Adaptive Proximal Causal Inference with Some Invalid Proxies

Eric Tchetgen Tchetgen; Prabrisha Rakshit; Xu Shi

REVIEW 6 major objections 5 minor 1 cited by

Adaptive Proximal Causal Inference with Some Invalid Proxies

T0 review · 6 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Adaptive LASSO selection makes proximal causal inference valid even when some proxy variables are invalid.

desk verdict A useful transplant of IV-LASSO into proximal causal inference, with a genuinely new median-over-OCP extension, but the majority-validity boundary is where the argument thins out. read the letter →

arxiv 2507.19623 v1 pith:WWNMRWLV submitted 2025-07-25 stat.ME

Prabrisha Rakshit , Xu Shi , Eric Tchetgen Tchetgen This is my paper

classification stat.ME MSC 62D2062J0762F12

keywords proximalcausalinferenceinvalidproxiesadaptiveLASSOmedianestimatorhiddenconfoundingnegativecontrolvariablessubsamplingeffectestimation

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

Proximal causal inference uses treatment- and outcome-inducing proxies to remove hidden confounding, but the standard framework requires every proxy to obey strict exclusion restrictions. This paper asks what happens when many candidate proxies are available and some of them are invalid: a treatment proxy that directly affects the outcome, or an outcome proxy affected by the treatment. It shows that, under a majority-validity condition, the causal effect is still identifiable and estimable at the parametric rate. The proposed estimator first uses a median of proxy-specific ratios to recover a nuisance parameter, then runs an adaptive LASSO to select valid treatment proxies and finishes with a two-stage least squares step whose limiting distribution matches an oracle that knows the invalid proxies. The same median idea extends to settings where the outcome proxies themselves may be invalid, with subsampling used for confidence intervals.

What carries the argument

The load-bearing object is a ratio-median identity. Projecting the outcome and proxy equations onto $M=(Z,D)$ gives coefficients $\Gamma^*$ and $\delta^*$ satisfying $\Gamma^*_{-} = \alpha + \delta^*_{-}\gamma$. For each proxy $j$, the ratio $\hat\pi_j = \hat\Gamma_j/\hat\delta_j$ converges to $\gamma + \alpha_j/\delta_j^*$. Because more than half of the $\alpha_j$ are zero under the majority condition, the median of these ratios converges to $\gamma$, identifying the nuisance parameter without knowing which proxies are valid. That estimate feeds inverse weights into the adaptive LASSO, whose selection consistency upgrades the final two-stage least squares step to oracle-equivalent inference. The extension to invalid outcome proxies replaces one central outcome proxy with a median over per-proxy estimates, protected by the same majority logic.

What would settle it

Rerun the simulation design of Section 6.1 with $p_z=10$ candidate treatment proxies but set $s_z=6$ invalid proxies, so that the majority condition fails. The paper's theory predicts that $\hat\beta_{\text{post}}$ is no longer consistent: the median ratio converges to $\gamma$ plus a nonzero $\alpha_j/\delta_j^*$ term, so the estimator should show a bias that persists as $n$ grows and confidence intervals should under-cover. A simulation that instead showed nominal coverage and vanishing bias would show that majority validity is not actually load-bearing.

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Extended reading notes

Core claim

The paper's central claim is that the exclusion restrictions required for proximal causal inference can be relaxed in a many-proxy linear structural model. Writing the outcome as $Y = D\beta + Z^\top \alpha + W\gamma + \varepsilon$ with $E[\varepsilon \mid D,Z]=0$, a candidate treatment proxy is invalid exactly when its direct effect $\alpha_j$ is nonzero. The paper proves that if fewer than half of the candidate treatment proxies are invalid, the ratio $\pi_j = \Gamma_j^*/\delta_j^*$ has probability limit $\gamma + \alpha_j/\delta_j^*$, so the median of the $\pi_j$ is a consistent estimator of $\gamma$. This yields a first-stage estimate of $\alpha$ and therefore data-dependent adaptive weights $1/|\hat\alpha_j|$. With these weights, the adaptive LASSO selects the invalid proxies consistently and the post-selection two-stage least squares estimator has the same asymptotic distribution as the oracle estimator: $\sqrt{n}(\hat\beta_{\text{post}} - \beta) \to N(0, \sigma^2_{\text{or}})$. When outcome-inducing proxies can also be invalid, the paper applies the procedure once per candidate outcome proxy and takes the median of the resulting estimates; under majority validity of the outcome proxies this median is $\sqrt{n}$-consistent and converges to an order statistic of the valid estimators, and subsampling provides confidence intervals.

Load-bearing premise

The method only works if fewer than half of the candidate treatment proxies (and, in the multi-outcome-proxy extension, fewer than half of the candidate outcome proxies) are invalid; if the invalid fraction reaches one half or more, the median-based selector targets the wrong quantity and the oracle-equivalence result collapses.

Editorial extensions

If this is right

If the majority condition holds, an analyst can include any number of candidate treatment proxies and let the adaptive LASSO decide which are invalid, without pre-specifying the invalid set.
Confidence intervals built from the oracle variance formula are asymptotically valid for the post-adaptive LASSO estimator, matching the efficiency of the oracle up to selection uncertainty.
In the invalid-outcome-proxy extension, the median over candidate outcome proxies remains $\sqrt{n}$-consistent even though the individual estimators are dependent and some are biased.
On the SUPPORT data, the method flags bilirubin and pH as likely invalid proxies and estimates a negative effect of right heart catheterization on 30-day survival, with all intervals excluding zero.
Simulations show that unadjusted two-stage least squares and ordinary least squares suffer severe bias, while the proposed estimator keeps coverage close to the nominal level.

Reading between the lines

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

The ratio-median logic could be turned into a sensitivity analysis: comparing the empirical distribution of $\hat\pi_j$ with its median would reveal a shift when the invalid fraction approaches one half, and trimming instead of taking the median would trade bias for variance.
Because the method needs only majority validity rather than known-valid anchors, it could combine with automated proxy-screening procedures to make proximal causal inference more routine in high-dimensional observational datasets, though the linearity of the outcome and proxy models is a genuine restriction.
Replacing the median with other robust location estimators should preserve consistency under majority validity while changing efficiency and the order-statistic limit, giving a family of estimators with different bias-variance trade-offs.
The subsampling approach used for the median estimator could also provide calibrated intervals for the post-adaptive LASSO estimator itself in settings where the limiting distribution is non-normal, which would extend its applicability beyond the single-valid-outcome-proxy case.

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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

Two things to know about this manuscript. First, it is a competent transplant of the IV-LASSO machinery (Kang et al. 2016; Windmeijer et al. 2019) into proximal causal inference, and the genuinely new piece is Section 5: when the outcome proxy itself may be invalid, they run the adaptive-LASSO routine with each candidate OCP and take the median of the resulting treatment-effect estimates. That's a sensible and useful extension. Second, the majority-validity condition does all the work, and the paper's own Table 4 shows the method stops working at the boundary where that condition fails (sz=5 for pz=10), yet the text doesn't analyze that regime or offer diagnostics.

What's good: the two-step estimator in Theorem 2 is elegant; the oracle decomposition in Proposition 2 is clean; the simulations are extensive and honestly report the boundary deterioration, even if they don't comment on it. The writing is clear. The extension to invalid OCPs is a real gap in the PCI literature, and the subsampling idea is a reasonable way to handle the non-regular median.

Soft spots: the proofs are largely borrowed or deferred. Theorem 3's proof is omitted, Propositions 1 and 2 follow from Zhou 2006, and Theorem 4's proof is sketched. That's acceptable for a preprint but means the new material isn't self-contained. The subsampling inference never discusses the sqrt(n/b) rescaling that standard subsampling theory requires; the intervals in Table 5 are conservative, which they attribute to hyperparameters but don't reconcile with theory. The real-data DPA assumption is introduced in a paragraph and changes the meaning of a valid OCP; it lets a proxy be both TCP and OCP, which sits awkwardly with the rest of the framework and is not evaluated.

The biggest substantive issue is the majority boundary. At sz=4, the estimator looks oracle-like; at sz=5, interval length jumps and bias appears. That's exactly where the identification argument breaks, so it's not a surprise, but the paper should say so explicitly and give users a way to detect or avoid that regime. As it stands, the central claim is conditionally supported: under the linear model, relevance, and strict majority, the estimator plausibly matches the oracle. But those assumptions are strong.

I would send this to a serious referee. The extension is useful, the framework is practical, and the weaknesses are fixable with clearer proofs and an honest discussion of the boundary.

Referee Report

6 major / 5 minor

Summary. The paper develops methods for proximal causal inference (PCI) when some candidate treatment-inducing confounding proxies (TCPs) and, in an extension, some candidate outcome-inducing confounding proxies (OCPs) may be invalid. Under a linear structural equations model, the authors show that the causal effect can be identified if a strict majority of TCPs are valid (Assumption 3 / Corollary 1) and propose a LASSO-based median estimator, then an adaptive-LASSO estimator with a post-selection oracle-like normal limit (Proposition 2). For invalid OCPs, they aggregate per-OCP estimates by the median and use subsampling for inference (Section 5). The paper includes simulations and an application to right heart catheterization in the SUPPORT study.

Significance. The paper addresses a relevant and timely problem—robustifying PCI against invalid proxies—and the reduction of the identification problem to the sparse coefficient vector alpha via the moment condition (6) is clean. The two-step algorithm in Theorem 2 is a useful practical contribution, and the simulations in Tables 3 and 5 show that in the strict-majority regime the proposed estimator tracks the oracle. However, the theoretical novelty is largely imported from Kang et al. (2016) and Windmeijer et al. (2019), and several load-bearing proofs in the manuscript are omitted or incomplete. In particular, the subsampling inference as described is not valid, and the real-data analysis relies on assumptions weaker than those proved. If the proof gaps and the inference procedure are fixed, the paper would be a solid contribution to the PCI literature.

major comments (6)

[Section 5, 'Inference via Subsampling'] The subsampling confidence interval is constructed directly from the empirical quantiles of the raw subsample estimates {\hat{\beta}^{(pw)}_{(i)}}, without centering at the full-sample estimate and without any rescaling. For \tau_n = \sqrt{n} and b = n^{4/5}, the standard subsampling approximation to the law of \sqrt{n}(\hat{\beta}^{(pw)} - \beta) is the law of \sqrt{b}(\hat{\beta}^{(pw)}_b - \hat{\beta}^{(pw)}_n), whose quantiles must then be rescaled by \sqrt{n/b}. As written, the interval does not have the claimed frequentist coverage; this is not a cosmetic detail because Theorem 4's limiting distribution is non-normal. The confidence interval reported in Section 7 is therefore not supported by the stated procedure.
[Appendix B.4 (proof of Proposition 2)] The proof contains the claim 'plim(1{\hat{A}_{ad}=A'})=1 if A'\neq A', which is false: for any fixed incorrect set A', the indicator converges in probability to 0, not 1. The subsequent conclusion that the wrong-model terms vanish requires an additional argument that \sqrt{n}(\hat{\beta}_{post}-\beta) is stochastically bounded on the wrong-model event, or that the selection error probability is o(n^{-1/2}). Without this, the claimed oracle limiting distribution for \hat{\beta}_{post} is not established. The proof also invokes the oracle properties of adaptive LASSO from Zhou (2006) without verifying the required conditions on the design P_{\tilde{D}\perp}\tilde{Z} and on the initial weights.
[Section 4.4, Theorem 3] The proof of Theorem 3 is omitted ('follows similarly'), yet this theorem is one of the main theoretical guarantees for the LASSO-based estimator. In addition, the stated RIP condition is on the raw matrix Z, whereas the actual design in the two-step algorithm (9) is P_{\tilde{D}\perp}\tilde{Z}; the relationship between the RIP constants of Z, P_{\tilde{W}}Z, P_{\tilde{D}}Z, and the design matrix is not derived. The bound (10) therefore does not follow from the stated assumptions in any transparent way. This gap needs to be filled or the theorem restated for the actual design.
[Appendix B.5 (proof of Theorem 4)] The proof asserts, without demonstration, that for an invalid OCP the limiting bias constants satisfy c_{1,k}\neq 0 and c_{2,k}\neq 0. Even if the adaptive LASSO fails to select the true invalid-TCP set A, the projection coefficient captured by c_{2,k} could be zero in the limit; similarly, an invalid OCP might still have c_{1,k}=0 in special but possible configurations. The identification of the median's limiting order statistic also requires the invalid estimators to have non-vanishing asymptotic bias relative to \sqrt{n}; this is assumed but not proved. The theorem as stated is therefore not fully supported.
[Section 7, final paragraphs] The real-data analysis introduces the Disconnected Proxy Assumption (DPA), under which a valid proxy has no direct effect on either D or Y. This assumption is not part of the formal model in Section 4, where valid TCPs may affect D and valid OCPs may affect Y, and no identification result is proved under DPA. Moreover, the paper states that 'at least five of the proxies ... are valid OCPs' before taking the median over the 10 candidate estimators. For pw=10, Theorem 4 and the majority-validity condition require strictly more than half of the OCPs to be valid, i.e., at least 6 valid OCPs. With exactly five valid, the median is not guaranteed to be \sqrt{n}-consistent. The reported estimate and CI in Section 7 therefore rely on a weaker assumption than the one proved.
[Tables 4 and 6] Table 4 includes rows with sz=5 and higher for pz=10, where the strict majority condition sz < pz/2 fails, and the text describes the results as demonstrating robustness to increasing sz. Table 6 similarly includes sw=5 and sw=6 with pw=10, where the OCP majority condition fails at sw=5. Since Lemma 1 and Proposition 2 require sz < pz/2 and Theorem 4 requires more than half of the OCPs to be valid, these entries are outside the theoretical scope. They should either be removed or explicitly labeled as sensitivity analyses outside the proved regime, with a warning that the method has no theoretical guarantee there.

minor comments (5)

[Lemma 1] The limiting distribution of \hat{\gamma}_m is described in terms of an unspecified index l determined by pz, sz, and the signs of \alpha_j/\delta_j; please provide the explicit formula for l, including the handling of ties for even pz.
[Equation (12) and Section 4.5] The tuning parameter \lambda_n is required to satisfy \lambda_n=o(\sqrt{n}) and \lambda_n\to\infty, but no practical choice or data-dependent rule is given. Also, the adaptive weights use 1/|\hat{\alpha}_{m,i}|, which is undefined if any initial estimate is exactly zero; state the convention used in the implementation.
[Tables 3 and 5] Table 3 is missing the n=10000 row mentioned in the text, and the first row of Table 3 is formatted as '15000.93' with missing separators. Please correct the table formatting and include all reported sample sizes.
[Appendix B.4] There is a typo in the last displayed line: 'bAkad' should be 'bAad'.
[Section 2] The claim of novelty relative to Yu et al. (2025) would be clearer if the paper explicitly compared the moment condition (6) with the IV moment condition in Kang et al. (2016) and stated which 'new challenges' for proxies are formalized beyond relabeling instruments as proxies.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the identification and oracle-selection results are imported from external IV/LASSO theory, and the target beta is defined independently of the fitted penalty or median ratios.

full rationale

This paper's derivation chain is an external dependency rather than a self-referential loop. The identification theorem (Theorem 1, Section 4.2) is stated as 'This result follows directly from Theorem 1 in Kang et al. 2016'; the adaptive-LASSO selection and oracle normality (Proposition 1) are explicitly imported from Zhou 2006; and Theorem 3's bound is said to follow the proof of Theorem 2 in Kang et al. 2016. None of these is a self-citation, and the target parameter beta is the structural coefficient in model (4), defined independently of any fitted penalty or median ratio, so no quantity is defined in terms of its own estimator. Lemma 1's median consistency is an assumption-driven result relying on the strict majority condition alpha_j = 0, not a fitted-input prediction. The only overlapping-author citation in the application is the Disconnected Proxy Assumption tied to DANCE (Kummerfeld et al. 2024, co-authored by Shi); however, DPA is explicitly introduced as an assumption, not derived from the citation, and the per-OCP estimates in Table 7 stand without it. A possible weakness in the proof of Proposition 2 is the asserted limit plim(sqrt(n)(bbeta_post - beta) - sqrt(n)(bbeta_or - beta) 1{bAkad=A}) = 0, which would require a stronger selection-consistency rate than the stated P(Ahat=A) -> 1; but that is a proof gap, not a circular definition or a fitted parameter renamed as a prediction. The boundary behavior at sz = pz/2 shown in Table 4 is an assumption violation, not circularity. Score 1 reflects one minor self-citation that is not load-bearing to the central theoretical claims.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The method is an adaptation of IV-LASSO to proximal settings; it inherits the linear SEM, relevance, and majority-validity assumptions from Kang et al. and Windmeijer et al. The additional assumptions, such as the RIP conditions, adaptive-LASSO tuning, and DPA, are either unverified or introduced only for the application.

free parameters (3)

LASSO tuning parameter lambda = not specified; theory gives lambda approximately sigma sqrt(log pz / n) and lambda_n = o(sqrt(n)), lambda_n -> infinity
Controls the trade-off between proxy selection and bias in equations (8) and (12); no operational selection rule is given in the paper.
Subsample size and number of subsamples (b, N) = b = n^(4/5), N = 1000
Chosen by guideline in Section 5; used for subsampling confidence intervals and not justified by sensitivity analysis.
Upper bound on invalid TCPs I = I at most pz/2 in practice
User-specified bound needed for Theorem 1 and Corollary 1; the paper adopts the majority rule in the rest of the paper.

assumptions (8)

domain assumption Linear structural equations (4): E(Y | D,Z,U) = beta D + alpha^T Z + beta_u U and E(W | D,Z,U) = eta_u U
All identification and estimation results are derived inside this linear model; nonlinear or nonparametric generalization is left to future work.
domain assumption Assumption 1: E(M^T M) is full rank
Needed for Gamma*, delta*, and the projections used in 2SLS to be well-defined.
domain assumption Assumption 2: delta*_j is nonzero for all j in 1 to pz+1
Relevance of every proxy and treatment for W; also avoids division by zero in the median estimator pi_j = Gamma_j / delta_j.
domain assumption Majority validity for TCPs: sz < I at most pz/2
Lemma 1 and Corollary 1 use this to force the median of pi to equal gamma and to guarantee uniqueness of the solution to (7).
domain assumption Majority validity for OCPs: more than half of candidate W_j are valid
Used in Theorem 4 so that the median of the candidate estimators is dominated by consistent estimates.
ad hoc to paper Restricted isometry property conditions in Theorem 3
The error bounds in (10) hold only under the stated inequalities involving Z, P_cW Z, and P_eD Z; the paper does not verify these conditions in simulations.
standard math Adaptive LASSO conditions from Zhou 2006: lambda_n = o(sqrt(n)), lambda_n -> infinity, and a sqrt(n)-consistent initial estimator
Invoked in Proposition 1 to obtain selection consistency and oracle normality; no proof is given beyond the citation.
ad hoc to paper Disconnected Proxy Assumption (DPA) in the real-data application
Introduced in Section 7 to allow a variable to be both a valid TCP and OCP, so that the median over 10 configurations is justified; not part of Sections 4-5 theory.

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Cite this review

Pith. "Pith review of Adaptive Proximal Causal Inference with Some Invalid Proxies." pith.science (2026). https://pith.science/paper/WWNMRWLV

@misc{pith2026250719623,
  author       = {Pith},
  title        = {Pith review of: Adaptive Proximal Causal Inference with Some Invalid Proxies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WWNMRWLV}},
  note         = {Machine review of arXiv:2507.19623}
}

read the original abstract

Proximal causal inference (PCI) is a recently proposed framework to identify and estimate the causal effect of an exposure on an outcome in the presence of hidden confounders, using observed proxies. Specifically, PCI relies on two types of proxies: a treatment-inducing confounding proxy, related to the outcome only through its association with unmeasured confounders (given treatment and covariates), and an outcome-inducing confounding proxy, related to the treatment only through such association (given covariates). These proxies must satisfy stringent exclusion restrictions - namely, the treatment proxy must not affect the outcome, and the outcome proxy must not be affected by the treatment. To improve identification and potentially efficiency, multiple proxies are often used, raising concerns about bias from exclusion violations. To address this, we introduce necessary and sufficient conditions for identifying causal effects in the presence of many proxies, some potentially invalid. Under a canonical proximal linear structural equations model, we propose a LASSO-based median estimator that jointly selects valid proxies and estimates the causal effect, with theoretical guarantees. Recognizing LASSO's limitations in consistently selecting valid treatment proxies, we develop an adaptive LASSO-based estimator with differential penalization. We show that it is root-n consistent and yields valid confidence intervals when a valid outcome proxy is available. We also extend the approach to settings with many potentially invalid outcome proxies. Theoretical results are supported by simulations and an application assessing the effect of right heart catheterization on 30-day survival in ICU patient.

Figures

Figures reproduced from arXiv: 2507.19623 by the authors.

**Figure 2.** Z2 is an invalid TCP, Z1 affects Z2. To address these challenges arising from violations of the irrepresentable condition, the adaptive LASSO offers a more reliable approach with better variable selection properties. As described below, we employ a √ n-consistent initial estimator for α in (13), followed by the adaptive LASSO algorithm to estimate the parameters: αb ad = arg min α,γ [PITH_FULL_IMAGE:figures/full_fi… view at source ↗

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Works this paper leans on

2 extracted references · 2 linked inside Pith · cited by 1 Pith paper

[1]

Semiparametric Proximal Causal Inference

Cui, Yifan, Hongming Pu, Xu Shi, Wang Miao, and Eric Tchetgen Tchetgen (2023). “Semiparametric Proximal Causal Inference”. In: Journal of the American Statistical Association 119, pp. 1348–1359. Deaner, Ben (2021). “Proxy Controls and Panel Data”. In: arXiv preprint arXiv:1810.00283 . Dukes, Oliver, Ilya Shpitser, and Eric Tchetgen Tchetgen (2023). “Proxi...

arXiv 2023
[112]

Minimax Kernel Machine Learning for a Class of Doubly Robust Functionals with Application to Proximal Causal Inference

Ghassami, Amiremad, Andrew Ying, Ilya Shpitser, and Eric Tchetgen Tchetgen (2022). “Minimax Kernel Machine Learning for a Class of Doubly Robust Functionals with Application to Proximal Causal Inference”. In: Proceedings of the International Conference on Artificial Intelligence and Statistics . Imbens, Guido, Nathan Kallus, Xiaojie Mao Mao, and Yuhao Wan...

arXiv 2022

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Semiparametric Proximal Causal Inference

Cui, Yifan, Hongming Pu, Xu Shi, Wang Miao, and Eric Tchetgen Tchetgen (2023). “Semiparametric Proximal Causal Inference”. In: Journal of the American Statistical Association 119, pp. 1348–1359. Deaner, Ben (2021). “Proxy Controls and Panel Data”. In: arXiv preprint arXiv:1810.00283 . Dukes, Oliver, Ilya Shpitser, and Eric Tchetgen Tchetgen (2023). “Proxi...

arXiv 2023

[2] [112]

Minimax Kernel Machine Learning for a Class of Doubly Robust Functionals with Application to Proximal Causal Inference

Ghassami, Amiremad, Andrew Ying, Ilya Shpitser, and Eric Tchetgen Tchetgen (2022). “Minimax Kernel Machine Learning for a Class of Doubly Robust Functionals with Application to Proximal Causal Inference”. In: Proceedings of the International Conference on Artificial Intelligence and Statistics . Imbens, Guido, Nathan Kallus, Xiaojie Mao Mao, and Yuhao Wan...

arXiv 2022