Adding covariates to bounds: What is the question?

Arvid Sj\"olander; Erin E Gabriel; Gustav Jonzon; Michael C Sachs

REVIEW 1 major objections 5 minor 1 cited by

Adding covariates to bounds: What is the question?

T0 review · 1 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The sharpness of covariate-averaged causal bounds reduces to uniform sharpness of the conditional bounds, and in the all-binary IV graph where S is independent of Z and a confounded parent of X and Y, the averaged Balke-Pearl bounds…

desk verdict A useful formal distinction for covariate-conditional sharpness, undermined by an overgeneralized equivalence that fails for continuous covariates. read the letter →

arxiv 2502.03156 v2 pith:26G4OVWZ submitted 2025-02-05 stat.ME

Gustav Jonzon , Erin E Gabriel , Arvid Sjölander , Michael C Sachs This is my paper · ORCID

classification stat.ME

keywords partialidentificationsharpboundsuniformsharpnesscovariateadjustmentinstrumentalvariablesBalke-Pearlcausalriskdifference

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 sharpens the notion of sharpness for partial identification bounds when an ancillary covariate is conditioned on. It distinguishes pointwise sharpness (each covariate stratum's bound can be attained separately) from uniform sharpness (one underlying distribution hits all strata's bounds at once), and proves that the covariate-averaged lower bound is sharp for the marginal causal effect if and only if the conditional bounds are uniformly sharp. In the all-binary instrumental variable model where the covariate is independent of the instrument and is a confounded parent of exposure and outcome, the paper proves the covariate-averaged Balke-Pearl bounds are valid and sharp, coinciding with the covariate-optimal bounds. In other IV graphs, such as when the covariate acts as an additional instrument or as an unconfounded mediator, covariate averaging can fail to be sharp even though every stratum-specific bound is sharp. The paper's practical message is that claims of sharpness after covariate conditioning must state the causal model containing the covariate.

What carries the argument

The load-bearing object is the definition of uniform sharpness: a pointwise valid set of conditional bounds $\{L_s:s\in S\}$ is uniformly sharp if for every observed joint distribution $P_O$ and every $\varepsilon>0$ there exists a single underlying distribution $P_V$ compatible with the model whose conditional estimands $\theta_s$ all lie within $\varepsilon$ of the corresponding $L_s$. The covariate-averaged bound $\bar L=\int L_s\,dP_S(s)$ is the mechanism that connects this global property to the marginal estimand $\theta=\int\theta_s\,dP_S(s)$. The proofs use the Balke-Pearl bounds, the sharp bounds on a binary causal risk difference under an instrumental variable, as the stratum-specific bounds in the IV setting, and a symbolic linear-programming computation of covariate-optimal bounds to compare against the averaged bounds.

What would settle it

Find a continuous covariate $S$ and any model in the Figure 5e graph for which the covariate-averaged Balke-Pearl bound equals the covariate-optimal bound for every observed distribution, yet no single compatible full distribution attains all conditional bounds within $\varepsilon$ for arbitrarily small $\varepsilon$; that would disprove Proposition 2 as printed. Concretely, simulate with $S$ continuous and check whether the only-if direction fails when stratum probabilities are zero or when the supremum over distributions of the average gap is zero but the infimum over distributions of the maximum stratum gap is positive.

Watch

Extended reading notes

Core claim

The central claim is Proposition 2: the covariate-averaged lower bound $\bar L(P_O)=\int L_s(P_{O|s})\,dP_S(s)$ is sharp for $\theta$ under $G$ if and only if the set of conditional bounds $\{L_s: s\in S\}$ is uniformly sharp for the set of conditional estimands $\{\theta_s: s\in S\}$. The paper also proves Proposition 4: in the all-binary IV model $G'$ of Figure 5e, where $S$ is independent of $Z$ and a confounded parent of $X$ and $Y$, the Balke-Pearl bounds conditioned on $S$ are uniformly sharp, so the covariate-averaged lower bound is valid and sharp for $\theta$. Together these claims identify the exact sense in which covariate averaging can be optimal: it inherits sharpness exactly when one distribution can realize all conditional bounds simultaneously, and in the binary IV setting this happens in the one graph where $S\perp\!\!\perp Z$ and $S$ confounds $X$ and $Y$.

Load-bearing premise

The equivalence between sharpness of the averaged bound and uniform sharpness of the conditional bounds requires that the covariate space is finite or atomic with positive probability in every stratum, a regularity condition the paper states only as a proof deferred to the supplement.

Editorial extensions

If this is right

If Proposition 2 holds, checking whether an averaged bound is sharp reduces to checking uniform sharpness of the conditional bounds; pointwise sharpness alone is not enough.
In the all-binary IV graph where $S$ is independent of $Z$ and is a confounded parent of $X$ and $Y$, covariate-averaged Balke-Pearl bounds coincide with the covariate-optimal bounds, so no information is lost by averaging.
When $S$ is an additional instrument or an unconfounded mediator, the covariate-averaged bounds are not sharp, and in the mediator case the full covariate-optimal model can even point-identify the effect.
Covariate-averaged bounds are never wider than covariate-marginal bounds when $S\perp\!\!\perp Z$, but can be wider than the marginal bounds when relevance of $Z$ is broken through $S$.
Averaging can still narrow valid bounds without being sharp, so width improvement and sharpness are separate claims that should be reported distinctly.

Reading between the lines

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

We infer that Proposition 2's only-if direction needs a regularity condition on $S$: the equivalence likely holds for finite or atomic covariate spaces, and for continuous $S$ a sequence of distributions could make the average gap vanish without a single distribution making all conditional gaps small.
A testable extension would be to approximate a continuous covariate by finer stratifications in the Figure 5e model and check whether the averaged bounds converge to the covariate-optimal bounds only under an added uniformity condition on the strata.
The paper's examples suggest a practical caution the authors leave implicit: covariates that are predictive of $X$ and $Y$ should not automatically be averaged over; the graph hosting $S$ decides whether averaging helps, and in some graphs it hurts.

Share X Bluesky LinkedIn Reddit HN

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

The paper gives a clear and genuinely useful distinction between pointwise and uniform sharpness for covariate-conditional bounds, and it is right that Levis et al. use the uniform concept without defining it. The definitions alone are worth having. The positive result for the graph where S is independent of Z and a confounded parent of X and Y (Proposition 4) is the kind of concrete sharpness result people will want to cite, and the simulations and examples map the territory well.

The soft spot is Proposition 2, which claims an if-and-only-if for 'all settings'. The only-if direction (average sharpness implies uniform sharpness) is false when S is continuous. A simple counterexample: S uniform on [0,1], U hidden, and for each u define θs(PV_u) as an indicator on a small-measure set that varies with u (e.g., A_u = [0,u] ∪ [1/2,1/2+u]) with Ls=0. The averaged bound 0 is sharp, and {Ls} is pointwise sharp for almost every s, but no single u makes θs < ε on almost every s. The equivalence needs S to be finite or to have a positive minimum stratum probability. The proof is in the supplement, so this missing condition is invisible in the main text. That is a genuine flaw, but a fixable one: the paper's applications use binary or categorical S, and the if direction (uniform implies average) holds.

The algebraic proof of Proposition 4 is also deferred; I could not inspect it. The simulation code appears to be available, and the simulation results are coherent.

Bottom line: this is a serious paper for people who build or use partial identification bounds. The central conceptual point is right, the examples are instructive, and the overreach in Proposition 2 is local and repairable. I would send it to a competent referee with a request to fix the statement of Proposition 2 and add the regularity condition. For a reading group it would generate good discussion.

Referee Report

1 major / 5 minor

Summary. The paper formalizes the distinction between pointwise and uniform sharpness for covariate-conditional bounds, proves general properties of covariate-averaged bounds, and applies these concepts to the binary instrumental-variable setting with covariates. The main positive result is Proposition 4, which states that in the DAG of Figure 5e—where S is independent of Z and is a confounded parent of X and Y—the covariate-conditional Balke-Pearl bounds are uniformly sharp and the covariate-averaged bound is sharp. The paper also gives two negative examples, in which covariate averaging is not sharp, and a simulation study over six DAGs comparing covariate-marginal, covariate-averaged, and covariate-optimal bounds.

Significance. If the results hold, the paper makes a useful conceptual contribution by carefully separating pointwise from uniform sharpness, a distinction that is often blurred in the partial-identification literature. The definitions are clear, the running examples are instructive, and the explicit algebraic comparison with causaloptim for Proposition 4 provides a concrete case where covariate averaging attains the optimal width. The reproducible simulation code is a further strength. However, the paper's most general theorem, Proposition 2, is false as stated for continuous covariate spaces, so the general framework needs a regularity condition or a restriction to finite/atomic S before the paper's broader claims can be accepted.

major comments (1)

[Section 4, Proposition 2] A second, related point: because Proposition 2 is invoked in the narrative that connects covariate-averaged sharpness to uniform sharpness, the authors should state clearly in the main text whether the result is intended for all covariate spaces or only for finite S. The examples and Proposition 4 use binary S, so they are not affected by the counterexample, but the general formulation in Section 4 and the abstract's 'general conditions' claim must be revised.

minor comments (5)

[Section 5.2] In the displayed definition of A|s, the third term is written as 'p10.0 + p01.0 − p00.1 − p01.1s'; the first three summands appear to be missing the s subscript and should presumably be p10.0s + p01.0s − p00.1s − p01.1s. As printed, the expression mixes conditional and unconditional probabilities and makes the subsequent algebra impossible to follow.
[Section 4.1] The sentence '¯L(PO) = P (X = 0, Y= 1) − P (X = 1, Y= 0)' has a sign error: the preceding line defines L(PO) = −P(X=0,Y=1) − P(X=1,Y=0), and the later covariate-optimal expression confirms that the averaged bound should be the negative sum, not the difference.
[Section 7] In the concluding paragraph, 'unformly sharp' should be 'uniformly sharp'.
[Section 6] The simulation section does not report Monte Carlo standard errors for the proportions in Table 1; with 10^5 replicates the error is likely small, but a brief statement would be helpful.
[Section 5.3.2] The phrase 'This example needs less explanation' is informal for a journal article and could be replaced by a more neutral transition.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: central results are proven from model assumptions; self-citations to causaloptim/Sachs et al. are independent computational/theorem support.

full rationale

The paper's derivation chain is self-contained. Proposition 2 is a general equivalence whose proof is deferred to the supplement; whether or not it requires a finite/atomic S regularity condition, it is a mathematical claim, not a definitional reduction. Proposition 4 is established by showing, algebraically in the supplement, that the covariate-averaged Balke-Pearl bound equals the covariate-optimal bound Lco computed by causaloptim. The use of causaloptim and Sachs et al. (2023) to certify sharpness is a self-citation, but it is not circular: Sachs et al. is an externally published, code-reproduced theorem with stated assumptions that do not include the covariate-averaging result, and the equality Lbar = Lco is checked algebraically rather than assumed. The examples in Section 5.3 use the same software to exhibit non-sharpness, which is again a computation, not a fitted prediction. No free parameter is fitted and later called a prediction; no estimand is defined in terms of the bound it is supposed to constrain; no uniqueness theorem from the authors' prior work is invoked to forbid alternatives. The only notable caveat is the possible failure of Proposition 2's only-if direction for continuous S without extra compactness or positivity conditions, but that is a correctness or regularity concern, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard causal assumptions (consistency, NPSEM, IV exclusion/latent independence) plus a black-box sharpness theorem for causaloptim. No free parameters are fitted. One regularity condition for Proposition 2 is unstated and is flagged as ad hoc.

assumptions (6)

domain assumption Consistency and no interference for counterfactuals (Section 2)
Standard causal inference assumptions used to define potential outcomes Y(x) and the target estimand.
domain assumption NPSEM/DAG encoding of the causal model for each setting (Figures 3-5)
The sharpness claims are relative to the assumed graph; the results depend on the specific connectivity of S.
domain assumption IV assumptions 1-2: Z ⊥⊥ {Y(z),X(z)}|S and Y(x,z)=Y(x), plus positivity (Section 5.1)
Defines the class G of IV models for Propositions 3, 4 and examples.
standard math Sharpness of causaloptim bounds proven in Sachs et al. 2023
Used to assert Lco is the sharp covariate-optimal bound; external published result taken as background.
domain assumption S ⊥⊥ Z and binary S in Proposition 4 (Figure 5e)
Needed for the equality pyx.z = E[pyx.zS] that underlies the algebraic identity Lbar = Lco; without it the averaged bounds need not be sharp.
ad hoc to paper Unstated regularity: S finite/atomic with positive stratum probabilities for the only-if direction of Proposition 2
The proposition as stated claims all settings; the only-if proof requires this discreteness to turn average sharpness into simultaneous conditional sharpness.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Adding covariates to bounds: What is the question?." pith.science (2026). https://pith.science/paper/26G4OVWZ

@misc{pith2026250203156,
  author       = {Pith},
  title        = {Pith review of: Adding covariates to bounds: What is the question?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/26G4OVWZ}},
  note         = {Machine review of arXiv:2502.03156}
}

read the original abstract

Symbolic nonparametric bounds for partial identification of causal effects now have a long history in the causal literature. Sharp bounds, bounds that use all available information to make the range of values as narrow as possible, are often the goal. For this reason, many publications have focused on deriving sharp bounds, but the concept of sharp bounds is nuanced and can be misleading. In settings with ancillary covariates, the situation becomes more complex. We provide clear definitions for pointwise and uniform sharpness of covariate-conditional bounds, that we then use to prove some general and some specific to the IV setting results about the relationship between these two concepts. As we demonstrate, general conditions are much more difficult to determine and thus, we urge authors to be clear when including ancillary covariates in bounds via conditioning about the setting of interest and the assumptions made.

Figures

Figures reproduced from arXiv: 2502.03156 by the authors.

**Figure 2.** Diagram illustrating the difference between pointwise and uniform sharpness. For uniform sharp [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** DAG models used for illustration. 4.1 Example Consider the DAG depicted in Figure 3a with X and Y binary, with assumptions encoded by NPSEM, given by the DAG. Given an underlying distribution PV over V = (X, Y, U), we observe the distribution PO = o(PV ) over O = (X, Y ) and L(PO) := −P(X = 0, Y = 1) − P(X = 1, Y = 0) is a valid lower bound on the population average causal risk difference θ = EPV [Y (X=1) − Y (X=0)]… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Causal IV setting with an unconfounded mediator [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Simulation settings 6 Simulations To evaluate methods for bounding causal effects in the presence of unmeasured confounding but where at least one measured covariate is available, we simulated true probabilities from various models over S, Z, X, Y with all variables bi…

**Figure 6.** Figure 6: The distributions of width(ca)/width(co), width(cm)/width(co) and width(cm)/width(ca) for mod [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Partial identification via conditional linear programs: estimation and policy learning
stat.ME 2025-06 conditional novelty 7.0 of 10

Two debiased estimators, one based on linear programming solutions and one on entropic smoothing, provide asymptotic confidence intervals for covariate-dependent partial identification bounds and support policy learning.

Reference graph

Works this paper leans on

13 extracted references · 12 canonical work pages · cited by 1 Pith paper

[1]

Bounds on treatment effects from studies with imperfect compliance

Alexander Balke and Judea Pearl. Bounds on treatment effects from studies with imperfect compliance. Journal of the American Statistical Association, 92 0 (439): 0 1171--1176, 1997. doi:10.1080/01621459.1997.10474074. URL https://doi.org/10.1080/01621459.1997.10474074

work page arXiv 1997
[2]

Nonparametric bounds on causal effects from partial compliance data

Alexander Balke and Judea Pearl. Nonparametric bounds on causal effects from partial compliance data. 2011. URL https://api.semanticscholar.org/CorpusID:142574577

work page 2011
[3]

Non-parametric bounds on treatment effects with non-compliance by covariate adjustment

Zhihong Cai, Manabu Kuroki, and Tosiya Sato. Non-parametric bounds on treatment effects with non-compliance by covariate adjustment. Statistics in Medicine, 26 0 (16): 0 3188--3204, 2007. doi:https://doi.org/10.1002/sim.2766. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/sim.2766

work page doi:10.1002/sim.2766 2007
[4]

Vanessa Didelez, Sha Meng, and Nuala A. Sheehan. Assumptions of IV Methods for Observational Epidemiology . Statistical Science, 25 0 (1): 0 22 -- 40, 2010. doi:10.1214/09-STS316. URL https://doi.org/10.1214/09-STS316

work page doi:10.1214/09-sts316 2010
[5]

Causal bounds for outcome-dependent sampling in observational studies

Erin E Gabriel, Michael C Sachs, and Arvid Sj \"o lander. Causal bounds for outcome-dependent sampling in observational studies. Journal of the American Statistical Association, 117 0 (538): 0 939--950, 2022

work page 2022
[6]

Sachs, and Erin E

Gustav Jonzon, Michael C. Sachs, and Erin E. Gabriel. Accessible computation of tight symbolic bounds on causal effects using an intuitive graphical interface. R JOURNAL, 15 0 (4): 0 53--68, DEC 2023. ISSN 2073-4859

work page 2023
[7]

Covariate-assisted bounds on causal effects with instrumental variables

Alexander W Levis, Matteo Bonvini, Zhenghao Zeng, Luke Keele, and Edward H Kennedy. Covariate-assisted bounds on causal effects with instrumental variables. arXiv preprint arXiv:2301.12106, 2023

arXiv 2023
[8]

Nonparametric bounds on treatment effects

Charles F Manski. Nonparametric bounds on treatment effects. The American Economic Review, 80 0 (2): 0 319--323, 1990

work page 1990

Show all 13 references

[9]

Causality

Judea Pearl. Causality. Cambridge University Press, New York, 2000

2000
[10]

Ramsahai

Roland R. Ramsahai. Causal bounds and instruments. In Proceedings of the Twenty-Third Conference on Uncertainty in Artificial Intelligence, UAI'07, page 310–317, Arlington, Virginia, USA, 2007. AUAI Press. ISBN 0974903930

2007
[11]

J.M. Robins. The analysis of randomized and non-randomized AIDS treatment trials using a new approach to causal inference in longitudinal studies. In L. Sechrest, H. Freeman, and A. Mulley, editors, Health service research methodology: a focus on AIDS, pages 113--159. US Publi...

1989
[12]

Sachs, Gustav Jonzon, Arvid Sjolander, and Erin E

Michael C. Sachs, Gustav Jonzon, Arvid Sjolander, and Erin E. Gabriel. A general method for deriving tight symbolic bounds on causal effects. JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS, 32 0 (2): 0 567--576, APR 3 2023. ISSN 1061-8600. doi:10.1080/10618600.2022.2071905

2023
[13]

Swanson, Miguel A

Sonja A. Swanson, Miguel A. Hernán, Matthew Miller, James M. Robins, and Thomas S. Richardson. Partial identification of the average treatment effect using instrumental variables: Review of methods for binary instruments, treatments, and outcomes. Journal of the American Stati...

2018

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Bounds on treatment effects from studies with imperfect compliance

Alexander Balke and Judea Pearl. Bounds on treatment effects from studies with imperfect compliance. Journal of the American Statistical Association, 92 0 (439): 0 1171--1176, 1997. doi:10.1080/01621459.1997.10474074. URL https://doi.org/10.1080/01621459.1997.10474074

work page arXiv 1997

[2] [2]

Nonparametric bounds on causal effects from partial compliance data

Alexander Balke and Judea Pearl. Nonparametric bounds on causal effects from partial compliance data. 2011. URL https://api.semanticscholar.org/CorpusID:142574577

work page 2011

[3] [3]

Non-parametric bounds on treatment effects with non-compliance by covariate adjustment

Zhihong Cai, Manabu Kuroki, and Tosiya Sato. Non-parametric bounds on treatment effects with non-compliance by covariate adjustment. Statistics in Medicine, 26 0 (16): 0 3188--3204, 2007. doi:https://doi.org/10.1002/sim.2766. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/sim.2766

work page doi:10.1002/sim.2766 2007

[4] [4]

Vanessa Didelez, Sha Meng, and Nuala A. Sheehan. Assumptions of IV Methods for Observational Epidemiology . Statistical Science, 25 0 (1): 0 22 -- 40, 2010. doi:10.1214/09-STS316. URL https://doi.org/10.1214/09-STS316

work page doi:10.1214/09-sts316 2010

[5] [5]

Causal bounds for outcome-dependent sampling in observational studies

Erin E Gabriel, Michael C Sachs, and Arvid Sj \"o lander. Causal bounds for outcome-dependent sampling in observational studies. Journal of the American Statistical Association, 117 0 (538): 0 939--950, 2022

work page 2022

[6] [6]

Sachs, and Erin E

Gustav Jonzon, Michael C. Sachs, and Erin E. Gabriel. Accessible computation of tight symbolic bounds on causal effects using an intuitive graphical interface. R JOURNAL, 15 0 (4): 0 53--68, DEC 2023. ISSN 2073-4859

work page 2023

[7] [7]

Covariate-assisted bounds on causal effects with instrumental variables

Alexander W Levis, Matteo Bonvini, Zhenghao Zeng, Luke Keele, and Edward H Kennedy. Covariate-assisted bounds on causal effects with instrumental variables. arXiv preprint arXiv:2301.12106, 2023

arXiv 2023

[8] [8]

Nonparametric bounds on treatment effects

Charles F Manski. Nonparametric bounds on treatment effects. The American Economic Review, 80 0 (2): 0 319--323, 1990

work page 1990