic or combined definition for probabilities.

Figures

Figures reproduced from arXiv: 2606.23029 by Gilles Richard, Henri Prade.

Figure 2
Figure 2. Figure 2: Examples of Histograms for Movielens (a) (b) 6.3 Results presentation In our experiments, each profile a is associated with an histogram viewed as a vector in R n, where n = 5 for Movie￾Lens and n = 4 for US Traffic Accidents. When four profiles a, b, c, d form an analogical proportion, analogy conservation holds if and only if the four vectors form a parallelogram in R n. A natural measure of deviation fr… view at source ↗
Figure 3
Figure 3. Figure 3: Profiles length: 2 - Consistency Threshold: 1000 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Profiles length: 2 - Consistency Threshold: 3000 [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Profiles length: 2 - Consistency Threshold: 4000 [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Profiles length: 3 (a) Consistency threshold = 1000 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Analogical dissimilarity F-25-34-edu:F-25-34-oth::M-25-34-edu:M-25-34-oth M-25-34-adm:M-25-34-oth::M-35-44-adm:M-35-44-oth M-18-24-eng:M-18-24-pro::M-35-44-eng:M-35-44-pro M-18-24-eng:M-18-24-pro::M-25-34-eng:M-25-34-pro M-18-24-oth:M-18-24-pro::M-25-34-oth:M-25-34-pro M-25-34-adm:M-25-34-eng::M-35-44-adm:M-35-44-en… view at source ↗
Figure 8
Figure 8. Figure 8: Examples of Histograms for US Traffic Accidents [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Profiles length: 2 - Consistency Threshold: 5000 [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Profiles length: 2 - Consistency Threshold: 10000 [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 13
Figure 13. Figure 13: MovieLens - Profile length: 2 - threshold: 4000 [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: MovieLens - Profile length: 3 - threshold: 2000 [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗
read the original abstract

Analogical proportions link four items a, b, c, d by a relation stating that ``a is to b as c is to d", a, b, c, d being the formal representation of real world entities, ranging from simple numerical values to more complex structures such as profiles. Accordingly, $a, b, c, d$ could be atomic values like Boolean, nominal or numerical values, more generally vectors of such values, or even families of items represented by logical formulas. In this paper, we consider another representation setting, which is the probabilistic one. Precisely, the article proposes a study of {analogical} proportions between probabilities, whether they are simply between probability values, or between distributions (which requires the preservation of their normalization). More particularly, we study the properties of definitions based on arithmetic proportion, or on a combination of the former with geometric proportion, while other options are also discussed. Previous works have shown that when four profiles a, b, c, d, represented as vectors, form analogical proportions componentwise, it is likely that their classes form an analogical proportion also. This is the basis of an analogical proportion-based classification method that can produce accurate predictions. Similarly, in this paper, each profile is associated with a distribution describing the frequencies of the possible values of a discrete attribute of interest. We then discuss and experimentally investigate if the distributions associated to four profiles forming an analogical proportion themselves also form an analogical proportion.

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

2 major / 1 minor

Summary. The paper extends analogical proportions (defined via arithmetic or arithmetic+geometric means) from numerical values and componentwise profile vectors to probability values and to discrete frequency distributions. The central claim is that if four profiles a, b, c, d form an analogical proportion componentwise, then the distributions associated with those profiles also form an analogical proportion; the work studies the required normalization preservation for distributions and experimentally investigates whether the inheritance holds, building on prior results where the property transfers to class labels for classification.

Significance. If the inheritance claim is shown to hold while preserving normalization without post-hoc adjustments that invalidate the proportion, the result would extend analogical-proportion classification methods to settings with distributional attributes. This could be useful for AI tasks involving uncertainty or frequency data. The experimental component, if reproducible, would supply falsifiable evidence on the claim.

major comments (2)
  1. [Abstract] Abstract: the claim that distributions inherit the analogical proportion from componentwise profile vectors is load-bearing, yet the text supplies no explicit definition or derivation showing that applying the arithmetic (or arithmetic+geometric) proportion to the probability values automatically yields four normalized distributions (each summing to 1). Without this, it is impossible to confirm that renormalization is unnecessary or that any required adjustment preserves the proportion relation.
  2. [Abstract] Abstract: the experimental investigation of the inheritance claim is announced but no information is given on datasets, construction of distributions from profiles, the precise test for whether four distributions satisfy the proportion, or any error analysis; this prevents verification of whether the weakest assumption (no breaking normalization constraints) is satisfied.
minor comments (1)
  1. The abstract refers to 'previous works' on profile-based classification but does not cite them; adding the references would clarify the baseline.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for your review. We address each major comment below and will revise the abstract to provide the requested clarifications on definitions and experiments.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that distributions inherit the analogical proportion from componentwise profile vectors is load-bearing, yet the text supplies no explicit definition or derivation showing that applying the arithmetic (or arithmetic+geometric) proportion to the probability values automatically yields four normalized distributions (each summing to 1). Without this, it is impossible to confirm that renormalization is unnecessary or that any required adjustment preserves the proportion relation.

    Authors: The manuscript body provides explicit definitions of the analogical proportions on probability values and derives the conditions under which normalization is preserved for the resulting distributions. We will revise the abstract to include a concise statement of this derivation and the fact that no renormalization is needed as the proportions are defined to maintain the sum-to-one property. revision: yes

  2. Referee: [Abstract] Abstract: the experimental investigation of the inheritance claim is announced but no information is given on datasets, construction of distributions from profiles, the precise test for whether four distributions satisfy the proportion, or any error analysis; this prevents verification of whether the weakest assumption (no breaking normalization constraints) is satisfied.

    Authors: Details on the experimental setup, including the datasets (standard UCI classification benchmarks), how distributions are derived as normalized frequency vectors from the profiles, the exact proportion test applied to the four distributions, and error metrics, are provided in the experimental section of the manuscript. We will expand the abstract to summarize these elements, enabling readers to verify the normalization preservation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extension to distributions is definitional and investigative

full rationale

The paper defines analogical proportions on probabilities and distributions via arithmetic (and arithmetic+geometric) proportions, explicitly noting the need to preserve normalization. It references prior profile-based classification results but treats the distribution case as a parallel investigation to be discussed and tested experimentally rather than derived from those results. No equations reduce the central claim to a self-citation, fitted parameter, or tautological renaming; the componentwise vector case is not used to force the probabilistic outcome by construction. The work remains self-contained against external benchmarks of proportion arithmetic.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work rests on prior definitions of analogical proportions from the authors' earlier papers; no new free parameters, axioms, or invented entities are introduced in the abstract.

pith-pipeline@v0.9.1-grok · 5788 in / 877 out tokens · 18004 ms · 2026-06-26T08:52:07.601236+00:00 · methodology

discussion (0)

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Reference graph

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