Title: PUATE: Efficient ATE Estimation from Treated (Positive) and Unlabeled Units
Venue: NeurIPS 2025
Accepted: yes
Keywords: Causal inference

PU learning

average treatment effect estimation
Abstract: The estimation of average treatment effects (ATEs), defined as the difference in expected outcomes between treatment and control groups, is a central topic in causal inference. This study develops semiparametric efficient estimators for ATE in a setting where only a treatment group and an unlabeled group—consisting of units whose treatment status is unknown—are observed. This scenario constitutes a variant of learning from positive and unlabeled data (PU learning) and can be viewed as a special case of ATE estimation with missing data. For this setting, we derive the semiparametric efficiency bounds, which characterize the lowest achievable asymptotic variance for regular estimators. We then construct semiparametric efficient ATE estimators that attain these bounds. Our results contribute to the literature on causal inference with missing data and weakly supervised learning.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a novel framework for estimating average treatment effects (ATE) using only treated and unlabeled data. It derives semiparametric efficiency bounds for two common missing data settings (censoring and case-control) and proposes estimators that are provably efficient. Theoretical results are supported by simulations showing improved performance over standard methods like inverse probability weighting.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**:\n\n- The claims made in the paper are well supported by both theoretical derivations and simulation studies\n- The writing is very clear and the contributions are well connected to the broader literature on treatment effect estimation (I appreciated the extended discussion in Appendix A)\n\n\n\n**Weaknesses**: \n\n-  The introduction does not clearly justify when and why the treated-plus-unlabeled scenario arises naturally in causal inference, beyond a brief and somewhat unclear recommendation-system example. It is not obvious to me how common this setting is in real-world causal inference\n- The censoring setting rests on the “selected completely at random” assumption that observation probability is constant within the treated class, an analogue of MCAR that is rarely satisfied\n\nQUESTIONS:\n- Asymptotic normality in this paper requires the *true* propensity score (or an auxiliary dataset), whereas standard doubly-robust AIPW only needs an estimated propensity score converging at a fast enough rate. What structural aspect of the treated-plus-unlabeled setup makes propensity-score error first-order, and do you see other ways to recover asymptotic normality using only the original data (i.e., without bringing in a separate dataset)?\n- Could you provide a concrete, real-world example where only treated and unlabeled units are observed, and where ATE is the relevant causal estimand?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-21"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates treatment effect estimation in settings where treatment assignments are missing for some observations, including both censoring and case-control scenarios. The authors establish identification and semiparametric efficiency theory, and demonstrate the performance of the proposed doubly robust estimators through theoretical analysis and empirical results.\n\nSTRENGTHS AND WEAKNESSES:\nOverall, this paper addresses an important and interesting problem in causal inference. The proposed methods are supported by solid theoretical guarantees and empirical evidence, and the presentation is clear and well-organized. The manuscript would benefit from a deeper discussion of the practical applications and real-world relevance of the proposed methods. More detailed comments on specific aspects are provided below.\nQuality: The theoretical foundations and proposed methods are solid. I have listed a few questions regarding them in the 'Questions' section below.\nClarity: The paper is clearly written and well organized. To maintain clarity and avoid excessive length, the details for the case-control setting are appropriately moved to the appendix. I have also included some comments on improving the notation in the 'Questions' section below.\nSignificance: While the theoretical development is solid, the authors should place greater emphasis on the practical applicability of their setting and methods.\nOriginality: The results in this paper are novel to the best of my knowledge.\n\nQUESTIONS:\n1. The significance of the work could be enhanced by incorporating additional application scenarios and concrete examples that align with the settings considered in the paper. This would better demonstrate the practical utility of the proposed methods.\n2. In Assumption 3.1, the authors should consistently use “o” as the realization of the variable \"O” rather than switching to “d,” which is used as the realization of “D.” A similar inconsistency appears in Assumption 3.3. It is recommended that the authors carefully review the manuscript and ensure consistent use of notation throughout to improve clarity.\n3. Assumption 3.1 posits that the missingness indicator O is independent of X for treated units, which corresponds to a Missing Completely At Random (MCAR) assumption. This is a strong assumption in many practical scenarios. It would be helpful for the authors to discuss why this assumption is necessary in the current framework and to explore whether a relaxation to a Missing At Random (MAR) assumption could be considered.\n4. Section 3 presents an IPW-style identification formula for the average treatment effect (ATE). It would be valuable to also include an identification formula based on outcome regression, such as $$E[Y(1)]=E[E(Y \\mid X, A=1)]$$ as commonly used in standard causal inference settings. Providing a derivation of the identification results—perhaps in an appendix—would also help readers better understand how the assumptions in Section 3 enable identification.\n5. The main result (Theorem 4.7) is currently stated under the assumption that the propensity score is known. Section 4.5 shows that the same result holds when the propensity score is estimated from an independent sample. It may improve the flow and clarity to directly present the main results under the sample-splitting framework (e.g., divide the sample into L+1 folds with one fold training the propensity score), which is widely adopted in practice. As long as the number of folds is fixed, the final convergence rates remain valid.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-22"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work studies the problem of estimating the average treatment effect in a setting analogous to the positive and unlabeled setting that has been previously studied in the supervised learning literature. Specifically, there is a set of known treated units and a second set of units for which the treatment status is unknown. The authors consider two variants of this problem, in the case-control setting the two groups are independent groups, in the censored setting there is a random process that dictates membership to the positive and unlabeled groups. The authors develop a semi-parametric estimator and derive the influence functions and asymptotic normality for both of these settings. In each, the authors leverage PU estimators from the literature. The propensity score amongst the unlabeled units is assume known in the main text for the proof of asymptotic normality. This is relaxed later in the paper by assuming the availability of an auxiliary dataset.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n* The problem studied in this paper is interesting, and novel to the causal setting. The authors do a nice job of making the connection between PU learning and semi-parametric estimation. \n* The derivation of $\\sqrt{n}$ consistency and asymptotic normality is nice to see. \n* The authors do a nice job of centering the proposed approach with the derivation of the IF. The assumptions are clearly laid out, and well stated throughout.\n* The proposed approach shows good empirical performance against sensible alternatives. \n\nWeaknesses: \n* The largest weakness with this paper is the assumption that the g function is known throughout. The authors do present a solution to this by having an auxiliary data set. However, it's not entirely clear to me how this will affect the results. My intuition is that the convergence rate is now going to be dominated by the smaller of the two data sets, either the one used to estimate the g function or the one used to estimate the rest of the estimator. \n\n* It would be really useful if the authors could contextualize the results and presented methodology within the larger framework of missing data. They do a nice job of setting up the missingness mechanism, SCAR. However, I think the paper would have greater impact if there was some discussion of this method and its relation to other approaches for missing data, such as the Heckman model or, e.g.Robins, Rotnitzky, and Zhao (1994). I do recognize that the proposed setting is unique, however I just think it would help gain intuition for the reader.\n\nQUESTIONS:\n* There is a fairly large set of methods that have been proposed for PU learning. It would be helpful if the authors could provide some intuition here for how the choice of PU learning method affects the proposed method, e.g., employing Elkon, vs du Plessis et al. \n\n* Can the authors provide some intuition on the effect of having to estimate the g function on the overall performance of the estimator? As I mentioned above, it seems like there would be an interaction between that and the results both simulation and theoretical presented, but it's not entirely clear to me from reading.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies a new setup of average treatment effect (ATE) estimation inspired by learning from positive and unlabeled data:\nwe have observations for which we know the treatment was given while we have other observations for which we do not know whether the treatment was given or not (i.e., in the control group). More specifically, the paper studies both censoring and case-control settings.\n\nFor each setting, the authors propose estimators based on the Inverse Probability Weighting (IPW), the Direct Method (DM), and the efficient (Eff) influence function.\nThe authors derive an explicit formula for the efficient influence function (Lemma 4.1) and show the efficiency bound (Theorem 4.2).\nThen, they propose the Eff estimator as the zero of the sample average of the influence function solved for the parameter of ATE (l.203).\nUnder several standard assumptions, they show their simplest form of double robustness result (Theorem 4.4).\n\nMoreover, assuming that the propensity score $g_0$ is known, they propose a cross-fitting-based estimator, for which they prove the asymptotic efficiency (Theorem 4.7) under the assumptions including those on some doubly-robust rates of convergence for the nuisance parameter estimates (Assumptions 4.6).\nThey also illustrate that the IPW and the DM estimators fail to achieve this asymptotic efficiency under those assumptions (Remarks in l.273 and l.282).\n\nThe authors then extend the result to the case where we need to estimate $g_0$, for which another dataset is assumed to be available.\nFor this methods, they show the asymptotic efficiency.\n\nFinally, the authors present simple simulation results confirming their theory.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n- The setup is interesting and seems useful.\n- The theoretical results seem sound. I checked the proofs for the main results. (However, I have a few questions for clarification.)\n- The proposed method and the results seem highly non-trivial and relevant.\n\n### Weaknesses\n- There is no real data experiment.\n- Some parts of the proofs are too brief.\n- The notation of the math symbols is a bit confusing.\n\nQUESTIONS:\nI will consider raising the score if these questions are addressed.\n\n### Major questions:\n- l.1300, I could not figure out how to obtain the first equality. Could the author give more details?\n- ll.155-156, \"the propensity scores $g_0(d | x)$ and $e_0(d | x)$ can be estimated using PU learning methods\"\nAre they identifiable? I thought we need to know the class prior in PU learning, and it requires some conditions for identifiability.\n- l.1250, what is $m$?\n- l.1260, where did $m$ come from? Where did $m$ and $\\frac{1}{\\bar{n}} \\sum_{i=1}{m}$ go after the equality?\n- l.1261--1262, how do I conclude that?\n\n### Minor questions:\n- l.1251, Is $C_{(\\ell)}$ defined? Is it $D_{(\\ell)}$?\n- l.1265, how did the authors deal with the random quantity $\\hat{\\pi}_n^{(\\ell)}$ in the denominator? A similar question applies to the last equality of l.1023.\n- l.1292, do we not need a boundedness of the functional for the Riesz representation theorem? (Do we really need the Riesz representation theorem if you anyway show Eq. (9) concretely?)\n- l.1298, did the authors use the fact that the scores have zero mean? Do we not need some regularity condition for this?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}