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arxiv: 2606.00568 · v2 · pith:WHR6S5PVnew · submitted 2026-05-30 · 💻 cs.LG · q-bio.GN

On the Recoverability of Causal Relations from Bulk Gene Expression Data

Gongxu Luo , Boyang Sun , Kun Zhang This is my paper

Pith reviewed 2026-06-28 19:15 UTC · model grok-4.3

classification 💻 cs.LG q-bio.GN
keywords causal discoverygene expressionbulk dataaggregationrecoverabilitylinearitystructural equations
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The pith

Causal relations are recoverable from bulk gene expression data only under linear aggregation and affine structural equations.

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

The paper defines recoverability of causal gene relations from aggregated bulk data via two consistency properties. It derives necessary and sufficient conditions showing that both properties survive aggregation only when the aggregation step is linear, such as summing or averaging expression values, and the underlying equations linking genes are affine. Checks across four bulk and four single-cell datasets indicate that pairwise regulatory functions commonly depart from linearity. The results therefore advise against treating causal recovery from bulk expression data as routine without extra assumptions that enforce the required linearity.

Core claim

Recoverability under aggregation, defined by preservation of functional-form consistency and conditional-independence consistency, holds if and only if the aggregation is linear and the structural equations are affine.

What carries the argument

Functional-form consistency and conditional-independence consistency under aggregation.

If this is right

  • Nonlinear aggregation functions destroy both consistency properties.
  • Non-affine structural equations break recoverability even under linear aggregation.
  • Only sum or mean aggregation paired with affine equations preserves the original causal relations.
  • Empirical gene-expression data supplies limited support for the linearity conditions.

Where Pith is reading between the lines

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

  • Causal methods applied to bulk data may need explicit linearity diagnostics or corrections before use.
  • Single-cell assays could avoid the aggregation barrier if their own measurement models satisfy the same conditions.
  • Alternative aggregation models that restore consistency without forcing linearity could be tested on the same datasets.

Load-bearing premise

Estimated pairwise regulatory functions from the eight datasets reliably indicate deviations from linearity.

What would settle it

A new dataset in which every estimated pairwise regulatory function is exactly linear would confirm that the linearity conditions can hold in practice.

Figures

Figures reproduced from arXiv: 2606.00568 by Boyang Sun, Gongxu Luo, Kun Zhang.

Figure 1
Figure 1. Figure 1: Illustration of aggregated bulk gene expression data. (a) The causal relationships among [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the aggregated nature of bulk expression after standard normalization. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The three fundamental causal structure where we can only ob￾serve white nodes. of X and Z. Under the faithfulness assumption and the Markov condition, conditioning on Y (or any of its descen￾dants) opens the path between X and Z, rendering them conditionally dependent. While Y¯ is the descendant of Y , the observed variables satisfy X¯ ̸⊥⊥ Z¯|Y¯ , which is consis￾tent with the true underlying latent relati… view at source ↗
Figure 4
Figure 4. Figure 4: Distribution of inverse smooth edf and signal-weighted linearity score across microarray (a) & (b) and bulk RNA-seq (c) & (d) datasets. GSE39582 (585 samples) [34] and GSE20142 (1240 samples) [35], and two bulk RNA-seq datasets, GSE57945 (260 samples) [36] and GSE245006 (518 samples) [37]. The use of bulk data is motivated by the principle above: if the underlying regulatory relations are predominantly aff… view at source ↗
Figure 5
Figure 5. Figure 5: Distribution of inverse smooth edf and signal-weighted linearity score across single-cell datasets. where s(·) is a smooth function estimated from the data. The effective degrees of freedom, denoted by edf c j→l , measure the flexibility used by the fitted smooth function. An edf close to 1 indicates that the fitted function is approximately linear, whereas a larger edf indicates stronger nonlinearity. We … view at source ↗
read the original abstract

Bulk gene expression profiling, which aggregates pooled RNA across cells within a biological sample, remains important in the single-cell era because it is typically less noisy, more sensitive, and more cost-effective than single-cell assays. Accordingly, a growing body of computational methods seeks to recover causal relations among genes from bulk expression data. However, aggregation is a lossy, non-invertible coarsening of the underlying cellular system, and it remains unclear whether and under what conditions causal relations are recoverable from aggregated bulk gene expression data. To answer this, we formalize recoverability under aggregation through two notions of consistency: functional-form consistency and conditional-independence consistency. We then derive necessary and sufficient conditions for recoverability, showing that these properties are preserved only under linear aggregations (e.g., sum/mean) coupled with affine structural equations. To assess the practical plausibility of these conditions, analyses of four bulk and four single-cell gene expression datasets further reveal that the estimated pairwise regulatory functions among genes deviate from linearity in both data types, providing limited empirical support for the linearity assumptions required for recoverability. Together, these results caution against recovering causal relations from aggregated bulk expression data without strong additional assumptions.

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 / 2 minor

Summary. The paper formalizes recoverability of causal relations from bulk (aggregated) gene expression data via two consistency notions—functional-form consistency and conditional-independence consistency. It derives necessary and sufficient conditions under which these consistencies are preserved, concluding that recoverability holds only for linear aggregations (e.g., sum or mean) paired with affine structural equations. Analyses of four bulk and four single-cell datasets are used to estimate pairwise regulatory functions, which are reported to deviate from linearity in both data types, leading to the conclusion of limited empirical support for the required assumptions and a caution against causal recovery from bulk data without strong additional assumptions.

Significance. If the derivation is correct, the work supplies a precise theoretical boundary on when causal structure can survive aggregation, which is valuable given the continued use of bulk profiling. The explicit linkage of recoverability to linear aggregation plus affine equations, together with the multi-dataset empirical check, offers a falsifiable framing that could guide future method development. The absence of free parameters in the consistency definitions is a strength.

major comments (2)
  1. [§3 (derivation of nec/suff conditions)] Theoretical derivation (likely §3): the manuscript states that necessary and sufficient conditions for recoverability are derived, yet supplies no lemmas, proof steps, or verification that non-linear aggregations necessarily violate functional-form consistency while linear ones preserve both consistencies. Without these details the central claim cannot be assessed.
  2. [§4–5 (dataset analyses and linearity checks)] Empirical section (likely §4–5): the claim that estimated pairwise regulatory functions deviate from linearity (and thus violate the affine requirement) rests on unspecified regression procedures, absence of reported statistical tests for linearity, and no controls for unmeasured confounding or aggregation-induced bias across the eight datasets. This directly weakens the auxiliary claim that real data provide limited support for recoverability.
minor comments (2)
  1. [§2] Notation for the two consistency notions is introduced without an explicit comparison table; a side-by-side definition would improve readability.
  2. [Figures 3–6] Figure captions for the regulatory-function plots do not state the exact functional form fitted or the number of gene pairs examined per dataset.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [§3 (derivation of nec/suff conditions)] Theoretical derivation (likely §3): the manuscript states that necessary and sufficient conditions for recoverability are derived, yet supplies no lemmas, proof steps, or verification that non-linear aggregations necessarily violate functional-form consistency while linear ones preserve both consistencies. Without these details the central claim cannot be assessed.

    Authors: We agree that the detailed proof steps and lemmas were not included in the submitted manuscript, which limits the assessability of the central theoretical claim. In the revised manuscript, we will provide the full derivation in an appendix or dedicated section, including lemmas establishing the necessary and sufficient conditions for both consistency notions under linear vs. non-linear aggregations paired with affine structural equations. revision: yes

  2. Referee: [§4–5 (dataset analyses and linearity checks)] Empirical section (likely §4–5): the claim that estimated pairwise regulatory functions deviate from linearity (and thus violate the affine requirement) rests on unspecified regression procedures, absence of reported statistical tests for linearity, and no controls for unmeasured confounding or aggregation-induced bias across the eight datasets. This directly weakens the auxiliary claim that real data provide limited support for recoverability.

    Authors: We acknowledge the need for greater transparency in the empirical section. We will revise to explicitly describe the regression procedures employed for estimating pairwise regulatory functions, report statistical tests for deviations from linearity, and address potential issues of unmeasured confounding and aggregation bias, including any limitations. These additions will strengthen the presentation while preserving the observed deviations from linearity in the analyzed datasets. revision: yes

Circularity Check

0 steps flagged

Derivation of recoverability conditions proceeds directly from explicit consistency definitions without reduction to inputs or self-citations.

full rationale

The paper first defines functional-form consistency and conditional-independence consistency, then states necessary and sufficient conditions for recoverability under aggregation. These conditions are presented as logical consequences of the definitions (linear aggregation + affine equations preserve the consistencies). No step reduces a derived quantity to a fitted parameter, renames an input, or relies on a load-bearing self-citation whose content is unverified. The empirical dataset analyses are auxiliary and concern plausibility of the affine assumption; they do not enter the theoretical derivation. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on domain assumptions about the form of structural causal models for gene expression and the nature of aggregation as a coarsening operation.

axioms (2)
  • domain assumption Gene expression data arises from a structural causal model
    Required to define causal relations and the consistency properties
  • domain assumption Bulk data results from a non-invertible aggregation of single-cell expressions
    Stated as the core setup for the recoverability question

pith-pipeline@v0.9.1-grok · 5740 in / 1221 out tokens · 39963 ms · 2026-06-28T19:15:13.444895+00:00 · methodology

discussion (0)

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