Technical Summary: Functional Estimation under Proxy-Based Full-Law Identification

Problem Setup and Motivation

This paper addresses the challenge of estimating target functionals in latent variable models where variables of interest ($\vec{H}$) are unobserved, but associated observed measurements ("proxies," denoted $\vec{O}$) are available. While existing literature has established conditions under which the full-data law (the joint distribution of observed and latent variables, $p(\vec{H}, \vec{O})$) is identifiable from the observed-data law ($p(\vec{O})$), estimation of target functionals derived from this law remains relatively nascent.

Current nonparametric approaches often rely on flexible model assumptions to recover the full-data law, yet they frequently lack robust estimation strategies that yield desirable statistical properties (such as $\sqrt{n}$-consistency and multiple robustness) when nuisance functions are estimated at slower-than-parametric rates. Furthermore, many existing methods focus on specific causal targets (e.g., via proximal causal inference) rather than recovering the full joint distribution, limiting their applicability to a broader class of functionals.

Methodology

1. Identification of the Full-Data Law

The authors establish a set of assumptions under which the full-data law $p(\vec{H}, \vec{O})$ is identified, potentially up to an arbitrary permutation of the support values of $\vec{H}$. The framework accommodates multiple latent variables and allows for both finite and continuous support.

Key assumptions include:

• Proxy Structure: For each latent variable $H \in \vec{H}$, there exist three observed proxies $\{W_{1H}, W_{2H}, W_{3H}\} \subseteq \vec{O}$.
• Conditional Independence: These proxies are mutually independent conditional on $H$ and the remaining observed variables $\vec{O}_{-W}$ (those not in the proxy sets).
• Completeness and Rank Conditions: Assumptions 5 and 6 impose completeness conditions on the conditional distributions of the proxies given the latent variable, ensuring sufficient variability to distinguish latent states. Specifically, $W_{1H}$ and $W_{2H}$ must satisfy completeness conditions, while $W_{3H}$ requires a distinctness condition.
• Label Fixing: To identify the law up to a specific labeling (rather than just up to permutation), an additional functional assumption (Assumption 7) or a conditional independence assumption (Assumption 8) is required.

The paper utilizes Acyclic Directed Mixed Graphs (ADMGs) to characterize graphical models compatible with these assumptions, extending previous results from single-latent-variable settings to multiple latent variables (e.g., tree-structured graphs).

2. Estimation via Proxy-Based Weighting

The core methodological contribution is a strategy to construct observed-data estimating equations and influence functions by replacing unobserved variables in full-data expressions with observed variables via a proxy-based weighting scheme.

• From Full-Data to Observed-Data: The authors start with a full-data estimating equation or influence function $\phi(\vec{H}, \vec{O}; \eta)$ that has mean zero. They construct an observed-data function $\phi(\vec{O}; \eta)$ such that its expectation matches that of the full-data function.
• Weight Construction: This is achieved by defining weight functions $\omega_{\vec{h}_i}$ (or $\lambda_{\vec{h}_i}$ for influence functions) that satisfy specific orthogonality conditions. For a latent state $\vec{h}_i$, the weight $\omega_{\vec{h}_i}$ is constructed such that $E[\omega_{\vec{h}_i} | \vec{H} = \vec{h}_j] = \mathbb{I}(i=j)$. These weights are derived using the conditional expectations of basis functions of the proxies given the latent variable.
• Implementation: The resulting estimator $\hat{\psi}$ is a consistent M-estimator obtained by solving the empirical estimating equation $P_n[\phi(O; \hat{\psi}, \hat{\eta})] = 0$.

3. Influence Functions and Robustness

The paper extends this construction to derive observed-data influence functions. These functions allow for the construction of estimators with:

• Multiple Robustness: Consistency is preserved if at least one of several sets of nuisance functions is correctly specified.
• $\sqrt{n}$-Consistency: The estimators achieve parametric convergence rates even when nuisance functions (e.g., conditional expectations of proxies) are estimated at slower, non-parametric rates, provided certain product-rate conditions are met (e.g., the product of estimation errors for pairs of nuisance functions is $o_p(n^{-1/2})$).

Two classes of target functionals are addressed:

1. Class 1: Functionals of $p(\vec{H}, \vec{O}_{-W})$, where the influence function depends on the full set of proxies.
2. Class 2: Functionals of $p(\vec{H}, \vec{O} \setminus \{W_{1H}, W_{2H}\})$, requiring a more complex weighting scheme involving products of weights derived from different proxy pairs.

Key Contributions and Results

1. General Identification Framework: The paper generalizes identification results for the full-data law to settings with multiple latent variables and continuous support, moving beyond the single-latent-variable or finite-support constraints of prior work (e.g., Hu and Schennach [2008], Zhou and Tchetgen Tchetgen [2024]).
2. Systematic Estimation Procedure: It provides a systematic procedure to translate full-data influence functions into observed-data influence functions using proxy-based weights. This bridges the gap between nonparametric identification theory and modern semiparametric estimation methodology.
3. Robust Estimators: The derived estimators possess multiple robustness properties. For instance, in the binary latent treatment case, consistency is maintained if any one of three specific pairs of conditional expectation functions (e.g., $E[W|H], E[Z|H]$) is correctly specified.
4. Efficiency under Slow Nuisance Convergence: The paper establishes conditions under which estimators remain $\sqrt{n}$-consistent despite the nuisance functions converging slower than parametric rates. This is achieved through specific product-rate conditions on the estimation errors of the nuisance functions.
5. Graphical Characterization: The authors provide sufficient graphical criteria (via ADMGs) to determine when a statistical model satisfies the required independence assumptions, offering practical guidance for model specification.

Significance and Claims

The authors claim that their work extends the scope of full-law identification to more complex settings involving multiple latent variables and continuous supports. By developing estimating equations and influence functions that rely on observed proxies, the paper enables the estimation of a broad class of target functionals (including latent means and average counterfactual outcomes) without requiring the full-data law to admit a simple closed-form representation in terms of observed data.

The paper positions itself as a bridge between theoretical identification results and practical estimation, offering a toolkit for "principled estimation" in scenarios where variables are missing not at random or where selection bias is present, provided suitable proxies exist. The authors note that while they provide the theoretical conditions for $\sqrt{n}$-consistency, the development of specific procedures (e.g., sieve-based or machine learning methods) to satisfy these product-rate conditions remains an important direction for future work. The approach is explicitly inspired by and overlaps with recent work on binary latent treatments (Zhou and Tchetgen Tchetgen [2024]) and finite support models (Egami and Shin [2026]), but generalizes these to broader contexts.