A dimension reduction for extreme types of directed dependence

Carsten Limbach; Sebastian Fuchs

REVIEW 3 major objections 4 minor 17 references

A dimension reduction for extreme types of directed dependence

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Complete separation is encoded as an ordinal sum in the Markov product.

desk verdict A useful extension of the Markov-product approach to directed dependence, with a genuinely new ordinal-sum characterization of complete separation whose proof needs patching. read the letter →

arxiv 2506.04825 v1 pith:JO6P5UVO submitted 2025-06-05 math.ST stat.TH

Sebastian Fuchs , Carsten Limbach This is my paper

classification math.STstat.TH

keywords directeddependenceMarkovproductcompleteseparationperfectordinalsumdimensionreductioncoefficientofconditionalindependence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies five extreme forms of directed dependence of a response Y on predictors X and asks how each can be seen in the Markov product (Y,Y'), where Y' is a conditionally independent copy of Y given X. It establishes that independence, zero-explainability, and stochastic comparability correspond, respectively, to independence, uncorrelatedness, and equality of concordance and discordance probabilities of (Y,Y'). For the maximum extremes, it proves that perfect dependence is equivalent to Y=Y' almost surely, and that complete separation is equivalent to (FY(Y),FY(Y')) admitting an ordinal sum structure whose blocks are indexed by the atoms of X and have lengths equal to the atoms' probabilities. The point of the translation is dimensional: these properties of (Y,Y') can be estimated by a single fully nonparametric nearest-neighbor principle without tuning parameters.

What carries the argument

The central object is the Markov product (Y,Y'), the joint law P(Y≤y,Y'≤y')=E[P(Y≤y|X)P(Y'≤y'|X)] formed from a conditionally independent copy of Y given X. The paper transfers extreme dependence concepts into properties of this two-dimensional distribution. For the maximum extreme it uses the ordinal sum of distributions on [0,1]2, defined by F(u,v)=Σ (bk-ak)Fk(u,v)+λ([0,min{u,v}]\∪(ak,bk]), and proves that complete separation corresponds to an ordinal sum whose blocks are tied to the atoms of X.

What would settle it

Find any distribution (X,Y) for which Λ(Y|X) and Λ(FY(Y)|X) differ, for example a Y with discrete atoms and two conditional distributions; since the proof of Theorem 3.2 invokes their equality in both directions, one such counterexample would invalidate the ordinal-sum characterization.

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Extended reading notes

Core claim

The central discovery is Theorem 3.2: Y is completely separated relative to X if and only if the bivariate distribution of (FY(Y),FY(Y')) is an ordinal sum over the intervals (az,bz] assigned to the atoms z of X, with bz-az=P(X=z) and each block distribution Fz satisfying Fz(u,v)=Fz(u,bz)Fz(bz,v). In words, the extreme case of pairwise disjoint conditional supports is exactly a layered, comonotone-like structure in the probability-transformed Markov product, and the layers are the atoms of the predictor. A corollary shows that when any coordinate of X is continuous, the ordinal sum is trivial: (FY(Y),FY(Y')) is comonotone, so complete separation forces perfect dependence and Λ=1 forces ξ=R2=1. Theorem 3.1 complements this by identifying perfect dependence itself with Y=Y' almost surely.

Load-bearing premise

The argument that complete separation becomes an ordinal sum depends on the invariance identity Λ(Y|X)=Λ(FY(Y)|X) taken from [9]; if that identity fails for distributions with atoms, the characterization of Theorem 3.2 no longer follows.

Editorial extensions

If this is right

If the theorem is right, complete separation can be detected by checking an ordinal sum structure in the Markov product rather than by comparing conditional distributions over Rp.
Perfect dependence is exactly equality of the Markov pair almost surely, so both maximum extremes become simple geometric properties of (Y,Y').
When at least one predictor has a continuous distribution, complete separation and perfect dependence coincide, making Λ=1 imply ξ=R2=1.
Independence, zero-explainability, and stochastic comparability also become properties of (Y,Y'), so all five extremes can be approached by one dimension-reduced nearest-neighbor estimation scheme.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An implicit condition for the proof is whether the invariance identity Λ(Y|X)=Λ(FY(Y)|X) taken from [9] can be proven directly for distributions with atoms; if it fails there, Theorem 3.2 would need a different argument.
The ordinal-sum formulation suggests a testable geometric criterion for complete separation: check whether the support of (FY(Y),FY(Y')) concentrates on a union of off-diagonal blocks indexed by the atoms of X.
Because the Markov product is symmetric in Y and Y', the same translation may apply to other asymmetric directed dependence measures by applying the distributional transform to the response in the same way.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the new content is Theorem 3.2 and Corollary 3.5. Theorem 3.2 says complete separation of Y relative to X is equivalent to an ordinal sum structure for the Markov product of FY(Y) and FY(Y'), with blocks indexed by the atoms of X. That is a clean structural insight: it lets you read separation off a two-dimensional distribution instead of conditional distributions on Rp. Corollary 3.5, the continuous-X special case, nicely complements [9, Thm 2.12]: when X has a continuous coordinate, complete separation forces perfect dependence, so the two extremes coincide in the fully continuous setting. The rest of the paper, Section 2 and Theorem 3.1, is mostly a routine extension of known continuous-cdf results to general distributions; useful but not surprising.

The proofs of Theorems 2.1 and 3.1 are correct and handle atoms via a standard countable-density extension. The examples are helpful, especially 3.3–3.6, and the scatterplots make the ordinal sum structure visible.

The main soft spot is the proof of Theorem 3.2. Two things need work. First, the construction of the az endpoints is mis-specified. As written, with two ordered atoms the upper atom can get az=0 and the intervals overlap, contradicting (14). This looks repairable by ordering the atoms and taking az as the previous bz (or 0 for the first), but as it stands the construction is wrong. Second, the proof leans on [9, Prop. 2.8], the invariance Λ(Y|X)=Λ(FY(Y)|X), without proof, and uses it in both directions. The cited companion paper may well contain a correct proof, but the identity is not obvious when Y has atoms, because FY is only nondecreasing and can tie previously separated supports. I would want the authors to state the lemma precisely and either prove it or give a citation that makes the flat-region behavior checkable. The step in (18) where Pu(X) is replaced by PFY(Y) is also sketched too quickly; the notation dPu(X) is a typo and the justification needs spelling out.

If the invariance lemma fails for some atom-valued distribution, the equivalence in Theorem 3.2 collapses. I do not think it fails, because relative effects are fairly robust to nondecreasing transforms when the original conditional supports are disjoint, but the paper should make this checkable. As it stands, the central theorem is plausible and likely true, but the written proof is incomplete.

Who this is for: people working on directed dependence measures, especially the Markov-product estimation framework. It is a useful paper, not a paradigm shift. I would send it to referees. The proof issues are repairable and the main structural result is worth having in the literature.

Referee Report

3 major / 4 minor

Summary. The paper proves equivalences between certain extreme directed-dependence properties of a pair (X,Y) and properties of the associated Markov product (Y,Y'), where Y' is a conditionally independent copy of Y given X. Specifically, Theorem 2.1 equates independence of X and Y with independence of Y and Y'; Corollaries 2.2 and 2.3 translate zero-explainability and stochastic comparability into uncorrelatedness and equality of concordance/discordance probabilities of (Y,Y'); Theorem 3.1 equates perfect dependence with comonotonicity and with Y=Y' almost surely; and Theorem 3.2 characterizes complete separation of Y relative to X by an ordinal sum structure of (F_Y(Y), F_Y(Y')) with block lengths equal to P(X=z). Several examples illustrate the non-implications among the concepts.

Significance. If Theorem 3.2 is correct, the paper delivers a genuine dimension-reduction principle: complete separation, a property defined through conditional distributions over R^p, can be read off the bivariate law of (F_Y(Y), F_Y(Y')). This would fit the authors' program of estimating directed dependence via nearest-neighbor methods on the Markov product. Theorems 2.1 and 3.1 are proven from first principles and appear correct, and the examples are useful and clearly presented. The main new result, however, currently rests on an incomplete proof and on an unproved external lemma, so the contribution is not yet ready in its present form.

major comments (3)

[§3, proof of Theorem 3.2, first step (paragraph containing (14))] The construction of the intervals (a_z,b_z] is defective. For z in M the proof sets b_z := u(z), and then sets a_z := u(x) only if there exists x in G such that u(w) <= u(x) <= l(z) for all w in G; otherwise a_z := 0. With two atoms z1,z2 whose supports are ordered with u(z1) < l(z2), the stated condition cannot be satisfied for z2 (and generally not for z1 either), so the proof assigns a_z=0 to both atoms. Then (a_{z1},b_{z1}] and (a_{z2},b_{z2}] overlap, contradicting the claimed disjointness in (14). The subsequent disintegration step in (17) depends on this disjointness. The construction is likely repairable by ordering the atoms and taking a_z to be the preceding interval's endpoint (or 0 for the first interval), but as written the central step of the proof is invalid.
[§3, statement before Theorem 3.2 and both directions of its proof] The proof relies essentially on the cited identity [9, Proposition 2.8], Lambda(Y|X) = Lambda(F_Y(Y)|X). This lemma is used twice: first to pass from complete separation of Y to complete separation of F_Y(Y), and again at the end to convert complete separation of F_Y(Y) back to complete separation of Y. No proof or statement of the lemma is given in the manuscript. Because Theorem 3.2 is asserted for general distributions, including atom-valued X and Y, one must verify that the distributional transform does not introduce ties on sets of positive P_X ⊗ P_X mass. If [9, Proposition 2.8] fails in that setting, the equivalence in Theorem 3.2 collapses. The authors should include a proof of the lemma or state precisely the conditions under which it holds and verify those conditions for the distributions treated in Theorem 3.2.
[§3, equations (17)-(19)] The identification of the distribution of u(X) with the distribution of F_Y(Y) on the complement of the union of the intervals is only sketched. Equation (15) gives distinctness of the values u(x1) and u(x2), but the equality P_{u(X)} = P_{F_Y(Y)} on [0,1] \ union (a_z,b_z] also requires the earlier observation that P(F_Y(Y)=u(x)|X=x)=1 for almost all x in G\M. As written, the underbrace in (18) suggests that (15) alone justifies the equality, which is not the case. Please spell out the argument or reorganize the derivation.

minor comments (4)

[§3, paragraph after (10)] The Lebesgue measure lambda is used in (10) and throughout the proof of Theorem 3.2 but is never defined; please introduce the notation explicitly.
[Example 3.4] The sentence 'which is ... the only permissible ordinal sum structure for Y to be stochastically comparable relative to X according to Theorem 3.2' appears to invoke Theorem 3.2, but that theorem characterizes complete separation, not stochastic comparability; the intended reference is likely Corollary 3.5 or the ordinal-sum size-zero characterization. Please correct the citation.
[§3, definition of ordinal sum] The phrase 'finite or countably finite subset of the natural numbers' is nonstandard; it should be 'finite or countably infinite subset' or simply 'countable subset'.
[Equation (17)] The underbraces containing '=1[0,u](u(x))' and '=1[0,v](u(x))' are assigned to (15), but (15) only states distinctness of the u(x) values; the equality with the indicator follows from the degeneracy of the conditional distributions established earlier. Please adjust the annotation.

Circularity Check

1 steps flagged · score 4.0 of 10

Theorem 3.2's bridge from Y to FY(Y) is imported from the authors' own [9, Prop. 2.8] in both directions, making the central equivalence partially self-citation-load-bearing; the ordinal-sum computation itself is independent.

self citation load bearing [Section 3, preamble to Theorem 3.2 and both directions of the proof of Theorem 3.2]
"Since Λ(Y |X) remains unchanged when replacing the random variable Y by its individual distributional transform due to [9, Proposition 2.8], i.e. Λ(Y |X) = Λ(FY (Y ) | X), in what follows we work with (X, FY (Y )) and its Markov product (FY (Y ), FY (Y ′))."

Theorem 3.2 characterizes complete separation of Y (statement b) by an ordinal sum condition on (FY(Y), FY(Y')) (statement c). The proof never derives the relation between Y and FY(Y) in this manuscript; it imports it from [9, Proposition 2.8], a companion paper by overlapping authors. The proposition is used at the start ('Then FY(Y) is completely separated relative to X due to [9, Proposition 2.8]') and again at the end ('this is equivalent to Y being completely separated relative to X due to [9, Proposition 2.8]'). Thus the two endpoints of the claimed equivalence reduce to a load-bearing self-citation that is not proved or independently verified here.

full rationale

The paper contains no fitted-input-renamed-as-prediction circularity: Corollaries 2.2 and 2.3 are explicitly immediate translations of the earlier representation formulas (7) and (8), and the main ordinal-sum derivation is carried out from the Markov product identity (5), not from any estimated parameter. The only true circular-adjacent issue is the load-bearing reliance on [9, Proposition 2.8] to pass from Y to FY(Y) in both directions of Theorem 3.2. That proposition comes from the same author group's companion paper and is not reproduced in this manuscript, so the central equivalence for Y rather than FY(Y) is inherited from a self-citation. This is a verifiability and correctness concern, especially for atom-valued X or Y, as is the apparently defective construction of az in the proof of Theorem 3.2 (the intervals (az,bz] are not shown to be disjoint as claimed in (14)). These issues are not, however, cases of a result being equivalent to its inputs by definition, and the ordinal-sum computation itself has independent mathematical content. The appropriate circularity score is therefore modestly above the no-circularity band.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented physical entities. The paper introduces a generalized ordinal sum definition as a mathematical tool, which is a definition rather than a postulated entity. The central claim rests on the Markov product framework and on prior representation results from the same research group.

assumptions (4)

domain assumption The Markov product (Y,Y') is defined via a regular conditional distribution of Y given X and conditional independence Y perpendicular Y' given X (Eq. (4)-(5)).
This is the probabilistic framework of the paper; standard but not derived within the paper.
domain assumption The representation formulas (6), (7), (8) express xi, R2, and Lambda in terms of (Y,Y').
Cited from [1], [8], [9] respectively; the paper's corollaries are immediate consequences.
domain assumption Lambda(Y|X)=Lambda(FY(Y)|X) ([9, Proposition 2.8]).
Used to replace Y by its distributional transform in Theorem 3.2; not proved in this paper.
standard math Holder's inequality, disintegration, and countable density arguments.
Used in proofs of Theorems 2.1 and 3.1.

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Pith. "Pith review of A dimension reduction for extreme types of directed dependence." pith.science (2026). https://pith.science/paper/JO6P5UVO

@misc{pith2026250604825,
  author       = {Pith},
  title        = {Pith review of: A dimension reduction for extreme types of directed dependence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JO6P5UVO}},
  note         = {Machine review of arXiv:2506.04825}
}

abstract

In recent years, a variety of novel measures of dependence have been introduced being capable of characterizing diverse types of directed dependence, hence diverse types of how a number of predictor variables $\mathbf{X} = (X_1, \dots, X_p)$, $p \in \mathbb{N}$, may affect a response variable $Y$. This includes perfect dependence of $Y$ on $\mathbf{X}$ and independence between $\mathbf{X}$ and $Y$, but also less well-known concepts such as zero-explainability, stochastic comparability and complete separation. Certain such measures offer a representation in terms of the Markov product $(Y,Y')$, with $Y'$ being a conditionally independent copy of $Y$ given $\mathbf{X}$. This dimension reduction principle allows these measures to be estimated via the powerful nearest neighbor based estimation principle introduced in [4]. To achieve a deeper insight into the dimension reduction principle, this paper aims at translating the extreme variants of directed dependence, typically formulated in terms of the random vector $(\mathbf{X},Y)$, into the Markov product $(Y,Y')$.

Figures

Figures reproduced from arXiv: 2506.04825 by the authors.

**Figure 2.** Scatterplots of sample size 1000 for the random vector ( [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Scatterplots of sample size 1000 for the random vectors ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Scatterplot of sample size 2000 of a distribution on [0 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: Scatterplots of sample size 1000 showing ( [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: Scatterplots of sample size 1000 showing ( [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: Scatterplots of sample size 1000 showing ( [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

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Reference graph

Works this paper leans on

17 extracted references · 15 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Ansari, J. and S. Fuchs (2023+). A direct extension of Azadkia & Chatterjee’s rank correlation to a vector of endogenous variables. Available at https: // arxiv. org/ abs/ 2212. 01621

work page 2023

[2] [2]

Ansari, J. and S. Fuchs (2025). On continuity of Chatterjee’s rank correlation and related depen- dence measures. Available at https: // arxiv. org/ abs/ 2503. 11390

work page 2025

[3] [3]

Langthaler, S

Ansari, J., P. Langthaler, S. Fuchs, and W. Trutschnig (2025). Quantifying and estimating depen- dence via sensitivity of conditional distributions. Bernoulli, to appear

work page 2025

[4] [4]

Azadkia, M. and S. Chatterjee (2021). A simple measure of conditional dependence. Ann. Stat. 49 (6), 3070–3102

work page 2021

[5] [5]

Birnbaum, Z. W. and O. M. Klose (1957). Bounds for the variance of the Mann-Whitney statistic. Ann. Math. Stat. 28 (4), 933–945

work page 1957

[6] [6]

Bathke, and F

Brunner, E., A. Bathke, and F. Konietschke (2019). Rank and Pseudo-Rank Procedures for Inde- pendent Oberservations in Factorial Design . Springer, Cham. 13

work page 2019

[7] [7]

Durante, F. and C. Sempi (2016). Principles of Copula Theory . CRC Press, Boca Raton FL

2016

[8] [8]

Fuchs, S. (2024). Quantifying directed dependence via dimension reduction. J. Multivariate Anal. 201, Article ID 105266

2024