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Debiased Inference for AI-Generated Data without Gold-Standard Labels: Identification via Multiple Imperfect Measurements
Naoki Egami, Sooahn Shin
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 91%
Last extracted: 8/20/2026, 4:16:26 AM
Summary
This paper introduces Debiased Inference with Multiple Imperfect Measurements (DMM), a framework for valid downstream statistical inference using AI-generated data without requiring gold-standard labels. DMM leverages multiple error-prone AI measurements (e.g., from different LLMs) that are conditionally independent given the true latent label and observed features. The authors prove the estimator is consistent and asymptotically normal, offering a solution to bias caused by ignoring measurement errors in automated annotation.
Entities (8)
Relation Signals (7)
Naoki Egami → authored → DMM
confidence 99% · Debiased Inference for AI-Generated Data without Gold-Standard Labels: Identification via Multiple Imperfect Measurements Naoki Egami
Sooahn Shin → authored → DMM
confidence 99% · Debiased Inference for AI-Generated Data without Gold-Standard Labels: Identification via Multiple Imperfect Measurements Naoki Egami ... Sooahn Shin
DMM → assumes → Conditional Independence
confidence 95% · DMM assumes that these measurements are independent conditional on the latent true label and observed unit-level features
DMM → uses → CP decomposition
confidence 90% · Building on the established results on CP decomposition, DMM assumes that these measurements are independent conditional on the latent true label
DMM → comparedto → PPI
confidence 85% · The proposed DMM makes no assumption about the gold-standard labels but makes stronger assumptions about measurement errors in AI, whereas methods using validation data, such as DSL (23) and PPI (3), make no assumption about measurement errors in AI but assume access to gold-standard labels.
DMM → comparedto → DSL
confidence 85% · The proposed DMM makes no assumption about the gold-standard labels but makes stronger assumptions about measurement errors in AI, whereas methods using validation data, such as DSL (23) and PPI (3), make no assumption about measurement errors in AI but assume access to gold-standard labels.
Dawid-Skene model → similarto → DMM
confidence 80% · The Dawid–Skene model uses the same core conditional-independence structure: it treats the true item label as latent
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Abstract
Abstract:An increasing number of scholars use AI to measure variables they subsequently include in downstream analyses. Although AI-measured variables are often analyzed as if observed without error, ignoring prediction errors in automated measurement leads to substantial bias and invalid confidence intervals in downstream analyses, even if AI measurement accuracy is high, e.g., above 90%. Existing solutions, such as design-based supervised learning and prediction-powered inference, combine error-prone AI-based measurements with gold-standard labels, which may be costly and difficult to obtain in some application areas. In this paper, we propose debiased inference with multiple imperfect measurements (DMM), a framework that combines multiple error-prone AI measurements to enable valid downstream inference without gold-standard labels. Building on the established results on CP decomposition, DMM assumes that these measurements are independent conditional on the latent true label and observed unit-level features, such as text features represented by embeddings. This framework allows for unknown misclassification rates to vary across annotation methods (e.g., large language models) and across units of annotation (e.g., texts). Under this assumption, we use semiparametric inference theory to prove that the DMM estimator is consistent and asymptotically normal, enabling valid inference for a wide range of downstream statistical analyses common in the social sciences. Our simulation results show that DMM yields valid inference and that adding accurate, though imperfect, measurements can improve efficiency. Focusing on common applications of large language model annotations, we also develop diagnostics to assess the conditional independence assumption.
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- Source: https://arxiv.org/abs/2608.18294v1
- Canonical: https://arxiv.org/abs/2608.18294v1
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Debiased Inference for AI-Generated Data without Gold-Standard Labels: Identification via Multiple Imperfect Measurements Naoki Egami Thanks: Associate Professor, Department of Political Science and Statistics and Data Science Center, Massachusetts Institute of Technology, Cambridge, MA 02139. Email: egami@mit.edu URL: https://naokiegami.com Sooahn Shin Thanks: Postdoctoral Associate, Department of Political Science, Massachusetts Institute of Technology, Cambridge, MA 02139. Email: sshin3@mit.edu URL: https://sooahnshin.com August 18, 2026 Abstract An increasing number of scholars use AI to measure variables they subsequently include in downstream analyses. Although AI-measured variables are often analyzed as if observed without error, ignoring prediction errors in automated measurement leads to substantial bias and invalid confidence intervals in downstream analyses, even if AI measurement accuracy is high, e.g., above 90%. Existing solutions, such as design-based supervised learning and prediction-powered inference, combine error-prone AI-based measurements with gold-standard labels, which may be costly and difficult to obtain in some application areas. In this paper, we propose debiased inference with multiple imperfect measurements (DMM), a framework that combines multiple error-prone AI measurements to enable valid downstream inference without gold-standard labels. Building on the established results on CP decomposition, DMM assumes that these measurements are independent conditional on the latent true label and observed unit-level features, such as text features represented by embeddings. This framework allows for unknown misclassification rates to vary across annotation methods (e.g., large language models) and across units of annotation (e.g., texts). Under this assumption, we use semiparametric inference theory to prove that the DMM estimator is consistent and asymptotically normal, enabling valid inference for a wide range of downstream statistical analyses common in the social sciences. Our simulation results show that DMM yields valid inference and that adding accurate, though imperfect, measurements can improve efficiency. Focusing on common applications of large language model annotations, we also develop diagnostics to assess the conditional independence assumption. 1 Introduction One of the most common applications of AI in the social sciences has been in measurement. Researchers have used AI to measure a wide range of variables, such as sentiment, tone, topics (26; 57, e.g.,), protests, political violence (30, e.g.,), and job types (31, e.g.,), among others. Such measurement tasks are only the first step. Scholars often use learned measurements as key variables of interest in downstream analyses. For example, 43 annotate whether each online post accuses local Chinese officials of corruption so that they can later study whether and how much such online complaints are censored. 25 annotate the tone of political ads and then analyze how politicians strategically change the tone of political advertising online and offline. When using such AI-measured variables in downstream analyses, it might be tempting to ignore measurement errors—the unknown and heterogeneous mismatch between the gold-standard label and the AI measurements—and analyze such variables as if they were observed without errors. However, recent papers theoretically and empirically demonstrate that ignoring errors in the first-stage measurement step can lead to substantial bias and invalid confidence intervals, even if the accuracy of the AI measurement is high, e.g., 90%. This is because such measurement errors are non-random and nonclassical and correlated with observed and unobserved variables that matter in downstream inference (46; 53; 23). In practice, this means that researchers can get substantively and statistically different results if they choose slightly different measurement methods, such as large language model (LLM) annotations with different prompts, temperature, and so on (6; 54, e.g.,). These biases can also limit the replicability of research based on AI-based measurements (49), as many AI models that are currently the state of the art will be deprecated in the near future. A popular existing solution is to combine a small number of gold-standard labels with error-prone AI-based measures via a doubly robust procedure (45; 18). Widely used variants include design-based supervised learning (23, DSL;), prediction-powered inference (3, PPI;), and MAR-S (10). They assume that researchers control the process through which each unit is sampled for gold-standard labeling (often simple random sampling but can also accommodate unequal sampling based on observed features). The key strength is that these methods make no assumptions about errors in AI-based measures. However, the biggest requirement is that they need high-quality gold-standard labels as the validation data, which might be costly and difficult to obtain in some application areas. In this paper, we propose a framework of debiased inference with multiple imperfect measurements (DMM) that combines three or more imperfect measurements in place of gold-standard labels to conduct valid downstream inference with AI-generated data. For identification, we build on the long-standing literature on nonparametric latent variable models and CP decomposition (37; 34; 1). The classical conditional independence assumption in this literature (20) assumes that multiple imperfect measurements, also known as proxies, are independent conditional on the true latent variable. This assumption is weaker than assuming that proxies have no measurement error and allows misclassification rates to vary across proxies. However, it does not allow for any shared source of errors (e.g., some documents might be more difficult to annotate for every LLM; 16). We instead assume that multiple proxies are independent conditional not only on the unobserved true labels but also on any observed features of each annotation task, such as the complexity and difficulty of texts captured by rich text embeddings. This assumption allows for unknown misclassification rates to vary not only across measurement methods (e.g., different LLMs) but also across units (e.g., texts). Our approach allows proxies to share sources of errors as long as such common causes are captured by conditioning variables, which can include rich text representation, for example. Importantly, if researchers wish to avoid assuming access to gold-standard labels, they have to make some assumptions about proxies, and our paper makes this assumption explicit and transparent. Recognizing its importance, Section 5 offers a series of strategies to make the conditional independence assumption more plausible and to statistically evaluate the observational implications of this assumption. For estimation and inference, the proposed DMM estimator builds on semiparametric causal inference, in particular 56 and 28, to perform debiased downstream inference. Simply estimating the finite mixture model (or variants of the Dawid-Skene model (20) or Kruskal decomposition (37; 1)) and including the learned latent variables directly in downstream inference is not sufficient. We need to use a tailored debiased moment estimator to achieve Neyman orthogonality (18) and allow for slow convergence rates of the estimation of conditional classification rate models, which are nuisance functions for downstream inference. The proposed DMM estimator is consistent and asymptotically normal, and its corresponding confidence interval is valid, under the conditional independence assumption and mild assumptions about convergence rates of the nuisance functions. By extending the existing literature, we allow for (a) a wide range of downstream analyses common in the social and biomedical sciences that can be written as the moment estimator (e.g., most maximum likelihood estimators); (b) settings where an error-prone variable is either a dependent or independent categorical variable in downstream analyses, and (c) more than three imperfect measurements (and their implications for efficiency). Thinking broadly, our proposed method and existing approaches based on gold-standard labels are complementary. The proposed DMM makes no assumption about the gold-standard labels but makes stronger assumptions about measurement errors in AI, whereas methods using validation data, such as DSL (23) and PPI (3), make no assumption about measurement errors in AI but assume access to gold-standard labels. Which method is more appropriate depends on applications, and we discuss when and how to combine these two methods. Our contributions can be summarized in three points. 1. Exploiting Multiple Imperfect Measurements in Place of Gold-Standard Labels: We develop a method that combines multiple proxies to conduct valid downstream inference for AI-generated data, without assuming access to gold-standard labels. This is in contrast to existing bias-correction methods, such as DSL and PPI, that rely on gold-standard data. 2. General Applicability: We extend existing theoretical results by allowing for (a) a wide range of downstream analyses common in practice, including linear regression, logistic regression, and most maximum likelihood estimators, (b) settings where an error-prone variable is either a dependent or independent categorical variable in downstream analyses, and (c) more than three proxies (e.g., more than three annotation methods). This general applicability is fundamental to support diverse downstream analyses conducted in empirical sciences. 3. Application to LLM annotations: While the method itself is applicable for downstream inference with any error-prone variables, we specifically focus on the most common applications of LLM annotations for AI-generated data. In particular, we discuss (a) how to make the conditional independence assumption more plausible by carefully choosing conditioning variables (e.g., estimated task difficulty and text embeddings), proxies (e.g., different families of LLMs), and prompts (e.g., randomly selecting different prompts for each LLM); (b) how to use more than three proxies to perform the overidentification test of the conditional independence assumption, and (c) how to combine DMM and DSL in a special case where some gold-standard labels are also available. After describing related work, we formally characterize the problem setting and existing methods (Section 2). In Section 3, we describe our proposed method and prove its theoretical properties. Here, for the sake of clear presentation, we focus on settings where an error-prone measurement is an independent variable in downstream analysis. We start with nonparametric identification and then derive asymptotic statistical properties of the DMM estimator in estimation and inference. In Section 4, we extend the framework to settings in which the annotated construct is a binary dependent variable and then discuss the most general case where a multicategory label can be either a dependent or independent variable. In Section 5, we come back to the core assumption of conditional independence and discuss how to make the assumption more plausible in practice. We also develop a series of statistical diagnostic tools to assess the conditional independence assumption. In Section 6, we provide extensive simulation and empirical validation studies to demonstrate the statistical properties of DMM. Section 7 concludes with a discussion. Related literature AI-Generated Data. With recent advances in AI and LLMs, a growing literature develops methods for valid downstream inference using AI- or machine-learning-generated measurements together with a gold-standard validation sample. Popular examples include design-based supervised learning (23; 22), prediction-powered inference (3), methods reviewed in 39, MAR-S (10), model-assisted impact analysis (41), and related control-variate approaches (35). They combine predictions available at scale with a smaller number of gold-standard labels to bias-correct downstream inference. All of these methods build on the longstanding literature on semiparametric inference and causal inference (45; 17; 14; 18, e.g.,). These methods assume access to gold-standard labels and knowledge of their sampling design, whereas DMM does not require such gold-standard labels. There are several recent papers that relax the assumption of gold-standard labels. 5 assume a new asymptotic regime where measurement errors and sampling errors are comparable and decrease as sample size grows. They use partial validation data where one only observes the gold-standard labels and error-prone AI measurements instead of the full validation where researchers need to observe not only the gold-standard labels and AI measurements but also all the downstream variables for each unit. 16 derive bounds on latent label prevalence and regression coefficients from panels of LLM reports under externally calibrated score and event restrictions (e.g., reporter-specific accuracy restrictions), while allowing arbitrary dependence across reports conditional on the latent truth. DMM complements these approaches by identifying the latent variable through the established array decomposition of multiple imperfect measurements. Compared to the classical conditional independence assumption (20, e.g.,) that only conditions on the true unobserved label and does not allow for any shared source of errors across multiple measurements, our conditional independence assumption explicitly allows for conditioning on a rich set of covariates that may contain observed or derived characteristics of the input (e.g., writing style, text length, language, or image quality) measured by rich text embeddings. Recognizing the importance of the assumption, we also develop a series of statistical diagnostics in Section 5. We also find that DMM performs well in our empirical validation study in Section 6 where the conditional independence assumption holds only approximately. Repeated Measurement Identification. Our work also builds on the literature on identifying latent variables from conditionally independent repeated measurements. A large classical measurement error literature assumes an additive error that is independent of the latent variable and mean zero, or at least mean zero conditional on it, and obtains identification using instrumental variables or repeated measurement identities (46). Such restrictions are ill suited to categorical labels: misclassification of a finite-support variable is inherently nonclassical, and for nominal categories an additive error representation is itself unnatural (46, Section 6.1). Accordingly, our identification strategy instead builds on the nonclassical measurement error literature. In foundational work, 37 studies three-way array decomposition and provides rank conditions under which the decomposition is unique up to a common permutation and scaling of its latent components. This result underlies the identification of discrete latent structures from conditionally independent measurements. 34 studies nonlinear models with a misclassified discrete independent variable, 33 develop related identification results for continuous variables, and 1 establish related results for discrete models with hidden variables. 13 and 46; 47 provide a broader and unifying review of this literature, including such identification strategies based on multiple imperfect measurements under classical and nonclassical measurement error. In contrast to this literature, we allow for more than three imperfect measurements, a more general class of downstream inference, and semiparametric estimation and inference methods that are Neyman orthogonal to the first-stage nuisance function estimation. The Dawid–Skene model uses the same core conditional-independence structure: it treats the true item label as latent, allows each annotator to have a distinct confusion matrix, and assumes that annotations are independent conditional on the true label (20). In the context of LLM annotations, 8 characterize conditions under which correlated annotations can be aggregated and propose a dependence-aware extension of the Dawid–Skene model. Proxy-based Causal Inference. Another related literature studies causal inference with unmeasured variables using proxies (38; 40; 24), in particular, 56; 28. Most importantly, 56 develop a method to make causal inference with a latent treatment. Specifically, they also build on the nonparametric latent variable model literature (37; 1) for nonparametric identification and derive a new semiparametric efficiency theory and semiparametric estimation strategies for causal effects with a hidden treatment. 28 consider causal inference with the latent outcome variable and derive an influence function-based semiparametric estimator. Similarly, 42 applies the array-decomposition ideas related to 56 to text and image annotations with a latent outcome. Theoretically, we extend this literature by allowing for (a) more than three proxies (with analytical results of how increasing the number of proxies affects efficiency); (b) a general class of downstream analyses (in contrast to causal effects these previous studies have focused on), and (c) settings where a latent variable is either a dependent or independent categorical variable (each of previous studies focuses on just one of them). 2 The Problem Setting and Existing Approaches Scholars often use machine or human annotations to measure key variables of interest that they want to analyze in downstream analysis. However, as recent papers theoretically and empirically show, ignoring measurement errors in the annotation step can bias downstream inference, even when the accuracy of the annotation step is high, e.g., more than 90%. For clarity of presentation, we first focus on settings where an error-prone variable is a binary independent variable in downstream analysis. In Section 4, we generalize our method to settings where an error-prone variable is a general categorical variable that is either an independent or dependent variable. 2.1 Setup For each unit i=1,…,ni=1,…,n, let Xi∗∈0,1X_i ∈\0,1\ denote an unobserved true label, YiY_i denote an observed downstream dependent variable, and WiW_i denote observed covariates in downstream analysis. Instead of observing the true label Xi∗X_i , researchers observe J≥3J≥ 3 error-prone proxy labels X~i=(Xi(1),…,Xi(J)),Xi(j)∈0,1. X_i= (X_i^(1),…,X_i^(J) ), X_i^(j)∈\0,1\. The labels may, for example, be produced by different human annotators, machine learning models, or large language models (LLMs). They need not have the same accuracy, and their errors may be systematic or heterogeneous across units. Especially when researchers use LLMs to measure X∗X , it is often easy to obtain multiple imperfect proxies, as each proxy can come from different LLMs. Let ψF(y,x,w,β)∈ℝdβψ^F(y,x,w;β) ^d_β be a user-specified full-data moment function for downstream analysis, where β∈ℬ⊂ℝdβ ^d_β denotes the finite-dimensional parameter of interest. If X∗X were observed for every unit, the oracle downstream parameter β∗β would be the unique solution to [ψF(Y,X∗,W,β)]=0.E\! [ψ^F(Y,X ,W;β) ]=0. (2.1) The moment estimator above can accommodate a large class of estimators as special cases, including most maximum likelihood estimators. For linear regression, ψlinF(y,x,w,β)=(1,x,w⊤)⊤[y−(1,x,w⊤)β] _lin^F(y,x,w;β)=(1,x,w ) [y-(1,x,w )β ] and the coefficients of interest β are the coefficients of the linear regression model that regresses Y on X∗X and W. When you have logistic regression, a moment function is ψlogisticF(y,x,w,β)=(1,x,w⊤)⊤[y−expit(1,x,w⊤)β]. _logistic^F(y,x,w;β)=(1,x,w ) [y-expit\(1,x,w )β\ ]. A moment function for generalized linear models can be written as ψglmF(y,x,w,β)=(1,x,w⊤)⊤[y−ℓ(1,x,w⊤)β], _glm^F(y,x,w;β)=(1,x,w ) [y- \(1,x,w )β\ ], (2.2) where ℓ(⋅) (·) is an inverse link function specific to each model. Our theory and methods do not assume the downstream regression model chosen by users is correctly specified. Under misspecification, coefficients β defined in equation (2.1) can be interpreted as the population projection parameter (9; 52, e.g.,). 2.2 Existing Approaches 2.2.1 Naive Approaches Ignoring Measurement Errors A common strategy first collapses the multiple labels to a single proxy, e.g., using majority voting Xˇi=∑jXi(j)>J/2 X_i=1\ _jX_i^(j)>J/2\. More generally, Xˇi∈0,1 X_i∈\0,1\ may be any function of the multiple labels, such as selecting one annotator or thresholding their average. The resulting Xˇi X_i is then treated as if it were Xi∗X_i . The naive estimator β^naive β_naive solves 1n∑i=1nψF(Yi,Xˇi,Wi,β)=0. 1n _i=1^nψ^F(Y_i, X_i,W_i;β)=0. This is exactly the downstream moment function one would obtain when users directly include Xˇ X in downstream analysis. Its population moment bias is [ψF(Y,Xˇ,W,β)−ψF(Y,X∗,W,β)]=[(Xˇ−X∗)ψF(Y,1,W,β)−ψF(Y,0,W,β)]. \! [ψ^F(Y, X,W;β)-ψ^F(Y,X ,W;β) ]\ =\ E\! [( X-X )\ψ^F(Y,1,W;β)-ψ^F(Y,0,W;β)\ ]. (2.3) High classification accuracy of Xˇ X does not guarantee that the right-hand side of equation (2.3) is zero. This is because classification errors are nonclassical and correlated with observed and unobserved variables relevant in downstream regression. Combining multiple imperfect measurements into one index may reduce unit-level classification errors, but it does not by itself justify treating the aggregated label as error-free in downstream analysis. 2.2.2 Gold-Standard-Only Estimation To address non-random, nonclassical measurement errors, the most dominant existing approach is to rely on gold-standard labels. Such methods assume that Xi∗X_i is observed for a small subset of units sampled according to a known sampling design. Let Ri∈0,1R_i∈\0,1\ indicate whether Xi∗X_i is observed, and define the sampling probability ρi≔Pr(Ri=1∣Yi,Wi,X~i)∈(0,1] _i (R_i=1 Y_i,W_i, X_i)∈(0,1]. We assume that the sampling probability is controlled by the researcher, so that ρi _i is known and Ri⟂Xi∗|Yi,Wi,X~iR_i\! \!\!\! X_i Y_i,W_i, X_i. The most common, simple random sampling is the special case where the sampling probability is constant and ρi=ρ _i=ρ for all i. The gold-standard-only estimation (GSO) β^GSO β_GSO uses only the gold-standard observations and solves 1n∑i=1nRiρiψF(Yi,Xi∗,Wi,β)=0. 1n _i=1^n R_i _iψ^F(Y_i,X_i ,W_i;β)=0. This is equivalent to running downstream regression only using a subset of data that have gold-standard labels. It is straightforward to show that this estimator is consistent for β∗β and asymptotically normal, allowing for valid statistical inference when the sampling probability is known (51). However, GSO tends to be inefficient as it discards information in imperfect labels and all units without gold-standard labels. 2.2.3 Debiased Inference with Gold-Standard Labels More recently, a broad class of methods combines predictions available at scale with a smaller sample of gold-standard measurements to bias-correct downstream analysis. This class of bias-correction methods includes design-based supervised learning (23; 22, DSL;), prediction-powered inference (3, PPI;), methods reviewed in 39, MAR-S (10), model-assisted impact analysis (41), and the control-variate approach of 35, among others. All of these methods build on the longstanding literature on semiparametric inference and causal inference (45; 17; 14; 18, e.g.,). Although these methods share a common prediction-and-correction strategy, they differ in their sampling designs, prediction-training procedures, and efficiency adjustments. Here, we use DSL as a representative benchmark. DSL uses the gold-standard observations to train a supervised predictor of the latent independent variable and then bias-correct prediction errors in the downstream moment function via a doubly robust procedure. Using cross-fitting, let X^i=g^−k(i)(Yi,Wi,X~i)∈[0,1] X_i= g_-k(i)(Y_i,W_i, X_i)∈[0,1] denote an out-of-fold prediction of Xi∗X_i where g−k(i)g_-k(i) is estimated using gold-standard data excluding the fold k(i)k(i) that unit i belongs to. The prediction model can naturally use all J proxies jointly, together with the observed outcome and covariates, to predict the true label X∗X . The DSL moment function is ψiDSL(β)≔ψF(Yi,X^i,Wi,β)+RiρiψF(Yi,Xi∗,Wi,β)−ψF(Yi,X^i,Wi,β). _i^DSL(β) ψ^F(Y_i, X_i,W_i;β)+ R_i _i \ψ^F(Y_i,X_i ,W_i;β)-ψ^F(Y_i, X_i,W_i;β) \. (2.4) More generally, the DSL estimator can use any generic out-of-fold prediction m^i(β) m_i(β) for ψF(Yi,Xi∗,Wi,β)ψ^F(Y_i,X_i ,W_i;β), and the expression above with m^i(β)=ψF(Yi,X^i,Wi,β) m_i(β)=ψ^F(Y_i, X_i,W_i;β) is a special case of the general DSL estimator. The DSL estimator β^DSL β_DSL solves 1n∑i=1nψiDSL(β)=0 1n _i=1^n _i^DSL(β)=0. This estimator makes a key assumption that the sampling probability ρi _i is controlled by and known to researchers. This scenario is common in many application areas where, for example, researchers choose which documents to be coded by experts. Under the known sampling design, [ψiDSL(β)|Xi∗,Yi,Wi,X~i,X^i]=ψF(Yi,Xi∗,Wi;β).E\! [ _i^DSL(β)\, |\,X_i ,Y_i,W_i, X_i, X_i ]\ =\ ψ^F(Y_i,X_i ,W_i;β). Consequently, DSL targets the oracle parameter even when the supervised prediction model and LLM annotations are arbitrarily misspecified. These classes of methods are popular and powerful as they do not require any assumption about errors made by LLMs or any methods used to generate X^i X_i. When X^i X_i is more accurate, DSL becomes more accurate too, while it is always valid without any assumption on X^i X_i. These existing strategies span a continuum between relying on proxy labels and relying on gold-standard labels. Naive approaches use proxy labels as if they are error-free and they are in general biased unless proxy labels are perfect. GSO is unbiased and provides valid inference but ignores proxy labels and can be inefficient. More recent bias-correction methods, such as DSL (23; 22) and PPI (3), provide valid inference under the known sampling probability for gold-standard labeling without making assumptions about errors in proxy variables. However, they assume access to gold-standard labels for a subset of units, which might be difficult or too costly for some applications. In the next section, we explore an alternative approach that does not assume access to gold-standard labels. 3 Debiased Inference with Multiple Imperfect Measurements We now show how to use multiple imperfect labels to perform valid downstream inference without requiring gold-standard labels. In particular, building on established results on identification of latent variables, we use the joint distribution of multiple imperfect labels to recover the unbiased downstream moment under a conditional independence assumption. In Section 3.1, we begin by reviewing this classical assumption and then relax it to accommodate modern applications of LLM annotations. We then discuss nonparametric identification (Section 3.2) and propose a consistent and asymptotically normal estimator to enable valid downstream inference (Section 3.3). 3.1 Assumptions As is clear from the previous section, if researchers assume no access to gold-standard labels, they have to instead make some assumptions about proxy labels (46). Here, we build on the established literature of nonparametric latent variable models (27; 37; 20; 34; 1, e.g.,). We begin with a brief review of the classical conditional independence assumption (see Section 6.1 of 46 for a detailed review). Specifically, the classical methods assume that multiple imperfect measurements are independent conditional on the true label: X(1)⟂⋯⟂X(J)|X∗.X^(1)\! \!\!\! ·s\! \!\!\! X^(J) X . (3.1) Importantly, the proxy labels may be biased, unequally accurate, and nonidentically distributed. In particular, their false-positive and false-negative rates may be nonzero and may differ across labels. This assumption is substantially weaker than assuming proxy labels are perfect and have no measurement errors. A large literature has developed methods for latent variable models under this conditional independence structure. Examples include nonparametric estimation using an algorithm and joint approximate diagonalization (29; 7, respectively), crowdsourcing and noisy label models (44; 55), and tensor based methods or canonical correlation analysis for multi-view latent variable learning (2; 12, the latter under a weaker conditional moment restriction). However, this classical assumption may be violated if human annotators make similar errors because of shared characteristics of the input or annotation task (e.g., some texts are much longer and more difficult to annotate). Similarly, machine annotators may make correlated errors because of shared model characteristics that interact with the input (e.g., overlapping training data or a common prompt). For example, recent papers emphasize that errors in LLMs are correlated even conditional on the true label (36; 16). To make this assumption more plausible, we allow for conditioning on auxiliary input- or annotation-level information DiD_i, the downstream covariates WiW_i, and the downstream dependent variable YiY_i. The vector DiD_i may account for heterogeneity in label accuracy or shared sources of dependence but need not enter the downstream regression. Assumption 3.1 (Independence across labels conditioning on input and annotation information). The multiple imperfect labels are mutually independent conditional on the latent label and the observed covariates: X(1)⟂⋯⟂X(J)|X∗,D~,X^(1)\! \!\!\! ·s\! \!\!\! X^(J) X , D, where D~=(D,W,Y). D=(D,W,Y). Assumption 3.1 allows for the misclassification rates to vary across annotators and across units. It specifically allows imperfect labels to share sources of errors as long as such common causes are captured by D~ D. The vector D may contain observed or derived characteristics of the input (e.g., writing style, text length, language, or image quality) measured by rich text embeddings. It may also contain annotation-level information that varies across units (e.g., prompt version, task duration, annotation order, or recorded fatigue). This assumption is significantly weaker than the classical conditional independence assumption stated in (3.1), which did not allow for any shared source of errors. Assumption 3.1 still fails if unobserved common sources of errors remain after conditioning. Importantly, if researchers want to avoid assuming access to gold-standard labels, they have to make some assumptions about proxies, and our paper makes the assumption explicit and transparent by building and extending the classical literature. Remark 1. As LLM annotations are one of the most common application areas, we will come back to this assumption and use Section 5 to specifically discuss (a) how to make this assumption more plausible by carefully selecting conditioning variables D~ D and (b) how to choose different families of LLMs and prompts to construct measurements. We also develop a series of diagnostic tools for this assumption. Please see Section 5. 3.2 Identification We now discuss how to identify the downstream moment function under the conditional independence assumption. Here we provide an overview of the two steps, which we elaborate in the subsequent sections. The first step is to identify the conditional classification rate defined as ηj,a∗(d)=Pr(X(j)=1∣X∗=a,D~=d),η _j,a(d)= (X^(j)=1 X =a, D=d), (3.2) where ηj,1∗(d)η _j,1(d) denotes the true positive rate and ηj,0∗(d)η _j,0(d) represents the false positive rate for a specific proxy j∈1,…,Jj∈\1,…,J\. Here, we build on general nonparametric identification results using array decomposition (37; 1). Second, given the identified conditional classification rate ηj,a∗(d)η _j,a(d), we construct a bridge function that connects the observed proxy X(j)X^(j) and the unobserved true label X∗X (56; 28): Mj(D~)=X(j)−ηj,0∗(D~)ηj,1∗(D~)−ηj,0∗(D~).M_j( D)= X^(j)-η _j,0( D)η _j,1( D)-η _j,0( D). This is a tailored function in that, by construction, it is unbiased for the true label, satisfying [Mj(D~)∣X∗,D~]=X∗.E[M_j( D) X , D]=X . More generally, as shown below, we can combine multiple proxy labels to construct a general bridge function H(X~,D~)H( X, D) that is unbiased for the true latent variable. [H(X~,D~)∣X∗,D~]=X∗.E[H( X, D) X , D]=X . We can then use this bridge function in place of the true label to build the unbiased, observed downstream moment. ψDMM(X~,D~,β)=1−H(X~,D~)ψF(Y,0,W,β)+H(X~,D~)ψF(Y,1,W,β). _DMM( X, D;β)\ =\ \1-H( X, D)\ψ^F(Y,0,W;β)\ +\ H( X, D)ψ^F(Y,1,W;β). Under Assumption 3.1, [ψDMM(X~,D~,β)]=[ψF(Y,X∗,W,β)],E[ _DMM( X, D;β)]=E[ψ^F(Y,X ,W;β)], which will allow for consistent estimation and valid inference as developed in Section 3.3. 3.2.1 Identification of Conditional Classification Rates We start with identification of conditional classification rate ηj,a∗(d)η _j,a(d). Assumption 3.1 implies that the joint distribution of imperfect proxies can be factorized as follows. Pr(X~=x~∣D~=d)=∑a=01Pr(X∗=a∣D~=d)∏j=1Jηj,a∗(d)x(j)1−ηj,a∗(d)1−x(j). ( X= x D=d)= _a=0^1 (X =a D=d) _j=1^Jη _j,a(d)^x^(j)\1-η _j,a(d)\^1-x^(j). (3.3) This equation mirrors the classical latent-class decomposition and its identification through array decomposition under Kruskal’s uniqueness condition (37). Adapting Theorem 4 of 1, we partition J proxies into three mutually disjoint and nonempty subsets S1,S2,S3S_1,S_2,S_3, i.e., 1,…,J=S1∪S2∪S3\1,…,J\=S_1∪ S_2∪ S_3. For each b∈1,2,3b∈\1,2,3\, define the corresponding composite measurement X~Sb≔(X(j))j∈Sb X_S_b (X^(j) )_j∈ S_b, which takes values in 0,1|Sb|\0,1\^|S_b| and therefore has 2|Sb|2^|S_b| possible label patterns. Note that the identified form does not depend on how we construct these partitions (1). When J=3J=3, this construction reduces to the singleton partition Sb=bS_b=\b\ for b∈1,2,3b∈\1,2,3\. For joint label patterns xb∈0,1|Sb|x_b∈\0,1\^|S_b|, b∈1,2,3b∈\1,2,3\, define the conditional probability tensor d∈ℝ2|S1|×2|S2|×2|S3|T_d ^2^|S_1|× 2^|S_2|× 2^|S_3| by [d]x1,x2,x3≔Pr(X~S1=x1,X~S2=x2,X~S3=x3∣D~=d).[T_d]_x_1,x_2,x_3 \! ( X_S_1=x_1, X_S_2=x_2, X_S_3=x_3 D=d ). For block SbS_b and latent class a∈0,1a∈\0,1\, define the block-level conditional probability matrix: [b(d)]xb,a≔Pr(X~Sb=xb∣X∗=a,D~=d),xb∈0,1|Sb|.[S_b(d)]_x_b,a \! ( X_S_b=x_b X =a, D=d ), x_b∈\0,1\^|S_b|. The conditional classification rate is then the corresponding marginals of the block distribution. In particular, for every j∈Sbj∈ S_b, ηj,a∗(d)=∑xb:xj=1[b(d)]xb,aη _j,a(d)= _x_b:x_j=1[S_b(d)]_x_b,a. Under Assumption 3.1, the tensor admits a two-component Candecomp/Parafac (CP) decomposition (11; 32; 50): d=∑a=01Pr(X∗=a∣D~=d)1,⋅a(d)⊗2,⋅a(d)⊗3,⋅a(d),T_d= _a=0^1 (X =a D=d)S_1,· a(d) _2,· a(d) _3,· a(d), (3.4) where ⊗ denotes the outer product of vectors. Provided that each latent class has positive conditional probability, Kruskal’s uniqueness theorem implies that the CP decomposition in equation (3.4) is unique if k1(d)+k2(d)+k3(d)≥2K∗+2k_1(d)+k_2(d)+k_3(d)≥ 2K_ +2 (3.5) where K∗K_ denotes the number of latent classes, and kb(d)≔krankb(d)k_b(d) \S_b(d)\ denotes the Kruskal rank of the bbth block level probability matrix. Because the columns of each b(d)S_b(d) are probability vectors, their scaling is fixed by the requirement that they sum to one. Therefore, under equation (3.5), b(d)b=13\S_b(d)\_b=1^3 are identified up to a common permutation of the latent classes (37; 1). In our binary setting, K∗=2K_ =2. If each block contains at least one proxy that is informative about X∗X , then b,⋅0(d)≠b,⋅1(d)S_b,· 0(d) _b,· 1(d) and hence kb(d)=2k_b(d)=2 for every b∈1,2,3b∈\1,2,3\. It follows that k1(d)+k2(d)+k3(d)=2+2+2=2K∗+2k_1(d)+k_2(d)+k_3(d)=2+2+2=2K_ +2, so Kruskal’s rank condition holds with equality. Each proxy-specific conditional classification rate ηj,a∗(d)η _j,a(d) can then be identified by marginalizing the corresponding identified block-level distribution. An additional anchoring rule resolves the remaining permutation ambiguity. For example, one may designate an anchor proxy j†j and assume ηj†,1∗(D~)>ηj†,0∗(D~)η _j ,1( D)>η _j ,0( D) a.s. This condition states that, for the chosen anchor proxy, the true positive rate is higher than the false positive rate, which is a mild requirement commonly satisfied in modern applications of AI-generated data. More general substantive or agreement-based anchoring rules are also possible. At the population level, the identified proxy-specific conditional classification rates do not depend on the choice of tripartition, provided that the same anchoring rule is used to orient the latent classes. For completeness, we reproduce the following known identification result. Result 1 (Identification of the measurement model; 1). Suppose Assumption 3.1 holds. For PD~P_ D-almost every d, suppose that both latent classes have positive conditional probability, there exists a tripartition into three nonempty blocks whose block-level probability matrices satisfy k1(d)+k2(d)+k3(d)≥2K∗+2,k_1(d)+k_2(d)+k_3(d)≥ 2K +2, and a prespecified anchoring rule uniquely orients the two latent classes. Suppose also that the resulting anchored conditional probabilities admit measurable versions. Then the conditional mixing probability and the conditional classification rates ηj,a∗(d):j=1,…,J,a∈0,1\η _j,a(d):j=1,…,J,\ a∈\0,1\\ are identified from the conditional distribution of X~ X given D~=d D=d, up to PD~P_ D-null sets. 3.2.2 Identification via the Robust Bridge Function Given the identified conditional classification rate ηj,a∗(D~)η _j,a( D), we now construct a tailored bridge function to obtain an unbiased downstream moment function by building on the recent literature in causal inference (56; 28). We first define a single-proxy bridge as Mj(D~,η∗)=X(j)−ηj,0∗(D~)ηj,1∗(D~)−ηj,0∗(D~),M_j( D;η )= X^(j)-η _j,0( D)η _j,1( D)-η _j,0( D), which by construction satisfies [Mj(D~;η∗)∣X∗,D~]=X∗E[M_j( D;η ) X , D]=X . Assumption 3.1 further implies that, for every nonempty set of distinct labels S, [∏j∈SMj(D~;η∗)|X∗,D~]=∏j∈S[Mj(D~;η∗)∣X∗,D~]=(X∗)|S|=X∗,E\! [ _j∈ SM_j( D;η )\; |\;X , D ]= _j∈ SE\! [M_j( D;η ) X , D ]=(X )^|S|=X , where the final equality follows because X∗∈0,1X ∈\0,1\. Thus, many functions of imperfect labels are valid bridges. Consider a class of polynomial bridge functions: H(X~,D~,η∗):=∑S∈ωS∏j∈SMj(D~,η∗)H( X, D;η ):= _S _S _j∈ SM_j( D;η ) (3.6) where S is a collection of nonempty subsets of 1,…,J\1,…,J\ and ωS _S is the coefficient assigned to the corresponding polynomial term. Any member of this class preserves the unbiasedness for X∗X whenever ∑S∈ωS=1 _S _S=1. To obtain desirable statistical properties within this class, we combine multiple proxies in a way that is symmetric between labels and Neyman orthogonal (45; 18) so that it is locally robust to the first-stage estimation error for the conditional classification rate ηj,a∗(d)η _j,a(d). Define the average pair and triple bridges as H¯2(X~,D~,η∗) H_2( X, D;η ) ≔(J2)−1∑j1<j2Mj1(D~,η∗)Mj2(D~,η∗), J2^-1 _j_1<j_2M_j_1( D;η )M_j_2( D;η ), H¯3(X~,D~,η∗) H_3( X, D;η ) ≔(J3)−1∑j1<j2<j3Mj1(D~,η∗)Mj2(D~,η∗)Mj3(D~,η∗). J3^-1 _j_1<j_2<j_3M_j_1( D;η )M_j_2( D;η )M_j_3( D;η ). One example of a symmetric robust bridge function is written as follows. HR(X~,D~,η∗)=3H¯2(X~,D~,η∗)−2H¯3(X~,D~,η∗).H^R( X, D;η )=3 H_2( X, D;η )-2 H_3( X, D;η ). (3.7) For a symmetric combination c2H¯2+c3H¯3c_2 H_2+c_3 H_3, the unbiasedness of the bridge function requires c2+c3=1c_2+c_3=1, while the Neyman orthogonality (i.e., cancellation of the first-order bridge error) under X∗=1X =1 requires 2c2+3c3=02c_2+3c_3=0, yielding c2=3c_2=3 and c3=−2c_3=-2 as a unique set of weights. Under X∗=0X =0, the pair and triple terms are already at least second order and achieve the Neyman orthogonality. Thus, while a single-proxy bridge is enough for identifying the downstream moment, at least three distinct proxies are needed for the Neyman orthogonality and the resulting robustness to the first-stage estimation error. For J=3J=3, equation (3.7) becomes HR=M1M2+M1M3+M2M3−2M1M2M3.H^R=M_1M_2+M_1M_3+M_2M_3-2M_1M_2M_3. This construction of the bridge function was first proposed by 28 for a setting in which the binary outcome is latent and the main estimand is the average treatment effect (see their Appendix B for details). We extend their construction to any J≥3J≥ 3 and apply it to both latent independent- and dependent-variable settings with a wide range of downstream moment estimators common in the social sciences. This is important as researchers can often construct many proxies from LLMs in the modern applications of AI-generated data. Below, we focus on a robust bridge function with pair and triple products for numerical stability and the sake of clear presentation. However, we emphasize that our method and proofs are applicable to a more general class of bridge functions that use higher-order interaction terms of single-proxy bridge functions (equation (3.6)). We now formally state the unbiasedness and local robustness of the symmetric bridge function. Proposition 3.1 (Unbiasedness and local robustness of the symmetric robust bridge function). We assume that imperfect measurements X(j)j=1J\X^(j)\_j=1^J are relevant, i.e., for some constant cΔ>0c_ >0, min1≤j≤J|ηj,1∗(D~)−ηj,0∗(D~)|≥cΔa.s. _1≤ j≤ J |η _j,1( D)-η _j,0( D) |≥ c_ .s. Under Assumption 3.1, [HR(X~,D~;η∗)∣X∗,D~]=X∗.E\! [H^R( X, D;η ) X , D ]=X . (3.8) Moreover, consider cross-fitted estimates of the single-label bridges M^j M_j and let the error term ej,a(D~i)=[M^ij(D~i;η^(−k(i)))∣Xi∗=a,D~i,ℐ−k(i)]−a,for a∈0,1,e_j,a( D_i)=E\! [ M_ij( D_i; η^(-k(i))) X _i=a, D_i,I_-k(i) ]-a, for a∈\0,1\, where ℐ−k(i)I_-k(i) is the data excluding the fold k(i)k(i) that unit i belongs to, and M^ij M_ij is constructed by the conditional classification rate η^(−k(i)) η^(-k(i)) estimated only using ℐ−k(i)I_-k(i). Conditional on the training sample ℐ−kI_-k, the fitted robust bridge only contains the second- or third-order bias. [H^R(X~,D~;η^(−k))∣X∗=1,D~,ℐ−k]−1 [ H^R( X, D; η^(-k)) X =1, D,I_-k]-1 =−3(J2)−1∑j1<j2ej1,1(D~)ej2,1(D~)−2(J3)−1∑j1<j2<j3ej1,1(D~)ej2,1(D~)ej3,1(D~), =-3 J2^-1 _j_1<j_2e_j_1,1( D)e_j_2,1( D)-2 J3^-1 _j_1<j_2<j_3e_j_1,1( D)e_j_2,1( D)e_j_3,1( D), [H^R(X~,D~;η^(−k))∣X∗=0,D~,ℐ−k] [ H^R( X, D; η^(-k)) X =0, D,I_-k] =3(J2)−1∑j1<j2ej1,0(D~)ej2,0(D~)−2(J3)−1∑j1<j2<j3ej1,0(D~)ej2,0(D~)ej3,0(D~). =3 J2^-1 _j_1<j_2e_j_1,0( D)e_j_2,0( D)-2 J3^-1 _j_1<j_2<j_3e_j_1,0( D)e_j_2,0( D)e_j_3,0( D). Consequently, the conditional bridge error contains no term that is linear in a single first-stage error. The proof is provided in Appendix A.1. As shown above, the coefficients 33 and −2-2 preserve the unbiasedness of the bridge function and eliminate first-order bridge error: at X∗=1X =1, 3(2/J)−2(3/J)=03(2/J)-2(3/J)=0, while at X∗=0X =0, no linear term arises because singleton products are excluded. Therefore, if each estimated conditional classification rate model converges in L2(P)L_2(P) at rate δn _n and the downstream moment contrast is uniformly bounded as assumed in Theorem 3.2, the resulting population remainder of the bridge-based moment function is of order Op(δn2)O_p( _n^2), rather than Op(δn)O_p( _n). This second-order remainder underlies the asymptotic inference results in the next subsection. Finally, we can now substitute the robust bridge into the binary decomposition of the full-data moment equation to obtain an unbiased downstream moment function. ψDMM(X~,D~,β,η∗)=1−HR(X~,D~,η∗)ψF(Y,0,W,β)+HR(X~,D~,η∗)ψF(Y,1,W,β). ψ^DMM( X, D;β,η )\ =\ \1-H^R( X, D;η )\ψ^F(Y,0,W;β)\ +\ H^R( X, D;η )ψ^F(Y,1,W;β). Theorem 3.1 (Identification of the downstream parameter). Suppose the conditions of Result 1 hold, and suppose that the class contrasts of all proxies used in the robust bridge are bounded away from zero. Suppose also that the oracle downstream moment in equation (2.1) has the unique solution β∗β . Then the robust bridge is identified from the observed-data law and, for every β, [ψDMM(X~,D~,β,η∗)]=[ψF(Y,X∗,W,β)]E[ψ^DMM( X, D;β,η )]=E[ψ^F(Y,X ,W;β)] where η∗(⋅)η (·) denotes the true conditional classification rate function. Consequently, β∗β is identified from the observed-data law as the unique solution to [ψDMM(X~,D~,β,η∗)]=0.E[ψ^DMM( X, D;β,η )]=0. Proof. By Result 1, η∗η is identified from the observed-data law. The class-contrast condition therefore makes the robust bridge HRH^R well defined and identified. By iterated expectation and equation (3.8), [ψDMM(X~,D~,β,η∗)] [ψ^DMM( X, D;β,η )] =[ψF(Y,0,W;β)+[HR(X~,D~;η∗)∣X∗,D~]ψF(Y,1,W;β)−ψF(Y,0,W;β)] =E [ψ^F(Y,0,W;β)+E[H^R( X, D;η ) X , D]\ψ^F(Y,1,W;β)-ψ^F(Y,0,W;β)\ ] =[(1−X∗)ψF(Y,0,W,β)+X∗ψF(Y,1,W,β)] =E [(1-X )ψ^F(Y,0,W;β)+X ψ^F(Y,1,W;β) ] =[ψF(Y,X∗,W,β)], =E [ψ^F(Y,X ,W;β) ], which completes the proof. ∎ 3.3 Estimation and Inference Given the identification results in the previous section, this section proposes the DMM (debiased inference with multiple imperfect measurements) estimator and proves its consistency and asymptotic normality. First, to estimate the conditional classification rate, we use cross-fitting (18) and separate estimation of the nuisance function from evaluation of the downstream moment function. Partition the observations into K folds ℐ1,…,ℐKI_1,…,I_K. For each fold k, estimate the conditional classification rate model ηj,0(d),ηj,1(d)j=1J\ _j,0(d), _j,1(d)\_j=1^J from (X~i,D~i)( X_i, D_i) using observations outside ℐkI_k. One option is to maximize the conditional likelihood ℒ−k=∑i∉ℐklog[∑a=01π(D~i)a1−π(D~i)1−a∏j=1Jηj,a(D~i)Xi(j)1−ηj,a(D~i)1−Xi(j)], _-k= _i _k [ _a=0^1π( D_i)^a\1-π( D_i)\^1-a _j=1^J _j,a( D_i)^X_i^(j)\1- _j,a( D_i)\^1-X_i^(j) ], where π(D~)≔Pr(X∗=1∣D~).π( D) (X =1 D). This likelihood can be maximized using the expectation-maximization (EM) algorithm (21), as in 28. More flexible nuisance functions can be accommodated by maximizing the same likelihood over sieve spaces (48; 33), or alternatively by sieve minimum distance based on conditional moment restrictions (15; 56). Each fold-specific fit is oriented using the same anchoring rule. As in debiased machine learning (18), the asymptotic results below do not depend on which particular fitting algorithm researchers use to estimate the conditional classification rates. Below, we provide high-level requirements for convergence rates that can be achieved by a wide range of fitting algorithms. Then, for i∈ℐki _k, we construct M^ij(−k)=Xi(j)−η^j,0(−k(i))(D~i)η^j,1(−k(i))(D~i)−η^j,0(−k(i))(D~i), M_ij^(-k)= X_i^(j)- η_j,0^(-k(i))( D_i) η_j,1^(-k(i))( D_i)- η_j,0^(-k(i))( D_i), (3.9) where η^(−k(i)) η^(-k(i)) is estimated only using ℐ−k(i)I_-k(i), the data excluding the fold k(i)k(i) that unit i belongs to. Then, the cross-fitted moment function for the DMM estimator is ψ^DMM(X~i,D~i,β,η^(−k(i)))≔1−H^iR(η^(−k(i)))ψF(Yi,0,Wi,β)+H^iR(η^(−k(i)))ψF(Yi,1,Wi,β), ψ^DMM( X_i, D_i;β, η^(-k(i)))\ \ \1- H_i^R( η^(-k(i)))\ψ^F(Y_i,0,W_i;β)\ +\ H_i^R( η^(-k(i)))ψ^F(Y_i,1,W_i;β), (3.10) where H^iR(η^(−k(i))) H_i^R( η^(-k(i))) is defined by equation (3.7) with estimated single-proxy bridge functions in equation (3.9). Our DMM estimator β^DMM β^DMM can then be written as the solution to the following equation. 1n∑i=1nψ^DMM(X~i,D~i,β,η^(−k(i)))=0. 1n _i=1^n ψ^DMM( X_i, D_i;β, η^(-k(i)))=0. Throughout this subsection, the observations are i.i.d., the numbers of proxies J and folds K are fixed, each fold contains a nonvanishing fraction of the sample, and all fold-specific nuisance estimates are oriented using the same anchoring rule. The following theorem presents the asymptotic properties of the DMM estimator. Theorem 3.2 (Consistency and asymptotic normality). Suppose the assumptions required in Theorem 3.1 hold. Let ℬ0⊂ℬB_0 be a compact neighborhood of β∗β , and let β^DMM β^DMM be a solution of the sample moment equation in ℬ0B_0, which exists with probability approaching one. We assume the following standard regularity conditions: (a) β∗β is the unique well-separated interior root on ℬ0B_0; (b) for a∈0,1a∈\0,1\, the moment functions ψF(Y,a,W,β)ψ^F(Y,a,W;β) and their Jacobians are Lipschitz in β on ℬ0B_0 with square-integrable envelopes, the population Jacobian 0A_0 defined below is nonsingular, and there exists a finite constant C such that supβ∈ℬ0‖ψF(Y,1,W,β)−ψF(Y,0,W,β)‖≤C _β _0 \|ψ^F(Y,1,W;β)-ψ^F(Y,0,W;β) \|≤ C a.s.; (c) the true and fitted class contrasts, ηj,1∗(D~)−ηj,0∗(D~)j=1J\η _j,1( D)-η _j,0( D)\_j=1^J and η^j,1(−k)(D~)−η^j,0(−k)(D~)j=1Jk=1K\\ η^(-k)_j,1( D)- η^(-k)_j,0( D)\_j=1^J\_k=1^K, are uniformly bounded away from zero over D~ D, the latter with probability approaching one, and (d) for every k, j, a, η^j,a(−k)(D~) η^(-k)_j,a( D) takes values in [0,1][0,1] almost surely. Define the maximum fold-specific nuisance error. δn=maxk,j,a‖η^j,a(−k)−ηj,a∗‖2,P, _n= _k,j,a \| η_j,a^(-k)- _j,a \|_2,P, where ηj,a∗ _j,a is the true classification rate and the norm is taken over the distribution of D~ D. If the nuisance function estimation is consistent, i.e., δn=op(1) _n=o_p(1), then β^DMM→β∗ β^DMM p→β . More generally, the DMM estimator achieves multiple robustness for consistency: consistency of the DMM estimator requires the convergence of the nuisance functions to the true functions for only J−1J-1 proxies, and one does not need to know which J−1J-1 classification rates are consistently estimated. If the nuisance function estimator converges at a slow nonparametric rate, i.e., δn=op(n−1/4), _n=o_p(n^-1/4), n(β^DMM−β∗)=1n∑i=1n0−1ψDMM(X~i,D~i,β∗,η∗)+op(1), n( β^DMM-β )= 1 n _i=1^nA_0^-1ψ^DMM( X_i, D_i;β ,η )+o_p(1), (3.11) and n(β^DMM−β∗)↝N(0,0−100−⊤) n( β^DMM-β ) N\! (0,A_0^-1 _0A_0^- ) (3.12) where 0 _0 =−[1−HR(X~,D~,η∗)∂β⊤ψF(Y,0,W,β∗)+HR(X~,D~,η∗)∂β⊤ψF(Y,1,W,β∗)], =-E\! [\1-H^R( X, D;η )\ ∂β ψ^F(Y,0,W;β )+H^R( X, D;η ) ∂β ψ^F(Y,1,W;β ) ], 0 _0 =[ψDMM(X~,D~,β∗,η∗)ψDMM(X~,D~,β∗,η∗)⊤]. =E\! [ψ^DMM( X, D;β ,η )ψ^DMM( X, D;β ,η ) ]. The rate requirement for the nuisance function estimation η η follows from Proposition 3.1: the population remainder caused by estimating the conditional classification rate is second order in the nuisance function estimation error. Cross-fitting controls the corresponding empirical process term without requiring restrictive Donsker conditions, following the logic of locally robust estimation (19). A proof is provided in Appendix A.2. To estimate the asymptotic variance, we evaluate the cross-fitted moment function at β^DMM β^DMM and define ^= A= −1n∑i=1n[1−H^iR(η^(−k(i)))∂β⊤ψF(Yi,0,Wi;β^DMM)+H^iR(η^(−k(i)))∂β⊤ψF(Yi,1,Wi;β^DMM)], - 1n _i=1^n [\1- H_i^R( η^(-k(i)))\ ∂β ψ^F(Y_i,0,W_i; β^DMM)+ H_i^R( η^(-k(i))) ∂β ψ^F(Y_i,1,W_i; β^DMM) ], (3.13) ^= = 1n∑i=1nψDMM(X~i,D~i,β^DMM,η^(−k(i)))ψDMM(X~i,D~i,β^DMM,η^(−k(i)))⊤, 1n _i=1^nψ^DMM( X_i, D_i; β^DMM, η^(-k(i)))ψ^DMM( X_i, D_i; β^DMM, η^(-k(i))) , (3.14) ^= V= ^−1^^−⊤. A^-1 A^- . (3.15) Then ^/n V/n estimates the covariance matrix of β^DMM β^DMM. For any fixed contrast c, an asymptotic (1−α)(1-α) confidence interval for c⊤β^DMMc β^DMM is c⊤β^DMM±z1−α/2c⊤^cn,c β^DMM\;±\;z_1-α/2 c Vcn, where z1−α/2z_1-α/2 is the (1−α/2)(1-α/2) percentile of the standard normal distribution. For the generalized linear model example in Section 2, c may select the coefficient on the latent regressor. Discussion on Efficiency. The robust bridge is designed to remove first-order nuisance bias. Here, we examine how the asymptotic precision of this estimator changes with the number and quality of the labels. Write H0,JRH_0,J^R to emphasize that the bridge uses J proxies. Then, the DMM moment function admits the decomposition ψJDMM(X~,D~,β∗,η∗)=ψF(Y,X∗,W,β∗)+H0,JR−X∗Δψ0(Y,W), _J^DMM( X, D;β ,η )=ψ^F(Y,X ,W;β )+\H_0,J^R-X \ _0(Y,W), (3.16) where Δψ0(Y,W)=ψF(Y,1,W,β∗)−ψF(Y,0,W,β∗) _0(Y,W)=ψ^F(Y,1,W;β )-ψ^F(Y,0,W;β ). The unbiasedness of the bridge function implies that the population bread matrix equals its oracle counterpart. 0=−[∂ψF(Y,X∗,W,β)∂β⊤|β=β∗].A_0=-E [ . ∂ψ^F(Y,X ,W;β)∂β |_β=β ]. (3.17) Thus, the number and quality of the labels affect the asymptotic variance only through the meat matrix: 0,JR=0F+[H0,JR−X∗2Δψ0(Y,W)Δψ0(Y,W)⊤] _0,J^R= _0^F+E [\H_0,J^R-X \^2 _0(Y,W) _0(Y,W) ] (3.18) where 0F≔[ψF(Y,X∗,W,β∗)ψF(Y,X∗,W,β∗)⊤] _0^F [ψ^F(Y,X ,W;β )ψ^F(Y,X ,W;β ) ]. We use 0,JR≔0−10,JR0−⊤V_0,J^R _0^-1 _0,J^RA_0^- to denote the asymptotic variance of the DMM estimator using J proxies, and we use 0F≔0−10F0−⊤V_0^F _0^-1 _0^FA_0^- to denote the asymptotic variance of an infeasible, oracle estimator that uses unobserved X∗X in downstream moments. For a fixed linear combination of coefficients c⊤β∗c β , we can write the asymptotic variance of the DMM estimator as follows. c⊤0,JRc=c⊤0Fc+[∑a=01π(D~i)a1−π(D~i)1−ava,JR(D~)(0−⊤c)⊤Δψ0(Y,W)2], c V_0,J^Rc\ =\ c V_0^Fc\ +\ E [ _a=0^1π( D_i)^a\1-π( D_i)\^1-av_a,J^R( D)\(A_0^- c) _0(Y,W)\^2 ], (3.19) where va,JR(d)≔(H0,JR∣X∗=a,D~=d)v_a,J^R(d) (H_0,J^R X =a, D=d ). Importantly, only va,JR(d)v_a,J^R(d) depends on the selection of proxies. Hence, decreasing va,JR(d)v_a,J^R(d) is sufficient to improve precision for downstream parameters of interest. For proxy j, define its conditional single bridge variance κj,a(d)≔ηj,a∗(d)1−ηj,a∗(d)ηj,1∗(d)−ηj,0∗(d)2. _j,a(d) η _j,a(d)\1-η _j,a(d)\\η _j,1(d)-η _j,0(d)\^2. Then, we can express the conditional variance of the bridge function as follows. va,JR(d)=9(J2)2∑j1<j2κj1,a(d)κj2,a(d)+4(J3)2∑j1<j2<j3κj1,a(d)κj2,a(d)κj3,a(d).v_a,J^R(d)\ =\ 9 J2^2 _j_1<j_2 _j_1,a(d) _j_2,a(d)\ +\ 4 J3^2 _j_1<j_2<j_3 _j_1,a(d) _j_2,a(d) _j_3,a(d). (3.20) Suppose an additional proxy has conditional bridge variance κJ+1,a(d) _J+1,a(d). Appendix B derives an explicit threshold κ¯J,a(d) κ_J,a(d) such that, except in the degenerate zero-variance case, va,J+1R(d)<va,JR(d)⟺κJ+1,a(d)<κ¯J,a(d).v_a,J+1^R(d)<v_a,J^R(d) _J+1,a(d)< κ_J,a(d). Thus, adding a proxy improves precision when the conditional variance of its single-proxy bridge function is small enough relative to those of the existing proxies. For example, a proxy with a very small class contrast, ηj,1(d)−ηj,0(d) _j,1(d)- _j,0(d), has a large κj,a(d) _j,a(d) and can increase the variance of the bridge function va,J+1R(d)v_a,J+1^R(d). More proxies therefore do not mechanically improve this particular estimator. It is straightforward to allow for different weights when constructing the robust bridge function so that it can guarantee the DMM estimator with J+1J+1 proxies is at least as efficient as the DMM estimator with J proxies. In a special case where every proxy has the same conditional variance κa(d) _a(d), va,JR(d)=18κa(d)2J(J−1)+24κa(d)3J(J−1)(J−2),v_a,J^R(d)= 18 _a(d)^2J(J-1)+ 24 _a(d)^3J(J-1)(J-2), which decreases with J at rate J−2J^-2, i.e., more proxies lead to higher accuracy for downstream parameter estimation. 4 Extensions We now extend the proposed framework to settings in which an error-prone variable is a binary dependent variable (Section 4.1). We then provide the most general results where we extend our method to settings where either an independent or dependent variable is multicategory (Section 4.2). 4.1 Latent Dependent Variable 4.1.1 Setup and Identification Let Yi∗∈0,1Y_i ∈\0,1\ denote the latent true outcome, and let WiW_i collect all observed variables entering the downstream analysis as before, including any focal independent variable and additional covariates. Instead of Yi∗Y_i , the researcher observes J≥3J≥ 3 imperfect labels Y~i=(Yi(1),…,Yi(J)),Yi(j)∈0,1. Y_i= (Y_i^(1),…,Y_i^(J) ), Y_i^(j)∈\0,1\. As in Section 3, let DiD_i collect auxiliary information that may help account for heterogeneity or shared dependence in the label errors. Define D~i=(Di,Wi) D_i=(D_i,W_i). Let ϕF(y,w,β)∈ℝdβφ^F(y,w;β) ^d_β be a user-specified full-data moment function. Reusing β∗β for the target parameter in this section, suppose that if Y∗Y were observed, β∗β would be the unique solution to [ϕF(Y∗,W,β)]=0.E\! [φ^F(Y ,W;β) ]=0. We impose an analog of Assumption 3.1 for the latent outcome setting. Assumption 4.1 (Independence across outcome labels conditioning on input and downstream information). The imperfect outcome labels are mutually independent conditional on the latent outcome and the measurement conditioning covariates: Y(1)⟂⋯⟂Y(J)|Y∗,D~.Y^(1)\! \!\!\! ·s\! \!\!\! Y^(J) Y , D. For a∈0,1a∈\0,1\, let π∗(d)≔Pr(Y∗=1∣D~=d),ηj,a∗(d)≔Pr(Y(j)=1∣Y∗=a,D~=d).π (d) (Y =1 D=d), η _j,a(d) (Y^(j)=1 Y =a, D=d). We impose the analogs of the overlap, relevance, and anchoring conditions in Section 3. Under these conditions, the joint distribution of (Y~,D~)( Y, D) identifies π∗(d)π (d) and ηj,a∗(d)η _j,a(d) by the same conditional latent-class argument, so we do not repeat the decomposition here. Define the outcome-proxy bridge MjY(D~)=Y(j)−ηj,0∗(D~)ηj,1∗(D~)−ηj,0∗(D~).M_j^Y( D)= Y^(j)-η _j,0( D)η _j,1( D)-η _j,0( D). Substituting this outcome-proxy bridge MjYM_j^Y into equation (3.7) defines the robust outcome bridge HYR(Y~,D~)H_Y^R( Y, D). The same argument as in Proposition 3.1 gives [HYR(Y~,D~)∣Y∗,D~]=Y∗.E\! [H_Y^R( Y, D) Y , D ]=Y . (4.1) The fitted bridge also retains the same local robustness: its conditional error contains no term that is linear in a single proxy-specific nuisance error. We can then define the observed-data moment function ϕYDMM(Y~,D~,β,η∗)=1−HYR(Y~,D~)ϕF(0,W,β)+HYR(Y~,D~)ϕF(1,W,β). _Y^DMM( Y, D;β,η )\ =\ \1-H_Y^R( Y, D)\φ^F(0,W;β)\ +\ H_Y^R( Y, D)φ^F(1,W;β). (4.2) Proposition 4.1 (Identification with a latent dependent variable). Suppose Assumption 4.1 and the analogs of the overlap, relevance, and anchoring conditions in Section 3.1 hold. Then, for every β, [ϕYDMM(Y~,D~,β,η∗)]=[ϕF(Y∗,W,β)].E\! [ _Y^DMM( Y, D;β,η ) ]=E\! [φ^F(Y ,W;β) ]. (4.3) Consequently, β∗β is identified as the unique solution to [ϕYDMM(Y~,D~,β,η∗)]=0E[ _Y^DMM( Y, D;β,η )]=0. Proof. Equation (4.3) follows directly from equations (4.1) and (4.2) by iterated expectation. ∎ 4.1.2 Estimation and Inference Estimation follows Section 3. Within each training fold, estimate the conditional latent-class model of Y~ Y given D~ D, apply the same anchoring rule, and construct the out-of-fold bridge H^Y,iR H_Y,i^R. Define ϕ^Y,iDMM(β)=(1−H^Y,iR)ϕF(0,Wi,β)+H^Y,iRϕF(1,Wi,β), φ_Y,i^DMM(β)=(1- H_Y,i^R)φ^F(0,W_i;β)+ H_Y,i^Rφ^F(1,W_i;β), and let β^YDMM β_Y^DMM solve 1n∑i=1nϕ^Y,iDMM(β)=0. 1n _i=1^n φ_Y,i^DMM(β)=0. Theorem 4.1 (Large-sample theory for a latent dependent variable). Suppose the direct analogs of the conditions in Theorem 3.2 hold. In particular, for the same compact neighborhood ℬ0B_0, suppose that there exists a finite constant C such that supβ∈ℬ0‖ϕF(1,W,β)−ϕF(0,W,β)‖≤C _β _0 \|φ^F(1,W;β)-φ^F(0,W;β) \|≤ C a.s. Let δY,n=maxk,j,a‖η^j,aY,(−k)−ηj,a∗‖2,P. _Y,n= _k,j,a \| η_j,a^Y,(-k)-η _j,a \|_2,P. If δY,n=op(1) _Y,n=o_p(1), then β^YDMM→β∗ β_Y^DMM p→β . If, in addition, δY,n=op(n−1/4) _Y,n=o_p(n^-1/4), then n(β^YDMM−β∗)↝N(0,Y,0−1Y,0Y,0−⊤), n( β_Y^DMM-β ) N\! (0,A_Y,0^-1 _Y,0A_Y,0^- ), where Y,0 _Y,0 =−[(1−HYR(Y~,D~,η∗))∂ϕF(Y∗=0,W,β)∂β⊤|β=β∗+HYR(Y~,D~,η∗)∂ϕF(Y∗=1,W,β)∂β⊤|β=β∗], =-E\! [(1-H_Y^R( Y, D;η )) . ∂φ^F(Y =0,W;β)∂β |_β=β +H_Y^R( Y, D;η ) . ∂φ^F(Y =1,W;β)∂β |_β=β ], Y,0 _Y,0 =[ϕYDMM(Y~,D~,β∗,η∗)ϕYDMM(Y~,D~,β∗,η∗)⊤]. =E\! [ _Y^DMM( Y, D;β ,η ) _Y^DMM( Y, D;β ,η ) ]. The result follows from Proposition 3.1 and Theorem 3.2 after replacing (X∗,X~,ψa)(X , X, _a) with (Y∗,Y~,ϕa)(Y , Y, _a). The covariance matrix can be estimated by the plug-in estimator for Y,0A_Y,0 and Y,0 _Y,0. 4.2 Multicategory Labels So far we have focused on settings with a binary independent or dependent variable. In this subsection, we discuss how our method can be generalized to the multicategory case. The downstream moment argument extends directly; the main additional challenge lies in identifying a vector of class-specific bridges under stronger rank and anchoring conditions. For the sake of clear presentation, here we present our method for a multicategory dependent variable, but an analogous extension applies to the multicategory latent independent variable setting. Let Y∗∈1,…,K∗Y ∈\1,…,K_ \ denote a multicategory latent dependent variable, and suppose that researchers observe Y~=(Y(1),…,Y(J)) Y= (Y^(1),…,Y^(J) ), Y(j)∈1,…,KjY^(j)∈\1,…,K_j\. The labels need not have the same number of categories as one another or as Y∗Y . This may occur, for example, when some human or machine annotators do not use all available categories for annotation. As before, let D~=(D,W) D=(D,W) collect the downstream covariates and any auxiliary information used to model the label distribution. Define ϕa(W,β)≔ϕF(a,W,β) _a(W;β) φ^F(a,W;β) to denote the full-data moment function evaluated at Y∗=aY =a, now for a∈1,…,K∗a∈\1,…,K_ \. Thus, it admits the exact decomposition: ϕF(Y∗,W;β)=∑a=1K∗Y∗=aϕa(W;β).φ^F(Y ,W;β)= _a=1^K_ 1\Y =a\ _a(W;β). Therefore, suppose that class-specific bridges Ha(Y~,D~)H_a( Y, D) satisfy [Ha(Y~,D~)∣Y∗=b,D~]=a=b,a,b∈1,…,K∗.E\! [H_a( Y, D) Y =b, D ]= 1\a=b\, a,b∈\1,…,K_ \. Then the observed-data moment function ϕH(Y~,D~,β)=∑a=1K∗Ha(Y~,D~)ϕa(W,β)φ^H( Y, D;β)= _a=1^K_ H_a( Y, D) _a(W;β) (4.4) is unbiased for the oracle downstream full-data moment function. To construct the class-specific bridges, let =∏j=1J1,…,KjY= _j=1^J\1,…,K_j\ denote the support of the joint proxy vector. At each value D~=d D=d, define the joint measurement matrix (d)=[Pr(Y~=y~∣Y∗=a,D~=d)]y~∈,a∈1,…,K∗.S(d)= [ ( Y= y Y =a, D=d) ]_ y ,\,a∈\1,…,K_ \. Under the multicategory analog of Assumption 4.1, the observed proxy distribution is a finite mixture of product distributions. Tensor decomposition identifies the class-specific label distributions, and hence (d)S(d), up to a common permutation of the latent classes under the corresponding rank and anchoring conditions (37; 1). Suppose that (d)S(d) has full column rank, and let †(d)≔(d)⊤(d)−1(d)⊤S (d) \S(d) S(d)\^-1S(d) be a left inverse satisfying †(d)(d)=K∗S (d)S(d)=I_K_ . Writing e(y)e(y) for the one-hot encoding of the observed joint proxy vector y and eae_a for the a th standard basis vector in ℝK∗R^K_ , define Ha(y,d)=ea⊤†(d)e(y).H_a(y,d)=e_a S (d)e(y). Then, this bridge function is unbiased for the latent variable of interest. [Ha(Y~,d)∣Y∗=b,D~=d]=ea⊤†(d)(d)eb=ea⊤eb= 1a=b, \! [H_a( Y,d) Y =b, D=d ]\ =\ e_a S (d)S(d)e_b\ =\ e_a e_b\ =\ 1\a=b\, where the first equality uses the identity (d)eb=[e(Y~)∣Y∗=b,D~=d]S(d)e_b=E[e( Y) Y =b, D=d]. As in the binary case, these bridges need not lie in [0,1][0,1] or sum to one for each realized proxy pattern. Their role is to reproduce the latent class indicators in conditional expectation. To retain the same local robustness to first-stage estimation error as in the binary case, we require at least three conditionally independent measurements. In this framework, several proxies may be combined into a composite measurement when a single proxy does not have enough categories to distinguish all K∗K_ latent classes. For each proxy j, define j(d)S_j(d) analogously to (d)S(d) as the conditional measurement matrix for Y(j)Y^(j), and let j†(d)S_j (d) denote its corresponding left inverse. Let Mj,a(d)≔ea⊤j†(d)e(Y(j))M_j,a(d) e_a S_j (d)e(Y^(j)). For example, if both Y∗Y and Y(j)Y^(j) are binary, then j(d)=(1−ηj,0∗(d)1−ηj,1∗(d)ηj,0∗(d)ηj,1∗(d)),S_j(d)= pmatrix1-η _j,0(d)&1-η _j,1(d)\\ η _j,0(d)&η _j,1(d) pmatrix, and Mj,1(d)=Y(j)−ηj,0∗(d)/ηj,1∗(d)−ηj,0∗(d)M_j,1(d)=\Y^(j)-η _j,0(d)\/\η _j,1(d)-η _j,0(d)\ which coincides with the single proxy bridge defined for the binary setting. We define the robust bridge as HaR=3(J2)−1∑j1<j2Mj1,aMj2,a−2(J3)−1∑j1<j2<j3Mj1,aMj2,aMj3,a,H_a^R=3 J2^-1 _j_1<j_2M_j_1,aM_j_2,a-2 J3^-1 _j_1<j_2<j_3M_j_1,aM_j_2,aM_j_3,a, analogous to the binary setting. Under conditional independence across the J proxies, this bridge recovers Y∗=a 1\Y =a\ and retains the same first-order cancellation property as the binary robust bridge. Thus, once the vector of class-specific bridges is identified, estimation and inference proceed as in the binary case by replacing the two-term bridge-based moment with equation (4.4). The principal new requirements therefore concern identification of the measurement matrix and, consequently, the nuisance components needed to construct the bridge. For example, with three binary proxy labels, each proxy measurement matrix has Kruskal rank at most two, whereas Kruskal’s condition for K∗K_ latent classes requires the three ranks to sum to at least 2K∗+22K_ +2. Multicategory latent outcomes consequently require richer proxy supports. In addition, the anchoring rule must match all K∗K_ recovered classes to their substantive meanings. The number of response categories alone does not guarantee identification; the corresponding class-conditional measurement matrices must also have sufficiently independent columns. A simple sufficient condition is that all three matrices have full column rank K∗K_ . Specifically, if three conditionally independent label groups have measurement matrices Sj(d)S_j(d) with full column rank K∗K_ , then each has Kruskal rank K∗K_ , and Kruskal’s condition reduces to 3K∗≥2K∗+23K_ ≥ 2K_ +2, which holds for K∗≥2K_ ≥ 2. More generally, if two groups have full column rank K∗K_ , the third need only have Kruskal rank at least two. Thus, one group may have fewer than K∗K_ response patterns if the other groups provide sufficient identifying variation; for example, when K∗=4K_ =4, the group support sizes (2,4,4)(2,4,4) can satisfy the rank condition at equality when the matrices reach their maximum Kruskal ranks. In the binary construction in Section 3, we showed how the rank condition underlying classical array decomposition is related to the nonzero contrast condition required for bridge-based identification. For a binary proxy j, detj(d)=ηj,1∗(d)−ηj,0∗(d) \S_j(d)\=η _j,1(d)-η _j,0(d). Thus, the nonzero-contrast condition ηj,1∗(d)≠ηj,0∗(d)η _j,1(d)≠η _j,0(d) is exactly the condition that j(d)S_j(d) be invertible, or equivalently have Kruskal rank two. In the multicategory setting, the corresponding requirement is that its joint measurement matrix j(d)S_j(d) have full column rank. This full column rank requirement is sufficient to construct a bridge; identification of the overall latent class may hold under weaker, asymmetric Kruskal rank conditions. The multicategory analog that requires the binary contrast to be bounded away from zero is a uniform lower bound on the smallest singular value, i.e., σminj(d) _ \S_j(d)\ is bounded away from zero. 5 Designing and Assessing Conditional Independence Conditional independence is a structural assumption about the residual errors of the measurements and is not guaranteed by high predictive accuracy of each imperfect measurement. We therefore first recommend explicitly designing multiple measurements to make the assumption as plausible as possible (Section 5.1). We then propose a diagnostic tool for the conditional independence assumption (Section 5.2). We finally propose a direct test of the assumption and a bias-correction estimator in a special case when a small amount of validation data is available (Section 5.3). 5.1 Making the Conditional Independence Assumption More Plausible The first approach is to judiciously create a conditioning set. Unlike the traditional conditional independence assumption that only conditions on the latent variable itself (equation (3.1)), our method explicitly allows for conditioning on auxiliary input- or annotation-level information DiD_i and the downstream variables WiW_i. Therefore, by incorporating rich information about each annotation task, the proposed method allows for measurement errors to be dependent through task difficulty, such as text length, complexity, language, writing style, or image quality. More generally, researchers can include any pre-specified features derived from the raw input like texts, images, and videos, such as text embeddings. One useful option is to include an estimate of task difficulty constructed from the level of agreement among additional proxies that are not used in the main analysis, e.g., quantifying the disagreement between multiple LLMs and including the disagreement score in a conditioning set. Second, researchers can also make the conditional independence assumption more plausible by carefully constructing multiple measurements themselves. In particular, researchers should seek measurements with substantively different error mechanisms rather than selecting proxies solely by predictive accuracy. For LLM annotations, this may involve using different model families (e.g., one from the GPT family and another from an open-source model) as errors are more likely to be correlated within the same model family. In general, our method only requires G≥3G≥ 3 groups of proxies that are conditionally independent, but we can allow for any dependence of errors between proxies within each group. Specifically, partition the proxies into G≥3G≥ 3 groups X~[1],…,X~[G] X^[1],…, X^[G] and generalize Assumption 3.1 to the following group-level conditional independence assumption. X~[1]⟂⋯⟂X~[G]|X∗,D~, X^[1]\! \!\!\! ·s\! \!\!\! X^[G] X , D, which allows for arbitrary dependence among proxies within the same group. Each group is then treated as a single multicategory measurement whose realizations are the joint label patterns. If each group has a full-rank joint measurement matrix, one can construct a group-specific bridge and apply the robust bridge construction across groups, as in Section 4.2. This formulation permits, for example, annotations from GPT-4 and GPT-5 to form one group, annotations from Claude Opus and Sonnet to form another group, and those from an open-source model to form the third group, while allowing for any error dependence within each group. In practice, independently produced human annotations can also provide another useful measurement group even if they are not treated as gold-standard labels and their accuracy might be lower than LLMs. This is because human errors may differ systematically from those of LLMs. Importantly, human annotators should be blinded to the other annotations. Finally, another strategy is to use different prompts across annotators, as a particular pattern in one prompt might create the same error across annotators. If researchers can commit to one clean and validated prompt or a codebook, that will not introduce any correlated error. But when researchers cannot pin down one particular prompt, as in many applications, they might be able to randomly sample prompts from an admissible pool of prompts so that different annotators share fewer sources of errors. More generally, prompts may also be adapted to different types of inputs, provided that the prompt version and characteristics are recorded and included in the conditioning covariates D~ D. 5.2 Diagnostics Without Gold-Standard Labels In most applications, researchers can also conduct a diagnostic test to evaluate the observable implications of the conditional independence assumption. With exactly three binary proxies and two latent classes, the unrestricted latent-class model is generically just identified at each fixed value D~=d D=d: the observed label distribution has 23−1=72^3-1=7 degrees of freedom, equal to the 2J+1=72J+1=7 latent-class and measurement parameters. Consequently, the three-proxy model supplies no generic overidentifying restrictions. With J≥4J≥ 4 proxies, however, the number of overidentifying restrictions at each fixed d is (2J−1)−(2J+1)=2J−2J−2(2^J-1)-(2J+1)=2^J-2J-2. These restrictions can be assessed by comparing the fitted product-mixture distribution in equation (3.3), with the empirical joint label distribution (i.e., ∑iX~i=x~,D~i=d/∑iD~i=d _i 1\ X_i= x, D_i=d\/ _i 1\ D_i=d\), using a parametric bootstrap deviance or an analogous conditional moment test. When D~ D is continuous or high-dimensional, cross-fitted residual interaction moments or held-out predictive checks are more practical than cell-by-cell tests. A complementary, target-specific diagnostic exploits the fact that every informative subset of at least three valid proxies identifies the same downstream parameter. Let β^SDMM β_S^DMM denote the DMM estimate obtained from label subset S. Under the identifying assumptions, β^SDMM β_S^DMM should agree across subsets up to sampling errors. A joint bootstrap or multiplier-bootstrap test can therefore assess heterogeneity among the subset-specific estimates. As usual, this is simply a specification diagnostic rather than a direct test of conditional independence: the failure to reject the null hypothesis of no difference across the DMM estimators based on different subsets of proxies does not imply the validity of the conditional independence assumption. 5.3 Testing and Bias-Correction with Small Gold-Standard Data When the conditional independence assumption is in serious doubt, it is generally recommended to collect the gold-standard labels, even for a small amount of data, for two reasons: (a) direct testing of the conditional independence assumption and (b) bias correction without assuming the conditional independence assumption. First, a gold-standard sample permits a direct assessment of the conditional independence assumption. Let R indicate that X∗X is observed and let ρ be its known sampling probability, as in Section 2. For a subset of labels S with |S|≥2|S|≥ 2, conditional independence implies the residual moment restriction [RρX∗=ah(D~)∏j∈SX(j)−ηj,a∗(D~)]=0,a∈0,1,E [ Rρ 1\X =a\h( D) _j∈ S \X^(j)-η _j,a( D) \ ]=0, a∈\0,1\, for suitable instrument functions h. Pair and higher-order residual products can be tested jointly using cross-fitted estimates of the class-specific label means. With low-dimensional discrete covariates, the same idea can be implemented using stratified log-linear or permutation tests. Again, because a small validation sample may have limited power, failure to reject these tests should not be interpreted as establishing the conditional independence assumption. Second, a small validation sample can also be used to combine DMM and methods that use validation data, such as DSL and PPI, to directly bias-correct downstream inference. The DSL correction can be applied to any cross-fitted imputation of the full-data moment function, not only one obtained from a supervised prediction of the latent label. We therefore use the DMM moment function as an imputation in the DSL framework. Let ψ^iDMM(β) ψ_i^DMM(β) denote the cross-fitted DMM moment function, which is available for every unit, and recall that ψF(Yi,Xi∗,Wi,β)ψ^F(Y_i,X_i ,W_i;β) denotes the full-data moment function, which is observed only when Ri=1R_i=1. The resulting DMM-assisted DSL moment function is ψ^iDMM-DSL(β)=ψ^iDMM(β)+RiρiψF(Yi,Xi∗,Wi,β)−ψ^iDMM(β). ψ_i^DMM -DSL(β)= ψ_i^DMM(β)+ R_i _i \ψ^F(Y_i,X_i ,W_i;β)- ψ_i^DMM(β) \. (5.1) Thus, DMM imputes the full-data moment function for all observations, and the validation observations estimate and correct any remaining imputation error. Under the known sampling design for obtaining gold-standard labels without assuming conditional independence of multiple imperfect measurements, [ψ^iDMM-DSL(β)∣Xi∗,D~i,X~i,ℐ−k(i)]=ψF(Yi,Xi∗,Wi;β),E [ ψ_i^DMM -DSL(β) X_i , D_i, X_i,I_-k(i) ]=ψ^F(Y_i,X_i ,W_i;β), where ℐ−k(i)I_-k(i) denotes the sample used to construct the cross-fitted DMM nuisance estimates. This identity does not require the DMM conditional-independence model to be correct. The validation design therefore provides validity, while DMM serves as an imputation or control variate that can improve precision when its bridge-based moment function is informative. More generally, the DMM-assisted DSL moment can be written as the following control-variate adjustment (or power-tuning version in 4): ψ^iDMM-DSL(β,C)=RiρiψF(Yi,Xi∗,Wi,β)⏟gold-standard-only moment−C(Riρi−1)ψ^iDMM(β)⏟mean-zero control variate, ψ_i^DMM -DSL(β;C)= R_i _iψ^F(Y_i,X_i ,W_i;β)_gold-standard-only moment-C ( R_i _i-1 ) ψ_i^DMM(β)_mean-zero control variate, (5.2) where C is a scalar for a scalar moment or a conformable matrix for a vector-valued moment. The coefficient C may be estimated on separate folds to reduce the asymptotic variance. Setting C=0C=0 yields gold-standard-only estimation, whereas setting C=C=I yields equation (5.1). 6 Simulation and Empirical Validation We use simulation studies (Section 6.1) and real-world empirical validation (Section 6.2) to evaluate DMM. First, a simulation study calibrated based on an empirical study of political advertisements (25) examines a latent dependent variable under a synthetic data-generating process that satisfies the conditional independence assumption. This simulation study evaluates the finite-sample performance of the DMM estimator under the required assumption. Second, an empirical validation based on 43 examines a latent independent variable using LLM annotations. This is an empirical validation study in that we do not impose the conditional independence assumption, and we empirically evaluate whether DMM without access to the gold-standard labels can recover the oracle benchmark estimate that relies on gold-standard labels. More details about each application are provided in Appendix C. 6.1 Simulation Study: Platform Differences in Political Advertising Tone 25 compare the tone of political advertisements on Facebook and television. We use the expert-coded “Promote” indicator as the latent dependent variable and target the coefficient on the Facebook indicator in an ad-level logistic regression adjusting for party, incumbency status, and office type. We hold the observed covariates for 12,97312,973 advertisements fixed and simulate the latent outcome and proxy label models so that the resulting distributions are similar to the distribution of the original expert labels and LLM annotations. Please see Section 5 of 22 for details on these data. In each simulation run, we generate imperfect measurements independently conditional on the latent outcome and covariates so that the conditional independence assumption holds by construction and the conditional classification rate model used by DMM is correctly specified. The target Facebook coefficient is 1.3141.314, and we conduct 500500 Monte Carlo replications. To evaluate the impact of increasing the number of proxies, we rank the proxy labels by their F1 scores against the original expert labels and construct nested proxy sets by adding labels one at a time in decreasing order of F1 score. For each number of labels J, the naive estimator replaces the latent outcome with the majority vote among the J proxy labels. When J is even, ties are resolved by random draw. DMM instead combines the multiple imperfect labels using the robust bridge function for downstream inference. We compare both estimators with the infeasible oracle estimator that assumes access to gold-standard labels. We report bias, root mean squared error (RMSE), and empirical coverage of nominal 95%95\% confidence intervals. Figure 1 shows the results. As repeatedly found in the literature, ignoring measurement errors leads to substantial bias even when the individual proxy labels have high F1 scores. With the top three proxies, the naive majority-vote estimator has bias 0.1860.186, coverage 0.0180.018, and RMSE 0.1910.191. However, using the same three imperfect labels, DMM can explicitly provide debiased inference: DMM has bias 0.0020.002, coverage 0.9480.948, and RMSE 0.0630.063, compared with an RMSE of 0.0450.045 for the oracle benchmark estimate. Thus, DMM reduces RMSE by approximately two thirds relative to majority voting and restores the valid confidence interval (i.e., the empirical coverage is nearly the nominal level). Figure 1 also shows how DMM behaves as additional proxies are introduced. Across the evaluated proxy sets, DMM’s bias remains between 0.0020.002 and 0.0030.003, and its coverage remains between 0.9460.946 and 0.9620.962. By contrast, the majority-vote estimator has bias between 0.0850.085 and 0.2260.226, with coverage ranging from 00 to 0.5380.538. Even with 1515 proxy labels, majority voting has bias 0.0850.085, coverage 0.5380.538, and RMSE 0.0960.096, while DMM has bias 0.0020.002, coverage 0.9460.946, and RMSE 0.0470.047. Figure 1: Simulation comparing DMM with naive majority voting as the number of proxy labels increases. Panels (a)–(c) report bias, RMSE, and 95%95\% confidence-interval coverage over 500500 replications; Panel (d) reports proxy F1 scores. Dashed lines mark the oracle benchmarks. Additional informative labels also improve the precision of DMM. Its RMSE falls from 0.0630.063 with three labels to 0.0530.053, 0.0480.048, and 0.0470.047 with the top 5, 7, and 10 labels, respectively, and reaches 0.0460.046 with 13 labels, close to the oracle RMSE of 0.0450.045. These gains eventually flatten as progressively weaker labels are added. Thus, additional sufficiently informative labels can narrow the precision gap between DMM and the oracle infeasible benchmark estimator that observes the latent variable for all units, but not every additional label is beneficial. 6.2 Empirical Validation: Accusations of Wrongdoing in China 43 study whether Chinese officials conceal online complaints that accuse local officials of wrongdoing. We treat the expert-coded indicator of prefecture-level wrongdoing as the latent independent variable and upward reporting as the downstream dependent variable. The main estimand is the coefficient on prefecture-level wrongdoing in a logistic regression of upward reporting on the latent indicator and a subset of the covariates (77 binary and 11 continuous) used in the original application. The analysis contains 1,4121,412 complaints and uses the LLM annotations as proxy labels. We use three LLM labels in this study: GPT-4.1 5-shot, GPT-4 5-shot, and Llama-4 0-shot. We compare (a) the DMM estimator using the robust bridge without access to the gold-standard labels, (b) three naive regressions that replace the latent variable with one proxy at a time, and (c) DSL using a simple random sample of 500500 gold-standard labels (35%35\% of the sample). We evaluate them against the oracle logistic regression that uses the gold-standard labels for all units. Importantly, in this empirical validation, we use the real-world data and do not simulate any data. Therefore, this is a realistic evaluation of the DMM estimator because we do not know whether the conditional independence assumption required in the DMM estimator holds, as in the real-world empirical application. Panel (a) of Figure 2 reports the point estimates and 95%95\% confidence intervals. The gold-standard benchmark for the prefecture-wrongdoing coefficient is −1.039-1.039. Despite their similarly high reported class-weighted F1 scores, the three naive estimates range from −1.869-1.869 to −0.388-0.388 and all of them are biased. DSL and DMM estimate the coefficient to be −1.279-1.279 and −0.844-0.844, respectively, and both confidence intervals contain the benchmark estimate. Importantly, unlike DSL, DMM obtains its estimate without using any gold-standard labels. Figure 2: 43 empirical estimates and coverage. Panel (a) reports coefficient estimates and 95% confidence intervals; the dashed line marks the full-sample expert-label benchmark. DMM uses no gold-standard labels, whereas DSL uses 500500. Panel (b) reports the proportion of intervals covering the fixed expert benchmark over 500500 bootstrap resamples. Panel (b) reports empirical coverage based on 500500 resamples. Specifically, it shows the proportion of 95%95\% confidence intervals that contain the benchmark estimate. The three naive estimators have coverage rates ranging from 0.1540.154 to 0.5720.572, whereas DSL and DMM have coverage rates of 0.9810.981 and 0.8620.862, respectively. 7 Discussion In this paper, we proposed debiased inference with multiple imperfect measurements (DMM), a framework to conduct valid downstream inference without gold-standard labels. Under the conditional independence assumption given the latent variable and any observed features of annotation tasks, such as annotation difficulty, captured by text embeddings, DMM identifies the conditional classification rate model required for valid downstream inference. Importantly, individual measurements may remain systematically biased, differ substantially in accuracy, and exhibit error rates that vary across units. By allowing the (mis)classification rate model to depend on input- and annotation-level information, our framework makes the conditional independence assumption more plausible and its substantive implications more transparent. We also provide practical guidance for designing measurement procedures that support this assumption and for assessing its plausibility in applied settings. Our main methodological contribution is to connect the multiple measurement model to general downstream analyses covering both settings in which the latent variable is either an independent or a dependent variable. We construct a robust bridge whose conditional expectation recovers the latent indicator and use it to reproduce the oracle full-data moment function. By carefully designing how the multiple measurements are combined within the bridge, we show that the resulting construction is robust to misspecification of any one label-specific nuisance model. Combined with cross-fitting, this yields a consistent and asymptotically normal estimator under standard product-rate conditions, allowing the nuisance estimators to converge at slow nonparametric rates n−1/4n^-1/4 while preserving root-n inference for the downstream parameter. We also discuss extensions to multicategory variables under the corresponding rank conditions. Our method accommodates any number of imperfect measurements, from as few as three to much larger collections, allowing researchers to adapt the measurement design to different annotation budgets and settings. The framework also clarifies how the number and quality of the available measurements affect statistical precision. Adding sufficiently informative measurements can reduce the measurement-induced component of the asymptotic variance, bringing DMM closer in efficiency to the oracle estimator. However, precision does not improve mechanically with the number of measurements: low-quality measurements can yield unstable bridges and offset the gains from averaging over a larger collection. We present simulation and empirical validation studies to demonstrate the performance of DMM. Using Monte Carlo simulations calibrated to data from 25, we show that DMM exhibits small bias and achieves nominal coverage across different numbers of labels, whereas naive estimators directly using imperfect labels remain biased and exhibit undercoverage even when individual proxies have high F1 scores. The empirical validation study based on 43 provides a complementary illustration using real LLM annotations. DMM produces an estimate and confidence interval close to the oracle benchmark without using expert labels in settings where the conditional independence assumption is not guaranteed to hold, as in many applications. The main limitation of DMM is that it requires the conditional independence assumption about measurement errors, even though we made it much more plausible than the classical counterpart by allowing for a rich conditioning set. Conditional independence is not guaranteed by high classification accuracy and may be violated when annotations interact with omitted features of the input. We therefore devote substantial attention to how these concerns can be mitigated and assessed in practice. In Section 5, we discuss designing measurement systems with diverse annotators (either human or machine), conditioning on input- and annotation-level information that may explain common errors, and examining subset stability and other observable implications of the fitted measurement model. When a limited validation sample is available, it can additionally be used to assess the conditional independence assumption and to combine DMM with the design-based bias correction of DSL. 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For each fold k, let ℐ−k=1,…,n∖ℐkI_-k=\1,…,n\ _k denote the corresponding training indices and let nk=|ℐk|n_k=|I_k|, where Oi=(Yi,X~i,D~i)O_i=(Y_i, X_i, D_i). Nuisance estimates indexed by (−k)(-k) are fitted using observations indexed by ℐ−kI_-k. We take the number of folds and the fold proportions to be fixed. For any measurable function f, write Pf=f(O),ℙn,kf=1nk∑i∈ℐkf(Oi),ℙnf=1n∑i=1nf(Oi).Pf=E\f(O)\, _n,kf= 1n_k _i _kf(O_i), _nf= 1n _i=1^nf(O_i). For a generic nuisance collection η≔ηj,0(⋅),ηj,1(⋅)j=1Jη \ _j,0(·), _j,1(·)\_j=1^J, let ψDMM(O,β,η)=1−HR(X~,D~,η)ψ0(Y,W,β)+HR(X~,D~,η)ψ1(Y,W,β)ψ^DMM(O;β,η)\ =\ \1-H^R( X, D;η)\ _0(Y,W;β)\ +\ H^R( X, D;η) _1(Y,W;β) where HR(X~,D~,η∗)=3H¯2(X~,D~,η∗)−2H¯3(X~,D~,η∗)H^R( X, D;η )=3 H_2( X, D;η )-2 H_3( X, D;η ) and ψa(Y,W,β)≔ψF(Y,X∗=a,W,β). _a(Y,W;β) ψ^F(Y,X =a,W;β). The fold-specific fitted objects are denoted by M^j,−k=Mj(D~,η^−k) M_j,-k=M_j( D; η_-k) and H^−kR=HR(X~,D~,η^−k) H_-k^R=H^R( X, D; η_-k), whereas Mj,0=Mj(D~,η∗)M_j,0=M_j( D;η ) and H0R=HR(X~,D~,η∗)H_0^R=H^R( X, D;η ) denote their population counterparts. A.1 Proof of Proposition 3.1: Bounds for the Robust Bridge For a∈0,1a∈\0,1\, define the fold-specific single-bridge error ej,a(−k)(D~)=[M^j,−k∣X∗=a,D~,ℐ−k]−a,e_j,a^(-k)( D)=E\! [ M_j,-k X =a, D,I_-k ]-a, and the corresponding robust bridge error Ba,−kR(D~)=[H^−kR∣X∗=a,D~,ℐ−k]−a.B_a,-k^R( D)=E\! [ H_-k^R X =a, D,I_-k ]-a. Lemma A.1 (Foldwise robust bridge error). Suppose Assumption 3.1 holds, J is fixed, and the true and fitted class contrasts, ηj,1∗(D~)−ηj,0∗(D~)j=1J\η _j,1( D)-η _j,0( D)\_j=1^J and η^j,1(−k)(D~)−η^j,0(−k)(D~)j=1Jk=1K\\ η^(-k)_j,1( D)- η^(-k)_j,0( D)\_j=1^J\_k=1^K , are uniformly bounded away from zero. Then, for every fold k, conditional on ℐ−kI_-k, B1,−kR(D~)= B_1,-k^R( D)= −3(J2)−1∑j1<j2ej1,1(−k)(D~)ej2,1(−k)(D~) -3 J2^-1 _j_1<j_2e_j_1,1^(-k)( D)e_j_2,1^(-k)( D) −2(J3)−1∑j1<j2<j3ej1,1(−k)(D~)ej2,1(−k)(D~)ej3,1(−k)(D~), -2 J3^-1 _j_1<j_2<j_3e_j_1,1^(-k)( D)e_j_2,1^(-k)( D)e_j_3,1^(-k)( D), (A.1) B0,−kR(D~)= B_0,-k^R( D)= 3(J2)−1∑j1<j2ej1,0(−k)(D~)ej2,0(−k)(D~) 3 J2^-1 _j_1<j_2e_j_1,0^(-k)( D)e_j_2,0^(-k)( D) −2(J3)−1∑j1<j2<j3ej1,0(−k)(D~)ej2,0(−k)(D~)ej3,0(−k)(D~). -2 J3^-1 _j_1<j_2<j_3e_j_1,0^(-k)( D)e_j_2,0^(-k)( D)e_j_3,0^(-k)( D). (A.2) Moreover, for a finite constant C that does not depend on n or k, maxk,j,a‖ej,a(−k)‖2,P _k,j,a \|e_j,a^(-k) \|_2,P ≤Cδn, ≤ C _n, (A.3) maxk‖H^−kR−H0R‖2,P _k \| H_-k^R-H_0^R \|_2,P ≤Cδn, ≤ C _n, (A.4) maxk,aP|Ba,−kR| _k,aP |B_a,-k^R | ≤Cδn2, ≤ C _n^2, (A.5) where δn=maxk,j,a‖η^j,a(−k)−ηj,a∗‖2,P. _n= _k,j,a \| η_j,a^(-k)- _j,a \|_2,P. Consequently, the conditional bridge error has no term that is linear in a single first-stage error. If all proxy-specific nuisance functions are correctly specified except possibly those for one proxy, both B1,−kRB_1,-k^R and B0,−kRB_0,-k^R are exactly zero. Proof. Because the fold-specific nuisance estimates are fixed conditional on ℐ−kI_-k, for a∈0,1a∈\0,1\, [M^j,−k∣X∗=a,D~,ℐ−k]=ηj,a∗(D~)−η^j,0(−k)(D~)η^j,1(−k)(D~)−η^j,0(−k)(D~) \! [ M_j,-k X =a, D,I_-k ]\ =\ η _j,a( D)- η_j,0^(-k)( D) η_j,1^(-k)( D)- η_j,0^(-k)( D) where ηj,a∗(d)≔Pr(X(j)=1∣X∗=a,D~=d).η _j,a(d) (X^(j)=1 X =a, D=d). Subtracting a yields ej,a(−k)(D~)=ηj,a∗(D~)−η^j,a(−k)(D~)η^j,1(−k)(D~)−η^j,0(−k)(D~).e_j,a^(-k)( D)= η _j,a( D)- η_j,a^(-k)( D) η_j,1^(-k)( D)- η_j,0^(-k)( D). Because the absolute value of the denominator is uniformly bounded away from zero, |ej,a(−k)(D~)|≤C|η^j,a(−k)(D~)−ηj,a∗(D~)|, |e_j,a^(-k)( D) |≤ C | η_j,a^(-k)( D)-η _j,a( D) |, which proves equation (A.3) after taking L2(P)L_2(P) norms. Assumption 3.1 implies that, conditional on (X∗,D~,ℐ−k)(X , D,I_-k), the fitted single-proxy bridges are mutually independent. When X∗=1X =1, [H¯^2,−k∣X∗=1,D~,ℐ−k]=(J2)−1∑j1<j21+ej1,1(−k)1+ej2,1(−k), \! [ H_2,-k X =1, D,I_-k ]= J2^-1 _j_1<j_2\1+e_j_1,1^(-k)\\1+e_j_2,1^(-k)\, [H¯^3,−k∣X∗=1,D~,ℐ−k]=(J3)−1∑j1<j2<j31+ej1,1(−k)1+ej2,1(−k)1+ej3,1(−k), \! [ H_3,-k X =1, D,I_-k ]= J3^-1 _j_1<j_2<j_3\1+e_j_1,1^(-k)\\1+e_j_2,1^(-k)\\1+e_j_3,1^(-k)\, where the arguments D~ D for e(⋅)e(·) are suppressed for readability. Each ej,1(−k)e_j,1^(-k) appears in J−1J-1 pairs and (J−12) J-12 triples, so the average linear terms are, respectively, 2J∑j=1Jej,1(−k)and3J∑j=1Jej,1(−k). 2J _j=1^Je_j,1^(-k) 3J _j=1^Je_j,1^(-k). Their coefficient in 3H¯^2,−k−2H¯^3,−k3 H_2,-k-2 H_3,-k is zero because 3(2/J)−2(3/J)=03(2/J)-2(3/J)=0. Each pair appears in J−2J-2 triples and (J−2)(J3)−1=3(J2)−1.(J-2) J3^-1=3 J2^-1. The coefficient on each pair product is therefore 3(J2)−1−23(J2)−1=−3(J2)−13 J2^-1-2\3 J2^-1\=-3 J2^-1, while each triple product has coefficient −2(J3)−1-2 J3^-1. This proves equation (A.1). When X∗=0X =0, the conditional mean of M^j,−k M_j,-k is ej,0(−k)e_j,0^(-k), so the pair and triple products directly give equation (A.2). By equation (A.3), Cauchy–Schwarz yields P|ej1,a(−k)ej2,a(−k)|≤‖ej1,a(−k)‖2,P‖ej2,a(−k)‖2,P≤Cδn2.P |e_j_1,a^(-k)e_j_2,a^(-k) |≤ \|e_j_1,a^(-k) \|_2,P \|e_j_2,a^(-k) \|_2,P≤ C _n^2. Because the fitted class contrasts are uniformly bounded away from zero, the single-bridge errors are uniformly bounded, so each cubic term is bounded by a constant times one of its pair products. Since J is fixed, summing the finite number of terms proves equation (A.5). It remains to prove equation (A.4). Write Δj(D~)=ηj,1∗(D~)−ηj,0∗(D~). _j( D)=η _j,1( D)-η _j,0( D). The difference between the fitted and true single-label bridges satisfies the following equality. M^j,−k−Mj,0=ηj,0∗−η^j,0(−k)η^j,1(−k)−η^j,0(−k)+X(j)−ηj,0∗(ηj,1∗−η^j,1(−k))−(ηj,0∗−η^j,0(−k))(η^j,1(−k)−η^j,0(−k))(ηj,1∗−ηj,0∗), M_j,-k-M_j,0\ =\ η _j,0- η_j,0^(-k) η_j,1^(-k)- η_j,0^(-k)\ +\ \X^(j)-η _j,0\ (η _j,1- η_j,1^(-k) )- (η _j,0- η_j,0^(-k) ) ( η_j,1^(-k)- η_j,0^(-k) ) (η _j,1-η _j,0 ), where all nuisance functions are evaluated at D~ D. Because X(j)X^(j) and the proxy means lie in [0,1][0,1] and both contrasts are bounded away from zero, |M^j,−k−Mj,0|≤Cmaxa∈0,1|η^j,a(−k)−ηj,a∗|. | M_j,-k-M_j,0 |≤ C _a∈\0,1\ | η_j,a^(-k)-η _j,a |. Hence maxk,j‖M^j,−k−Mj,0‖2,P≤Cδn. _k,j \| M_j,-k-M_j,0 \|_2,P≤ C _n. For any S=j1,…,jmS=\j_1,…,j_m\ with m∈2,3m∈\2,3\, inserting one fitted factor at a time gives ∏t=1mM^jt,−k−∏t=1mMjt,0=∑t=1m(∏s<tM^js,−k)(M^jt,−k−Mjt,0)(∏s>tMjs,0). _t=1^m M_j_t,-k- _t=1^mM_j_t,0= _t=1^m ( _s<t M_j_s,-k )( M_j_t,-k-M_j_t,0) ( _s>tM_j_s,0 ). Because the true and fitted class contrasts are uniformly bounded away from zero, all factors multiplying each difference are uniformly bounded. Taking L2(P)L_2(P) norms, applying the triangle inequality, and averaging the finitely many pair and triple products proves equation (A.4). ∎ Proof of Proposition 3.1. The conditional error identities follow from Lemma A.1. Applying the lemma at the true nuisance functions gives ej,a(−k)=0e_j,a^(-k)=0 for every j and a∈0,1a∈\0,1\, yielding the unbiasedness of the bridge function. The displayed identities contain only products involving errors from at least two distinct labels and therefore contain no term that is linear in a single first-stage error. Moreover, if all proxy-specific nuisance functions are correctly specified except possibly those associated with one proxy, every pair and triple product of the errors vanishes, yielding the stated multi-proxy robustness. ∎ A.2 Proof of Theorem 3.2: Second-Order Remainder for the DMM Moment Function and Large-Sample Theory Let ℬ0⊂ℬB_0 be the compact neighborhood of β∗β in Theorem 3.2. Under the bounded-envelope condition in Theorem 3.2, there exists a finite constant C such that supβ∈ℬ0‖ψF(Y,1,W,β)−ψF(Y,0,W,β)‖≤Ca.s. _β _0 \|ψ^F(Y,1,W;β)-ψ^F(Y,0,W;β) \|≤ C .s. The remaining smoothness and envelope conditions are those stated in Theorem 3.2. For each fold, define the conditional population remainder Rn,k(β)=[ψDMM(O,β,η^−k)−ψDMM(O,β,η∗)∣Oi:i∈ℐ−k]R_n,k(β)=E\! [ψ^DMM(O;β, η_-k)-ψ^DMM(O;β,η ) \O_i:i _-k\ ] where O=(Y,X~,D~)O=(Y, X, D), and its fold-weighted average R¯n(β)=∑k=1KnknRn,k(β). R_n(β)= _k=1^K n_knR_n,k(β). Lemma A.2 (Second-order remainder of the DMM moment function). Under the conditions of Lemma A.1 and the envelope conditions above, for every fold k and β∈ℬ0β _0, Rn,k(β)=[X∗B1,−kR(D~)+(1−X∗)B0,−kR(D~)×Δψ(Y,W,β)|ℐ−k]. R_n,k(β)=E [\X B_1,-k^R( D)+(1-X )B_0,-k^R( D)\\ ×\ ψ(Y,W;β)\; |\;I_-k ]. (A.6) Consequently, maxsupβ∈ℬ01≤k≤K‖Rn,k(β)‖2≤Cδn2, _1≤ k≤ K _β _0 \|R_n,k(β) \|_2≤ C _n^2, (A.7) and the same bound holds with Rn,kR_n,k replaced by R¯n R_n. Proof. The difference between the fitted and population estimating functions has the exact form ψDMM(O,β,η^−k)−ψDMM(O,β,η∗)=(H^−kR−H0R)Δψ(Y,W,β) ψ^DMM(O;β, η_-k)-ψ^DMM(O;β,η )\ =\ ( H_-k^R-H_0^R) ψ(Y,W;β) where H^−kR≔HR(X~,D~,η^−k)=3(J2)−1∑j1<j2M^j1,−kM^j2,−k−2(J3)−1∑j1<j2<j3M^j1,−kM^j2,−kM^j3,−k, H_-k^R H^R( X, D; η_-k)=3 J2^-1 _j_1<j_2 M_j_1,-k M_j_2,-k-2 J3^-1 _j_1<j_2<j_3 M_j_1,-k M_j_2,-k M_j_3,-k, and H0R≔HR(X~,D~,η∗)H_0^R H^R( X, D;η ) is defined analogously by replacing M^j,−k M_j,-k with Mj,0M_j,0. These are the robust bridges evaluated at the fold-specific estimated and true nuisance functions, respectively. Conditioning on (X∗,D~,ℐ−k)(X , D,I_-k) and using [H0R∣X∗,D~]=X∗E[H_0^R X , D]=X gives [H^−kR−H0R∣X∗,D~,ℐ−k]=X∗B1,−kR(D~)+(1−X∗)B0,−kR(D~).E\! [ H_-k^R-H_0^R X , D,I_-k ]=X B_1,-k^R( D)+(1-X )B_0,-k^R( D). Substitution proves equation (A.6). By that identity and the bounded moment-contrast condition, supβ∈ℬ0‖Rn,k(β)‖2 _β _0 \|R_n,k(β) \|_2 ≤[|B1,−kR(D~)|+|B0,−kR(D~)|supβ∈ℬ0‖Δψ(Y,W,β)‖|ℐ−k] \! [ \ |B_1,-k^R( D) |+ |B_0,-k^R( D) | \ _β _0 \| ψ(Y,W;β) \|\, |\,I_-k ] ≤C∑a=01P|Ba,−kR|≤Cδn2, ≤ C _a=0^1P |B_a,-k^R |≤ C _n^2, where the last inequality follows from equation (A.5). This proves equation (A.7). Because K is fixed and the weights nk/n_k/n sum to one, the same rate holds for R¯n R_n. ∎ Proof of Theorem 3.2. Define the cross-fitted sample moment and its population counterpart by Ψ^n(β)=∑k=1Knknℙn,kψDMM(O,β,η^−k),Ψ(β)=PψDMM(O,β,η∗). _n(β)= _k=1^K n_knP_n,kψ^DMM(O;β, η_-k), (β)=Pψ^DMM(O;β,η ). By Theorem 3.1, Ψ(β∗)=0 (β )=0 and β∗β is its unique root. For every β∈ℬ0β _0, Ψ^n(β)−Ψ(β)= _n(β)- (β)= ∑k=1Knkn(ℙn,k−P)ψDMM(O,β,η^−k)−ψDMM(O,β,η∗) _k=1^K n_kn(P_n,k-P) \ψ^DMM(O;β, η_-k)-ψ^DMM(O;β,η ) \ +(ℙn−P)ψDMM(O,β,η∗)+R¯n(β). +(P_n-P)ψ^DMM(O;β,η )+ R_n(β). (A.8) Consistency. We first establish consistency. We begin with the first term in equation (A.8). Conditional on ℐ−kI_-k, the observations in ℐkI_k are independent of η^−k η_-k. By the exact moment-difference identity, the bounded moment-contrast condition, and equation (A.4), maxsupβ∈ℬ0k‖ψDMM(O,β,η^−k)−ψDMM(O,β,η∗)‖2,P _k _β _0 \|ψ^DMM(O;β, η_-k)-ψ^DMM(O;β,η ) \|_2,P =maxsupβ∈ℬ0k‖(H^−kR−H0R)Δψ(Y,W,β)‖2,P = _k _β _0 \|( H_-k^R-H_0^R) ψ(Y,W;β) \|_2,P ≤Cmaxk‖H^−kR−H0R‖2,P≤Cδn=op(1). ≤ C _k \| H_-k^R-H_0^R \|_2,P≤ C _n=o_p(1). For any r>0r>0, let β1,…,βN\ _1,…, _N\ be a finite r-net of ℬ0B_0. For each fold k and ϵ>0ε>0, conditional Chebyshev’s inequality and a union bound give Pr[max1≤ℓ≤N‖(ℙn,k−P)ψDMM(O,βℓ,η^−k)−ψDMM(O,βℓ,η∗)‖2>ϵ|ℐ−k] [ _1≤ ≤ N \|(P_n,k-P) \ψ^DMM(O; _ , η_-k)-ψ^DMM(O; _ ,η ) \ \|_2>ε\, |\,I_-k ] ≤1nkϵ2∑ℓ=1N‖ψDMM(O,βℓ,η^−k)−ψDMM(O,βℓ,η∗)‖2,P2=op(1). ≤ 1n_kε^2 _ =1^N \|ψ^DMM(O; _ , η_-k)-ψ^DMM(O; _ ,η ) \|_2,P^2=o_p(1). Because K is fixed, this also holds uniformly over k. Moreover, class-contrast separation uniformly bounds the bridge difference, and the Lipschitz-envelope condition for the moment contrast implies maxsupβ∈ℬ0k‖(ℙn,k−P)ψDMM(O,β,η^−k)−ψDMM(O,β,η∗)‖2 _k _β _0 \|(P_n,k-P) \ψ^DMM(O;β, η_-k)-ψ^DMM(O;β,η ) \ \|_2 ≤max1≤k≤K1≤ℓ≤N‖(ℙn,k−P)ψDMM(O,βℓ,η^−k)−ψDMM(O,βℓ,η∗)‖2+Op(r) ≤ _ subarrayc1≤ k≤ K\\ 1≤ ≤ N subarray \|(P_n,k-P) \ψ^DMM(O; _ , η_-k)-ψ^DMM(O; _ ,η ) \ \|_2+O_p(r) =op(1)+Op(r). =o_p(1)+O_p(r). Letting n→∞n→∞ and then r↓0r 0 yields maxsupβ∈ℬ0k‖(ℙn,k−P)ψDMM(O,β,η^−k)−ψDMM(O,β,η∗)‖2=op(1). _k _β _0 \|(P_n,k-P) \ψ^DMM(O;β, η_-k)-ψ^DMM(O;β,η ) \ \|_2=o_p(1). Since the weights nk/n_k/n sum to one, the first term in equation (A.8) is therefore uniformly op(1)o_p(1). Now, we consider the second term in equation (A.8). The oracle moment function class is finite-dimensional and has an integrable Lipschitz envelope, so a uniform law of large numbers makes the second term uniformly op(1)o_p(1). Finally, we consider the third term in equation (A.8). Lemma A.2 and δn=op(1) _n=o_p(1) make the third term uniformly op(1)o_p(1). Therefore, supβ∈ℬ0‖Ψ^n(β)−Ψ(β)‖2=op(1). _β _0 \| _n(β)- (β) \|_2=o_p(1). Because β∗β is a well-separated root and Ψ^n(β^DMM)=0 _n( β^DMM)=0, the standard Z-estimation argument yields β^DMM→β∗ β^DMM p→β . The robustness claim follows similarly. If the nuisance functions for all but one proxy are consistently estimated, every pair or triple in the conditional bridge-error formulas contains at least one op(1)o_p(1) factor, while the other factors remain bounded under the regularity condition. Hence, the remainder term is uniformly op(1)o_p(1). For the centered empirical term, equation (A.4) need not hold when one proxy-specific nuisance limit is misspecified. Nevertheless, class-contrast separation uniformly bounds both the fitted and true bridges, and the downstream moment contrast is uniformly bounded. Conditional on the training sample, the corresponding centered foldwise average is therefore Op(n−1/2)O_p(n^-1/2), and hence op(1)o_p(1). The same Z-estimation argument yields consistency even if one proxy-specific nuisance model is misspecified. Asymptotic Normality. We next establish asymptotic normality. At β∗β , Ψ^n(β∗)=(ℙn−P)ψDMM(O,β∗,η∗)+∑k=1Knkn(ℙn,k−P)ψDMM(O,β∗,η^−k)−ψDMM(O,β∗,η∗)+R¯n(β∗) _n(β )\ =\ (P_n-P)ψ^DMM(O;β ,η )+ _k=1^K n_kn(P_n,k-P) \ψ^DMM(O;β , η_-k)-ψ^DMM(O;β ,η ) \+ R_n(β ) (A.9) because PψDMM(O,β∗,η∗)=0Pψ^DMM(O;β ,η )=0. Conditional on the training observations Oi:i∈ℐ−k\O_i:i _-k\, [‖nk(ℙn,k−P)ψDMM(O,β∗,η^−k)−ψDMM(O,β∗,η∗)‖22|Oi:i∈ℐ−k] \! [ \| n_k(P_n,k-P) \ψ^DMM(O;β , η_-k)-ψ^DMM(O;β ,η ) \ \|_2^2\; |\;\O_i:i _-k\ ] ≤C‖ψDMM(O,β∗,η^−k)−ψDMM(O,β∗,η∗)‖2,P2 ≤ C \|ψ^DMM(O;β , η_-k)-ψ^DMM(O;β ,η ) \|_2,P^2 ≤C‖H^−kR−H0R‖2,P2≤Cδn2=op(1), ≤ C \| H_-k^R-H_0^R \|_2,P^2≤ C _n^2=o_p(1), where the second inequality uses the exact moment-difference identity and the bounded moment-contrast condition, and the third inequality follows from equation (A.4). Conditional Markov’s inequality and fixed K therefore imply that the second term in equation (A.9) is op(n−1/2)o_p(n^-1/2). Lemma A.2 and δn2=op(n−1/2) _n^2=o_p(n^-1/2) imply that the third term is also op(n−1/2)o_p(n^-1/2). Hence Ψ^n(β∗)=(ℙn−P)ψDMM(O;β∗,η∗)+op(n−1/2). _n(β )=(P_n-P)ψ^DMM(O;β ,η )+o_p(n^-1/2). (A.10) The integral form of the mean-value expansion gives 0= 0= Ψ^n(β∗) _n(β ) +[∫01∑k=1Knknℙn,k1−H^−kR∂β⊤ψF(Y,0,W;β∗+t(β^DMM−β∗)) + [ _0^1 _k=1^K n_knP_n,k \\1- H_-k^R\ ∂β ψ^F\! (Y,0,W;β +t( β^DMM-β ) ) +H^−kR∂β⊤ψF(Y,1,W;β∗+t(β^DMM−β∗))dt](β^DMM−β∗). 156.49014pt+ H_-k^R ∂β ψ^F\! (Y,1,W;β +t( β^DMM-β ) ) \\,dt ]( β^DMM-β ). Consistency, the uniform law of large numbers, smoothness of the moment functions, and equation (A.4) imply that the matrix in brackets converges in probability to −0-A_0. Since 0A_0 is nonsingular, combining this result with equation (A.10) yields n(β^DMM−β∗)=0−11n∑i=1nψDMM(Oi,β∗,η∗)+op(1), n( β^DMM-β )=A_0^-1 1 n _i=1^nψ^DMM(O_i;β ,η )+o_p(1), which is equation (3.11). The multivariate central limit theorem gives 1n∑i=1nψDMM(Oi,β∗,η∗)↝N(0,0), 1 n _i=1^nψ^DMM(O_i;β ,η ) N(0, _0), and Slutsky’s theorem proves equation (3.12). Finally, consistency and equation (A.4) imply that the cross-fitted moment function evaluated at β^DMM β^DMM converges in L2(P)L_2(P) to ψDMM(O,β∗,η∗)ψ^DMM(O;β ,η ). Foldwise laws of large numbers then give ^→0 p→ _0 and ^→0 A p→A_0. Therefore, ^→0−100−⊤. V p→A_0^-1 _0A_0^- . ∎ Appendix B Details for the Efficiency Discussion Throughout this section, write H0,JRH_0,J^R when emphasizing that the robust bridge uses J proxies. Equation (3.16) gives ψJDMM(X~,D~,β∗,η∗)=ψ0F+H0,JR−X∗Δψ0, _J^DMM( X, D;β ,η )= _0^F+\H_0,J^R-X \ _0, where ψ0F _0^F is a shorthand for the oracle full-data moment function evaluated at the true parameter. The unbiasedness of the bridge function [H0,JR∣X∗,D~]=X∗E [H_0,J^R X , D ]=X implies equation (3.17). Expanding the product defining the meat matrix yields 0,JR= _0,J^R= [ψ0Fψ0F⊤]+[(H0,JR−X∗)ψ0FΔψ0⊤] [ _0^F _0^F ]+E [(H_0,J^R-X ) _0^F _0 ] +[(H0,JR−X∗)Δψ0ψ0F⊤] +E [(H_0,J^R-X ) _0 _0^F ] +[(H0,JR−X∗)2Δψ0Δψ0⊤]. +E [(H_0,J^R-X )^2 _0 _0 ]. The two cross terms vanish after conditioning on (X∗,D~)(X , D), which proves equation (3.18). We may write [(H0,JR−X∗)2Δψ0Δψ0⊤] [(H_0,J^R-X )^2 _0 _0 ] =[[(H0,JR−X∗)2∣X∗,D~]Δψ0Δψ0⊤] =E [E[(H_0,J^R-X )^2 X , D] _0 _0 ] =[∑a=01Pr(X∗=a∣D~)[(H0,JR−a)2∣X∗=a,D~]Δψ0Δψ0⊤] =E [ _a=0^1 (X =a D)E[(H_0,J^R-a)^2 X =a, D] _0 _0 ] =[∑a=01π(D~i)a1−π(D~i)1−ava,JR(D~)Δψ0Δψ0⊤]. =E [ _a=0^1π( D_i)^a\1-π( D_i)\^1-av_a,J^R( D) _0 _0 ]. Premultiplying and postmultiplying by 0−1A_0^-1 and 0−⊤cA_0^- c, respectively, gives equation (3.19). Define the centered single-proxy bridge noise εj=Mj(D~,η)−X∗. _j=M_j( D;η)-X . Conditional on (X∗,D~)=(a,d)(X , D)=(a,d), the variables ε1,…,εJ _1,…, _J are mutually independent and satisfy [εj∣X∗=a,D~=d]=0E[ _j X =a, D=d]=0, [εj2∣X∗=a,D~=d]=κj,a(d)E[ _j^2 X =a, D=d]= _j,a(d). Direct expansion of the pair and triple averages gives H0,JR−X∗=3(1−2X∗)(J2)∑j1<j2εj1εj2−2(J3)∑j1<j2<j3εj1εj2εj3.H_0,J^R-X = 3(1-2X ) J2 _j_1<j_2 _j_1 _j_2- 2 J3 _j_1<j_2<j_3 _j_1 _j_2 _j_3. (B.1) For any nonempty subset S⊆1,…,JS \1,…,J\, let εS=∏j∈Sεj _S= _j∈ S _j. If S≠TS≠ T, then [εSεT∣X∗=a,D~=d] [ _S _T X =a, D=d ] =∏j∈S∩T[εj2∣X∗=a,D~=d]∏j∈(S∖T)∪(T∖S)[εj∣X∗=a,D~=d]=0, = _j∈ S∩ TE [ _j^2 X =a, D=d ] _j∈(S T)∪(T S)E [ _j X =a, D=d ]=0, because (S∖T)∪(T∖S)(S T)∪(T S) is nonempty. Thus, distinct pair and triple monomials are conditionally orthogonal: all pair–pair, triple–triple, and pair–triple covariance terms vanish. For S=TS=T, [εS2∣X∗=a,D~=d]=∏j∈Sκj,a(d)E [ _S^2 X =a, D=d ]= _j∈ S _j,a(d). Squaring equation (B.1) and taking conditional expectations therefore yields va,JR(d)=9(J2)−2∑j1<j2κj1,a(d)κj2,a(d)+4(J3)−2∑j1<j2<j3κj1,a(d)κj2,a(d)κj3,a(d),v_a,J^R(d)=9 J2^-2 _j_1<j_2 _j_1,a(d) _j_2,a(d)+4 J3^-2 _j_1<j_2<j_3 _j_1,a(d) _j_2,a(d) _j_3,a(d), which is equation (3.20). Our main question is: when does an additional label improve precision? Define the elementary symmetric sums S1,a,J(d) S_1,a,J(d) =∑j1=1Jκj1,a(d), = _j_1=1^J _j_1,a(d), S2,a,J(d) S_2,a,J(d) =∑j1<j2κj1,a(d)κj2,a(d), = _j_1<j_2 _j_1,a(d) _j_2,a(d), S3,a,J(d) S_3,a,J(d) =∑j1<j2<j3κj1,a(d)κj2,a(d)κj3,a(d). = _j_1<j_2<j_3 _j_1,a(d) _j_2,a(d) _j_3,a(d). Suppose a new label has conditional bridge variance κJ+1,a(d) _J+1,a(d). The elementary symmetric sums update according to S2,a,J+1(d)=S2,a,J(d)+κJ+1,a(d)S1,a,J(d),S_2,a,J+1(d)=S_2,a,J(d)+ _J+1,a(d)S_1,a,J(d), S3,a,J+1(d)=S3,a,J(d)+κJ+1,a(d)S2,a,J(d).S_3,a,J+1(d)=S_3,a,J(d)+ _J+1,a(d)S_2,a,J(d). Consequently, va,J+1R(d)−va,JR(d) v_a,J+1^R(d)-v_a,J^R(d) =κJ+1,a(d)9S1,a,J(d)(J+12)−2+4S2,a,J(d)(J+13)−2 = _J+1,a(d) \9S_1,a,J(d) J+12^-2+4S_2,a,J(d) J+13^-2 \ −[9S2,a,J(d)(J2)−2−(J+12)−2+4S3,a,J(d)(J3)−2−(J+13)−2]. - [9S_2,a,J(d) \ J2^-2- J+12^-2 \+4S_3,a,J(d) \ J3^-2- J+13^-2 \ ]. (B.2) If the denominator below is positive, define κ¯J,a(d)≔9S2,a,J(d)(J2)−2−(J+12)−2+4S3,a,J(d)(J3)−2−(J+13)−29S1,a,J(d)(J+12)−2+4S2,a,J(d)(J+13)−2. κ_J,a(d) 9S_2,a,J(d)\ J2^-2- J+12^-2\+4S_3,a,J(d)\ J3^-2- J+13^-2\9S_1,a,J(d) J+12^-2+4S_2,a,J(d) J+13^-2. (B.3) Equation (B.2) then implies va,J+1R(d)<va,JR(d)⟺κJ+1,a(d)<κ¯J,a(d)v_a,J+1^R(d)<v_a,J^R(d) _J+1,a(d)< κ_J,a(d). If this denominator is zero, then all current κj,a(d) _j,a(d) are zero, and both conditional bridge variances remain zero after adding a single additional label. For a particular coefficient contrast, subtracting equation (3.19) for J and J+1J+1 gives the exact condition c⊤(0,J+1R−0,JR)c c (V_0,J+1^R-V_0,J^R )c =[∑a=01π(D~i)a1−π(D~i)1−ava,J+1R(D~)−va,JR(D~)(0−⊤c)⊤Δψ0(Y,W)2]. =E [ _a=0^1π( D_i)^a\1-π( D_i)\^1-a\v_a,J+1^R( D)-v_a,J^R( D)\\(A_0^- c) _0(Y,W)\^2 ]. Therefore, satisfying the threshold for both latent classes almost surely is a simple sufficient condition for weakly improving every coefficient contrast. For a fixed contrast, it is sufficient and necessary that the displayed weighted average be negative. A special case. If the existing proxies have common conditional bridge variance κa(d)>0 _a(d)>0 and the added proxy has conditional bridge variance qa(d)q_a(d), equation (B.3) reduces to qa(d)<2κa(d)J(J−2)+(2J−1)κa(d)(J−2)J−1+2κa(d).q_a(d)<2 _a(d) J(J-2)+(2J-1) _a(d)(J-2)\J-1+2 _a(d)\. (B.4) In particular, an additional proxy of the same quality, qa(d)=κa(d)q_a(d)= _a(d), always satisfies this condition. If all J proxies have common variance κa(d) _a(d), then va,JR(d)=18κa(d)2J(J−1)+24κa(d)3J(J−1)(J−2),v_a,J^R(d)= 18 _a(d)^2J(J-1)+ 24 _a(d)^3J(J-1)(J-2), which is strictly decreasing in J for κa(d)>0 _a(d)>0 and is of order J−2J^-2. Moreover, the right-hand side of equation (B.4) converges to 2κa(d)2 _a(d) as J→∞J→∞. Thus, in this case, an added proxy may be somewhat noisier than the existing labels and still improve precision through the additional averaging. All formulas in this subsection concern the population bridge evaluated at the true nuisance functions. If conditional independence across proxies fails, additional covariance terms generally enter the conditional variance, and the threshold above no longer applies. These comparisons also hold across designs with different fixed values of J; a formal analysis with J=Jn→∞J=J_n→∞ would require additional control of the growing nuisance dimension and the minimum class contrast. Appendix C Additional Details on the Experiments This appendix provides additional information on the two studies in Section 6. We first clarify how our analyses relate to the original applications, then describe the logistic latent-class model and the expectation–maximization (EM) algorithm used in both studies, and finally give application-specific details. Throughout, an “expert label” denotes the benchmark annotation. C.1 Scope and Relationship to the Original Applications The two analyses are numerical illustrations of DMM rather than exact replications of the original substantive studies. In the first application, the original analysis and the DSL validation study examine differences in political-advertising tone between Facebook and television using the original expert-coded corpus and a candidate-adjusted downstream analysis (25; 22). We instead use a transparent ad-level logistic regression for a binary Promote indicator. The original annotation file contains 13,04013,040 expert-coded ads. After merging it with the platform and candidate covariates and retaining complete records, our analysis contains 12,97312,973 unique ads. Importantly, our Facebook coefficient is not the same estimand as the candidate-fixed-effects coefficient in the original study. In the second application, the original study codes both prefecture-level and county-level wrongdoing and includes both variables in a richer downstream specification (43). We focus on prefecture-level wrongdoing because it enters the simplified downstream model as a single main effect. This isolates one latent independent variable and avoids introducing a second latent variable and the county-wrongdoing interaction used in the original analysis. The resulting coefficient should therefore be interpreted as the coefficient from the working model defined below, not as an exact replication of every specification in 43. Expert labels serve three different roles across the exercises. First, they define the full-sample benchmark in both applications. Second, they are used to calibrate the synthetic Fowler data-generating process. Third, they are used retrospectively to rank the available proxy labels by F1 score. They do not enter the feasible DMM moment equations within a Monte Carlo replication or the DMM fit. The F1-based ranking should consequently be viewed as a device for constructing interpretable nested proxy sets for evaluation. C.2 Model Specification Both applications use the same low-dimensional logistic latent-class model for the measurement nuisance functions. To describe the common implementation, let Li∗∈0,1L_i ∈\0,1\ denote a generic latent binary label, let L~i=(Li(1),…,Li(J)) L_i=(L_i^(1),…,L_i^(J)) collect its binary proxy labels, and let ZiZ_i denote the nuisance-model design vector, including an intercept. The correspondence is Fowler:Li∗=Yi∗,Zi=(1,Ti,Wi⊤)⊤,Pan–Chen:Li∗=Xi∗,Zi=(1,Wi⊤)⊤ array[]lFowler:&L_i =Y_i ,&Z_i=(1,T_i,W_i ) ,\\[2.0pt] Pan--Chen:&L_i =X_i ,&Z_i=(1,W_i ) array where TiT_i is the main independent variable, an indicator for Facebook advertisements. We posit pi p_i ≔Pr(Li∗=1∣Zi)=expit(Zi⊤α), (L_i =1 Z_i)=expit(Z_i α), ηij,a _ij,a ≔ηj,a(Zi)≔Pr(Li(j)=1∣Li∗=a,Zi)=expit(Zi⊤γj,a),a∈0,1, _j,a(Z_i) (L_i^(j)=1 L_i =a,Z_i)=expit(Z_i _j,a), a∈\0,1\, where expit(u)=1+exp(−u)−1expit(u)=\1+ (-u)\^-1. Conditional independence implies the observed-data likelihood ℒ(θ)=∏i=1n[(1−pi)∏j=1Jηij,0Li(j)(1−ηij,0)1−Li(j)+pi∏j=1Jηij,1Li(j)(1−ηij,1)1−Li(j)],L(θ)= _i=1^n [(1-p_i) _j=1^J _ij,0^L_i^(j)(1- _ij,0)^1-L_i^(j)\\ +p_i _j=1^J _ij,1^L_i^(j)(1- _ij,1)^1-L_i^(j) ], (C.1) where θ=(α,γj,aj,a)θ=(α,\ _j,a\_j,a). The downstream dependent variable is not included in this likelihood. In particular, in the Pan–Chen analysis, upward reporting is not used to infer the latent wrongdoing label. Thus, we impose two substantively reasonable conditional independence assumptions and use the corresponding nuisance models: X(1)⟂⋯⟂X(J)|X∗,WX^(1)\! \!\!\! ·s\! \!\!\! X^(J) X ,W and X~⟂Y|X∗,W X\! \!\!\! Y X ,W, which jointly imply Assumption 3.1. C.2.1 Expectation–Maximization Algorithm Let ξi(t)=Prθ(t)(Li∗=1∣L~i,Zi) _i^(t)= _θ^(t)(L_i =1 L_i,Z_i) denote the posterior responsibility at iteration t. For compactness, define fij,a(t)=ηij,a(t)Li(j)1−ηij,a(t)1−Li(j).f_ij,a^(t)=\ _ij,a^(t)\^L_i^(j)\1- _ij,a^(t)\^1-L_i^(j). The E-step is ξi(t)=pi(t)∏j=1Jfij,1(t)(1−pi(t))∏j=1Jfij,0(t)+pi(t)∏j=1Jfij,1(t). _i^(t)= p_i^(t) _j=1^Jf_ij,1^(t)(1-p_i^(t)) _j=1^Jf_ij,0^(t)+p_i^(t) _j=1^Jf_ij,1^(t). (C.2) The production code evaluates equation (C.2) on the log scale using a log-sum-exp calculation. Given ξi(t) _i^(t), the M-step separates into standard logistic regression problems. The latent prevalence model is updated by a fractional-response logistic regression of ξi(t) _i^(t) on ZiZ_i: α(t+1)=argmaxα∑i=1n[ξi(t)logpi(α)+1−ξi(t)log1−pi(α)].α^(t+1)= *arg\,max_α _i=1^n [ _i^(t) p_i(α)+\1- _i^(t)\ \1-p_i(α)\ ]. For each proxy j, the class-specific response models are updated by weighted logistic regressions: γj,1(t+1) _j,1^(t+1) =argmaxγ∑i=1nξi(t)logfjLi(j)∣1,Zi;γ, = *arg\,max_γ _i=1^n _i^(t) f_j\L_i^(j) 1,Z_i;γ\, γj,0(t+1) _j,0^(t+1) =argmaxγ∑i=1n1−ξi(t)logfjLi(j)∣0,Zi;γ. = *arg\,max_γ _i=1^n\1- _i^(t)\ f_j\L_i^(j) 0,Z_i;γ\. Thus, ξi(t) _i^(t) and 1−ξi(t)1- _i^(t) act as the effective class-one and class-zero weights. The M-step produces updated pi(t+1)p_i^(t+1) and ηij,a(t+1) _ij,a^(t+1), which are then used in the next E-step. The algorithm is initialized from several posterior-responsibility vectors. The candidate starts include the row-wise proxy mean, its complement, single-proxy starts of the form 0.1+0.8Li(j)0.1+0.8L_i^(j), and random draws from Uniform(0.2,0.8)Uniform(0.2,0.8) when additional starts are needed. We use six starts and retain the solution with the largest final observed-data log-likelihood in equation (C.1). Because the latent classes are identified only up to a common permutation, we orient each fitted solution after its final EM update. All proxies are coded so that one denotes the substantive positive class. We therefore require the average fitted proxy contrast to be positive: 1nJ∑i=1n∑j=1J(η^ij,1−η^ij,0)>0. 1nJ _i=1^n _j=1^J( η_ij,1- η_ij,0)>0. (C.3) If the fitted contrast is negative, we replace ξ^i ξ_i by 1−ξ^i1- ξ_i and refit the latent-prevalence and class-specific proxy regressions. This is the implementation counterpart of the anchoring condition in Section 3. Convergence and numerical safeguards. The settings used in both applications are summarized in Table C.1. Convergence requires both a small maximum change in posterior responsibilities and a small absolute change in the observed log-likelihood per observation. Coefficients and probabilities are clipped only for numerical stability. Table C.1: Shared settings for the low-dimensional logistic EM implementation. Quantity Setting Number of starts 66 Maximum EM iterations 500500 Posterior-change tolerance maxi|ξi(t)−ξi(t−1)|<10−5 _i| _i^(t)- _i^(t-1)|<10^-5 Per-observation likelihood tolerance |logℒ(θ(t))−logℒ(θ(t−1))|/n<10−7| (θ^(t))- (θ^(t-1))|/n<10^-7 Logistic coefficient clipping [−20,20][-20,20] Probability clipping [10−8,1−10−8][10^-8,1-10^-8] Bridge denominator floor 0.050.05 The general theory is stated using cross-fitting so that flexible nuisance learners can be accommodated without restrictive empirical-process conditions. The application studies use the low-dimensional parametric implementation above; for this finite-dimensional smooth nuisance class, the usual parametric M-estimation conditions replace the role of outer cross-fitting. Specifically, the shared EM engine is fit once on the analysis sample in each Fowler replication or Pan–Chen bootstrap resample. The two application adapters differ only in the construction of the proxy matrix and ZiZ_i; both call the same EM algorithm with the settings in Table C.1. C.2.2 From the EM Nuisance Fit to DMM Let Δ^ij≔η^ij,1−η^ij,0 _ij η_ij,1- η_ij,0. In the implementation, the denominator is stabilized as Δ^ij,ϵ=max(Δ^ij,ϵ),Δ^ij≥0,min(Δ^ij,−ϵ),Δ^ij<0,ϵ=0.05, _ij,ε= cases ( _ij,ε),& _ij≥ 0,\\ ( _ij,-ε),& _ij<0, cases ε=0.05, and the fitted single-proxy bridge is M^ij=Li(j)−η^ij,0Δ^ij,ϵ. M_ij= L_i^(j)- η_ij,0 _ij,ε. The robust bridge is then H^iR=3(J2)−1∑j1<j2M^ij1M^ij2−2(J3)−1∑j1<j2<j3M^ij1M^ij2M^ij3. H_i^R=3 J2^-1 _j_1<j_2 M_ij_1 M_ij_2-2 J3^-1 _j_1<j_2<j_3 M_ij_1 M_ij_2 M_ij_3. The floor is a finite-sample stabilization device rather than part of the population identification argument. For the Fowler latent-outcome analysis, let ri=(1,Ti,Wi⊤)⊤r_i=(1,T_i,W_i ) . The final estimate solves the generated-outcome quasi-score 1n∑i=1nriH^iR−expit(ri⊤β)=0. 1n _i=1^nr_i \ H_i^R-expit(r_i β) \=0. (C.4) For the Pan–Chen latent-independent-variable analysis, the final estimate solves 1n∑i=1n[1−H^iRψ0(Yi,Wi,β)+H^iRψ1(Yi,Wi,β)]=0. 1n _i=1^n [\1- H_i^R\ _0(Y_i,W_i;β)+ H_i^R _1(Y_i,W_i;β) ]=0. (C.5) Sandwich standard errors are computed from the empirical moment function variance and Jacobian, as in equations (3.13)–(3.15). C.3 Monte Carlo Simulations Data, annotation task, and downstream target. The annotation task follows the Wesleyan Media Project codebook used in 25: an ad is classified according to whether its primary purpose is to promote a candidate, attack a candidate, or contrast candidates. We convert each LLM response into the binary indicator that the ad is classified as Promote. The observed covariates are the Facebook indicator TiT_i, party, incumbency status, and office type. Categorical variables are represented by dummy indicators in both the calibration models and the EM model. The complete-case expert Promote prevalence is 0.7560.756. The three highest-ranked labels are GPT-4 multi 6-shot, GPT-4 multi 3-shot, and GPT-4 instruction 6-shot, with F1 scores of approximately 0.9500.950, 0.9420.942, and 0.9330.933, respectively. Let Yi∗Y_i denote the expert-coded Promote indicator and define ri=(1,Ti,Wi⊤)⊤.r_i=(1,T_i,W_i ) . The downstream working model is Pr(Yi∗=1∣Ti,Wi)=expit(ri⊤β∗), (Y_i =1 T_i,W_i)=expit(r_i β ), (C.6) and the coefficient on TiT_i is the target. Fitting equation (C.6) to all expert labels in the complete-case sample gives the calibration target βFB∗=1.3141 _FB =1.3141. Calibration of the data-generating process. We hold the 12,97312,973 observed covariate rows fixed. We first fit equation (C.6) to obtain π^ical=expit(ri⊤β^cal). π_i^cal=expit(r_i β^cal). For each proxy label j and expert class a∈0,1a∈\0,1\, we separately fit Pr(Yi(j)=1∣Yi∗=a,Ti,Wi)=expit(ri⊤γj,a), (Y_i^(j)=1 Y_i =a,T_i,W_i)=expit(r_i _j,a), (C.7) using the original expert and LLM labels. These calibration models are ordinary logistic regressions with probability clipping for numerical stability. Let η^ij,acal η_ij,a^cal denote the fitted probabilities. In replication b, we draw Yi∗,b Y_i ,b ∼Bernoulli(π^ical), ( π_i^cal), Yi(j),b|Yi∗,b=a,Ti,Wi Y_i^(j),b Y_i ,b=a,T_i,W_i ∼Bernoulli(η^ij,acal),j=1,…,J, ( η_ij,a^cal), j=1,…,J, with the proxy draws mutually independent across j conditional on (Yi∗,b,Ti,Wi)(Y_i ,b,T_i,W_i). Hence the conditional-independence restriction holds exactly in the simulated population. Moreover, the EM estimator uses the same logistic family as the calibration model, so its nuisance working models are correctly specified. The feasible estimators use only (Y~ib,Ti,Wi)( Y_i^b,T_i,W_i); Yi∗,bY_i ,b is supplied only to the infeasible oracle. Proxy sets and comparison estimators. The binary Promote proxy labels are ranked by their F1 score against the original expert labels. We use the nested proxy counts displayed in Figure 1. Because both the number and the quality of the included proxies change as the set expands, this exercise should be interpreted as adding progressively weaker proxies, not as holding proxy quality fixed while varying J. For a set of size J, define Sib=∑j=1JYi(j),bS_i^b= _j=1^JY_i^(j),b. The naive majority-vote outcome is Yˇib=Sib>J/2+BibSib=J/2,Bib∼Bernoulli(1/2), Y_i^b=1\S_i^b>J/2\+B_i^b1\S_i^b=J/2\, B_i^b (1/2), where the auxiliary tie-breaking draws are independent across tied samples and Monte Carlo replications. The naive estimator fits the same logistic regression as equation (C.6) after replacing Yi∗,bY_i ,b with Yˇib Y_i^b. DMM uses the same J proxies but constructs H^iR H_i^R based on nuisance components fitted with the aforementioned EM algorithm and solves equation (C.4). The oracle fits the downstream logit using Yi∗,bY_i ,b. Each proxy-count cell uses 500500 Monte Carlo replications. For an estimator m, the reported summaries are Bias(m) (m) =1500∑b=1500(β^m,b−βFB∗), = 1500 _b=1^500( β_m,b- _FB ), RMSE(m) (m) =1500∑b=1500(β^m,b−βFB∗)21/2, = \ 1500 _b=1^500( β_m,b- _FB )^2 \^1/2, Coverage(m) (m) =1500∑b=1500βFB∗∈C^m,b95%. = 1500 _b=1^5001\ _FB ∈ C_m,b^95\%\. Additional results. The three expert-coded tone indicators differ substantially in prevalence: Promote accounts for 75.6%75.6\% of advertisements in the complete-case sample, whereas Contrast accounts for 17.2%17.2\% and Attack for only 7.1%7.1\%. We repeat the same application-calibrated simulation separately for the two less prevalent indicators, recalibrating the latent-outcome and proxy models for each outcome. The target Facebook coefficients are −1.029-1.029 for Contrast and −1.334-1.334 for Attack. Table C.2 shows that DMM performs especially well for Attack: across the top-3 through top-7 sets, its bias remains close to zero, coverage ranges from 0.9420.942 to 0.9620.962, and its RMSE falls from 0.1130.113 to 0.0850.085, approaching the oracle RMSE of 0.0760.076. The corresponding naive estimators are substantially biased and severely undercover. Contrast shows similar results. Table C.2: Fowler simulation results for the alternative Attack and Contrast tone indicators. Each cell uses 500500 Monte Carlo replications. For naive rows, F1 is the individual proxy’s positive-class F1 score; for DMM rows, it is the median [minimum, maximum] F1 score among the included proxies. Outcome Estimator / set F1 summary Bias Coverage RMSE Attack Oracle — −0.002-0.002 0.9540.954 0.0760.076 Naive: GPT-4 multi, 3-shot 0.6510.651 0.5910.591 0.0000.000 0.5940.594 Naive: GPT-4 multi, 0-shot 0.5950.595 0.8900.890 0.0000.000 0.8920.892 Naive: Llama-2 multi, 6-shot 0.5810.581 0.1130.113 0.4100.410 0.1240.124 DMM: Top 3 0.5950.595 [0.5810.581, 0.6510.651] −0.005-0.005 0.9580.958 0.1130.113 DMM: Top 5 0.5810.581 [0.5650.565, 0.6510.651] −0.001-0.001 0.9620.962 0.0870.087 DMM: Top 7 0.5770.577 [0.5480.548, 0.6510.651] −0.001-0.001 0.9420.942 0.0850.085 Contrast Oracle — 0.0000.000 0.9520.952 0.0500.050 Naive: GPT-4 multi, 3-shot 0.6650.665 −0.275-0.275 0.0000.000 0.2800.280 Naive: GPT-4 multi, 0-shot 0.6580.658 −0.308-0.308 0.0000.000 0.3120.312 Naive: Llama-2 instruction, 6-shot 0.5110.511 −0.144-0.144 0.3400.340 0.1550.155 DMM: Top 3 0.6580.658 [0.5110.511, 0.6650.665] −0.083-0.083 0.9580.958 0.5980.598 DMM: Top 5 0.5110.511 [0.4890.489, 0.6650.665] 0.0050.005 0.9500.950 0.1550.155 DMM: Top 7 0.4940.494 [0.3750.375, 0.6650.665] 0.0030.003 0.9600.960 0.1630.163 C.4 Empirical Validation Data and downstream model. The analysis uses 1,4121,412 Chinese citizen complaints. The latent independent variable is the expert-coded indicator Xi∗=prefecture-level wrongdoingX_i =1\prefecture-level wrongdoing\, and the downstream outcome Yi=SendOrNotiY_i= SendOrNot_i indicates whether the complaint was sent to higher-level authorities. The expert-coded positive-class prevalence is 0.0550.055, and the mean of YiY_i is 0.4180.418. The downstream model is logitPr(Yi=1∣Xi∗,Wi)=α+τXi∗+γ⊤Wi,logit\ (Y_i=1 X_i ,W_i)\=α+τ X_i +γ W_i, (C.8) where WiW_i contains connect2b, prevalence, regionj, groupIssue, realWorldCollectiveAction, petitioning, sentiment_indico, and personal_experience. Seven controls are binary and sentiment_indico is continuous. The coefficient τ is the main estimand. The full-sample expert-label fit gives τ^expert=−1.0388 τ_expert=-1.0388. Proxy labels. We use three LLM labels: GPT-4.1 5-shot, GPT-4 5-shot, and Llama-4 0-shot. Table C.3 reports several descriptive metrics against the expert labels. Table C.3: Descriptive quality of the three Pan–Chen proxy labels. The class-weighted F1 is the quantity displayed in Figure 2; positive-class F1 and sensitivity focus on the rare wrongdoing class. Proxy Weighted F1 Positive F1 Accuracy Sensitivity Specificity GPT-4.1 5-shot 0.944 0.545 0.938 0.667 0.954 GPT-4 5-shot 0.942 0.511 0.939 0.577 0.960 Llama-4 0-shot 0.949 0.492 0.953 0.410 0.985 DMM and benchmark estimators. The expert-label benchmark fits equation (C.8) using Xi∗X_i . Each naive estimator replaces Xi∗X_i with one LLM label and otherwise fits the same model. DMM estimates nuisance components using the aforementioned EM algorithm, constructs the robust bridge, and solves equation (C.5). The DSL benchmark uses a simple random sample of 500500 expert labels, or about 35.4%35.4\% of the sample. It supplies the three LLM labels jointly, together with WiW_i, to the supervised prediction step. In the implementation, dsl::dsl uses the logit downstream model and its default generalized random forest learner with cross-fitting. Thus, DSL neither selects one proxy nor converts the three proxies to a majority vote; it learns a joint predictor of Xi∗X_i and then applies the design-based correction using the sampled expert labels. Fixed-target bootstrap diagnostic. We estimate empirical coverage by asking how often each method’s interval contains the fixed full-sample expert-label coefficient under nonparametric resampling of the observed complaints. For bootstrap draw b, we sample n=1,412n=1,412 complaints with replacement, refit the complete estimator, and record the interval [τ^b,lo,τ^b,hi][ τ_b,lo, τ_b,hi]. The displayed fixed-target rate is 1500∑b=1500τ^b,lo≤τ^expert≤τ^b,hi,τ^expert=−1.0388. 1500 _b=1^5001\ τ_b,lo≤ τ_expert≤ τ_b,hi\, τ_expert=-1.0388.