arxiv: 2605.13550 · v1 · submitted 2026-05-13 · 📊 stat.ME

Recognition: 2 theorem links

· Lean Theorem

Causal Discovery via Statistical Power (CDSP)

Shreya Prakash , Fan Xia , Elena A. Erosheva

Authors on Pith no claims yet

Pith reviewed 2026-05-14 17:44 UTC · model grok-4.3

classification 📊 stat.ME

keywords causal discoverystatistical powereffect sizebivariate datauncertainty quantificationfalse discovery ratedirection estimationobservational data

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The pith

Effect-size asymmetry lets statistical power identify causal direction with uncertainty from bivariate observations.

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

The paper develops CDSP to treat causal direction inference as a comparison of statistical power between the two possible directions. It defines an effect-size asymmetry assumption under which the probability of correctly identifying the true direction exceeds the probability of selecting the reverse. This supplies both a decision rule for direction and a measure of uncertainty in the choice. Simulations indicate the procedure tolerates mild and moderate model errors, while tests on 100 real benchmark pairs show roughly 18 percent fewer false discoveries than a standard competing method.

Core claim

Considering the foundational setting of bivariate observational data, we show how quantities analogous to statistical power and effect size can be used in causal discovery to determine when data contain sufficient information to favor one direction over the other. We introduce the effect-size asymmetry assumption that characterizes when the probability of correctly detecting the causal direction exceeds that of incorrectly favoring the reverse direction. We show that the effect-size asymmetry assumption can be used for causal direction estimation with uncertainty quantification. Simulations show that CDSP direction estimation is robust to mild and moderate model misspecifications.

What carries the argument

The effect-size asymmetry assumption, which states that the power to detect the correct causal direction exceeds the power to detect the incorrect reverse direction.

If this is right

Direction can be estimated only when the observed data supply enough information for one direction to have strictly higher power.
Uncertainty about the chosen direction can be reported directly from the power calculations.
The procedure remains reliable under mild and moderate departures from the assumed functional form.
False discovery rates drop by approximately 18 percent relative to a common existing method on standard cause-effect benchmarks.

Where Pith is reading between the lines

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

The same power-comparison logic could be adapted to decide when to stop collecting more observations in a causal study.
Domain-specific data such as gene-expression pairs could be used to test how often the asymmetry assumption holds in practice.
Combining CDSP with existing functional causal methods might produce hybrid procedures that inherit both statistical guarantees and flexibility.
Exact sample-size formulas derived from the asymmetry condition would let experimenters plan studies that reach a target power level for direction recovery.

Load-bearing premise

An effect-size asymmetry must exist such that the probability of correctly detecting the causal direction exceeds the probability of incorrectly favoring the reverse direction.

What would settle it

On pairs whose true direction is known independently, the estimated power for the correct direction is not higher than for the reverse, or the method fails to show an 18 percent reduction in false discoveries on the 100 benchmark pairs.

Figures

Figures reproduced from arXiv: 2605.13550 by Elena A. Erosheva, Fan Xia, Shreya Prakash.

**Figure 1.** Figure 1: Settings of linearity ¶Using the test statistics of Sen and Sen [2014], the critical values satisfy c α Y,n = c α Y /n and c α X,n = c α X/n. Consequently, the corresponding standardized critical values satisfy q α X,n − q α Y,n = o(1), so asymptotically the contribution of the critical values is negligible, and population effect-size asymmetry is determined by IX and IY . 10 [PITH_FULL_IMAGE:figures/full… view at source ↗

**Figure 2.** Figure 2: Distribution of the estimated directional detectability indices [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: Rows 1–3 correspond to the Population and Dietary Consumption, Balltrack, and Ozone–Temperature [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

**Figure 4.** Figure 4: Distribution of the estimated directional detectability indices [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗

read the original abstract

Causal discovery methods aim to infer causal direction from observational data. Functional causal discovery approaches use structural asymmetries to identify causal directionality but rely on strong modeling assumptions and provide limited tools for uncertainty quantification. We introduce Causal Discovery via Statistical Power (CDSP), a statistical inference framework that connects causal direction estimation with statistical power and enables uncertainty quantification. Considering the foundational setting of bivariate observational data, we show how quantities analogous to statistical power and effect size can be used in causal discovery to determine when data contain sufficient information to favor one direction over the other. We introduce the effect-size asymmetry assumption that characterizes when the probability of correctly detecting the causal direction (i.e., the power of causal discovery) exceeds that of incorrectly favoring the reverse direction. We show that the effect-size asymmetry assumption can be used for causal direction estimation with uncertainty quantification. Simulations show that CDSP direction estimation is robust to mild and moderate model misspecifications. Real data analyses on 100 cause-effect benchmark pairs further demonstrate that CDSP reduces false discovery rates by approximately 18% relative to a commonly used existing method.

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

CDSP ties statistical power ideas to bivariate causal direction via a new effect-size asymmetry assumption, with decent simulation robustness and an 18% FDR drop on benchmarks, but the power ordering lacks explicit regularity conditions.

read the letter

The main thing to know is that this paper frames bivariate causal discovery around statistical power quantities and introduces an effect-size asymmetry assumption to support direction estimates plus uncertainty quantification. It reports robustness in simulations under mild and moderate misspecification and an 18% FDR reduction versus a standard method on 100 benchmark pairs. That empirical piece is the clearest contribution. The connection between power analogs, effect sizes, and causal direction is not in the literature they cite, so the framing itself counts as new. They treat the asymmetry assumption as a domain premise rather than something fitted or derived from self-citations, which keeps the setup clean and lets the reported gains stand somewhat independently. The simulations and benchmark results give a practical sense of when the method favors one direction over the reverse. The soft spot is exactly the one the stress-test flags: no theorem spells out the regularity conditions on the functional class, noise moments, or identifiability that would guarantee the claimed power ordering. Without those conditions the uncertainty quantification rests on an unverified premise, and mild misspecifications that flip the ordering would break both the point estimate and the FDR claim. The abstract states the results but does not supply the derivations or exact definitions needed to check calibration. This is for people working on observational causal discovery who want uncertainty tools in the bivariate case. A reader focused on statistical framing of discovery methods would get value from the simulations and the concrete benchmark comparison. I would send it for peer review. The empirical improvement is concrete enough to justify referee time even if the theoretical support needs tightening.

Referee Report

2 major / 2 minor

Summary. The paper introduces Causal Discovery via Statistical Power (CDSP), a framework for bivariate causal discovery that analogizes statistical power and effect size to determine when observational data suffice to favor one causal direction over the reverse. It posits an effect-size asymmetry assumption under which the probability of correctly detecting the true direction exceeds that of favoring the reverse, and uses this to perform direction estimation together with uncertainty quantification. Simulations are reported to show robustness under mild-to-moderate misspecification, and an analysis of 100 cause-effect benchmark pairs is claimed to yield an approximately 18% FDR reduction relative to a standard existing method.

Significance. If the central assumption can be placed on a rigorous footing, CDSP would supply a statistically grounded route to uncertainty quantification in causal discovery, a feature largely absent from existing functional approaches. The reported empirical gains on benchmarks and the explicit link to power analysis constitute a potentially useful contribution, provided the mapping from observed statistics to calibrated inferences is shown to be valid rather than merely observed.

major comments (2)

[effect-size asymmetry assumption and surrounding derivation] The effect-size asymmetry assumption (introduced after the abstract and developed in the main theoretical section) asserts that P(correct direction | data) > P(reverse | data) under the assumption, yet no theorem supplies the regularity conditions (e.g., on the class of structural functions, noise moments, or identifiability of the effect-size measure) that guarantee the inequality rather than merely illustrate it in simulations. Without such conditions the subsequent uncertainty quantification and FDR claims rest on an unverified premise.
[real data analyses] Section on real-data experiments: the reported 18% FDR reduction on the 100 benchmark pairs is presented without an explicit statement of the data-exclusion rules, the precise definition of the power-analog statistics, or the baseline method's implementation details, making it impossible to verify that the gain is attributable to the new framework rather than to preprocessing choices.

minor comments (2)

[abstract and introduction] The abstract refers to 'quantities analogous to statistical power and effect size' without a concise one-sentence definition; a short clarifying sentence in the introduction would improve readability.
[simulation results] Simulation figures lack error bars or explicit sample-size annotations in several panels, which obscures the precision of the reported robustness claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below, indicating the revisions we will make to provide greater rigor and reproducibility.

read point-by-point responses

Referee: The effect-size asymmetry assumption asserts that P(correct direction | data) > P(reverse | data) under the assumption, yet no theorem supplies the regularity conditions (e.g., on the class of structural functions, noise moments, or identifiability of the effect-size measure) that guarantee the inequality rather than merely illustrate it in simulations. Without such conditions the subsequent uncertainty quantification and FDR claims rest on an unverified premise.

Authors: We appreciate the referee's point on the need for explicit regularity conditions. The manuscript introduces the effect-size asymmetry assumption via power analogies and supports it with derivations and simulations, but we acknowledge that a formal theorem stating the precise conditions was omitted. In the revised manuscript we will add a theorem that specifies the required conditions, including Lipschitz continuity and monotonicity of the structural functions, finite second moments of the noise terms, and identifiability of the effect-size measure. Under these conditions we will prove that the probability of correctly identifying the true direction exceeds that of the reverse, thereby placing the uncertainty quantification on a rigorous footing. revision: yes
Referee: the reported 18% FDR reduction on the 100 benchmark pairs is presented without an explicit statement of the data-exclusion rules, the precise definition of the power-analog statistics, or the baseline method's implementation details, making it impossible to verify that the gain is attributable to the new framework rather than to preprocessing choices.

Authors: We agree that the real-data section requires additional detail to ensure reproducibility and to confirm that the reported gain is attributable to CDSP. In the revised manuscript we will add an explicit description of the data-exclusion rules applied to the 100 benchmark pairs, the precise mathematical definitions of the power-analog statistics, and the full implementation details of the baseline method together with all preprocessing steps. These additions will allow independent verification of the 18% FDR reduction. revision: yes

Circularity Check

0 steps flagged

No significant circularity in CDSP derivation

full rationale

The paper introduces the effect-size asymmetry assumption explicitly as a new domain premise characterizing when power for correct causal direction exceeds that for the reverse. This premise is not derived from or reduced to fitted parameters, self-citations, or prior results by the same authors; it is posited directly and then used to motivate direction estimation and uncertainty quantification. Empirical claims (simulation robustness and 18% FDR reduction on benchmarks) are presented as independent validations rather than tautological consequences of the assumption. No load-bearing step in the abstract or described framework equates a prediction or result to its own inputs by construction, so the derivation chain remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the newly introduced effect-size asymmetry assumption; no free parameters or invented entities are described in the abstract.

axioms (1)

domain assumption Effect-size asymmetry assumption
States that the probability of correctly detecting the causal direction exceeds the probability of favoring the reverse direction under the observed effect-size properties.

pith-pipeline@v0.9.0 · 5485 in / 1167 out tokens · 43384 ms · 2026-05-14T17:44:47.676433+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We introduce the effect-size asymmetry assumption that characterizes when the probability of correctly detecting the causal direction exceeds that of incorrectly favoring the reverse direction.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Directional power is the probability that the causal discovery method correctly favors X→Y

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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