Unified Framework for Partial Identification

• The paper introduces a unified framework that rigorously formalizes partial identification by characterizing the set of parameter values consistent with incomplete data.
• It leverages convex geometry and support functions to transform moment inequality models into tractable convex programs, enabling efficient computation.
• Applications span causal inference, fairness evaluation, and panel data analysis, with robust inference achieved via bootstrap and de-biasing techniques.

A unified framework for partial identification provides a rigorous mathematical and algorithmic structure for characterizing and estimating the set of parameter values compatible with incomplete data or limiting assumptions. Unlike point identification, where the data and maintained assumptions suffice to determine a unique value for the parameter of interest, partial identification recognizes—and systematically exploits—the inherent ambiguity of the information set, often yielding convex sets characterized by statistically or computationally tractable representations. This article surveys the core concepts, key mathematical underpinnings, leading computational frameworks, and practical applications of the unified approach to partial identification, drawing on advances in optimal transport, random set theory, conditional linear programming, and modern statistical inference.

1. Foundational Concepts and Mathematical Formalism

Partial identification generalizes classical identification analysis by replacing the goal of unique parameter recovery with the explicit characterization of the "identified set"—the set of parameter values consistent with both the observed data and maintained assumptions. This universal perspective subsumes the classical parametric scenario and extends to nonparametric, design-based, and incomplete-data settings. The typical setup involves:

• A statistical universe $S$ of data-generating objects $s$, with observation map $\lambda: S \to \Lambda$ (the observed data law) and estimand map $\theta: S \to \Theta$.
• The identification set at observed data $\ell_0$ is $$H\{\theta; \ell_0\} = \{ \theta(s): \lambda(s) = \ell_0 \}$$. Point identification holds if $|H\{\theta;\ell_0\}| = 1$, otherwise partial identification (Basse et al., 2020).

This formalism is sufficiently general to recover known identification sets in econometrics, causal inference, nonparametric models, and missing data.

Random set theory provides a natural language for encapsulating the random bundle of parameter values consistent with each realization of the observed data. The identified set $C$ is typically the Aumann expectation or selection expectation of a random closed set $\Theta(\omega)$, and its geometry can be fully characterized via its support function: $$h_C(u) = E[\sup_{\theta \in \Theta(\omega)} u^T \theta], \quad u \in S^{d-1}$$ where $d$ is the dimension of the parameter space (Molinari, 2020).

2. Unified Characterization via Convex Geometry and Support Functions

Under linear or moment-inequality structures, many identified sets arising in partial identification problems are convex and admit a full description via their support functions. The support function $h_C(u)$ encodes the maximal value of the directional linear functional $u^T \theta$ over the set $C$. This perspective yields several advantages:

• The set $C$ can be recovered as the intersection over supporting halfspaces:

$$C = \bigcap_{u \in S^{d-1}} \{ x \in \mathbb{R}^d: u^T x \le h_C(u) \}$$

• Plug-in empirical estimators for $h_C(u)$ are consistent and amenable to CLT and bootstrap-based inference (Molinari, 2020).
• Computational approaches leverage off-the-shelf convex programming, reducing high-dimensional partial identification to tractable support function evaluation.

In optimal transport-based approaches for incomplete-data moment models, the support function for the identified set $\Theta_I$ can be computed by solving a sequence of lower-dimensional optimal transport (OT) problems. When the moment function $m$ is affine in $\theta$, the set $\Theta_I$ is convex, and the support function is given explicitly as: $$h_{\Theta_I}(q) = -\int \mathrm{KT}_{q^T E[m_1]^{-1} m_2} (\mu_{1|x}, \mu_{0|x}; x) \, d\mu_X(x)$$ where $\mathrm{KT}$ denotes the OT cost with respect to a specified cost function (Fan et al., 20 Mar 2025).

3. Algorithmic and Computational Frameworks

Modern unified frameworks operationalize partial identification through a blend of convex/linear programming, optimal transport, and duality, often exploiting structure in the problem for scalable computation. Notable strategies include:

• Optimal Transport Characterization: Transform moment (in)equality models with incomplete data into a continuum of OT cost inequalities. Each direction on the unit sphere corresponds to an OT problem, and the intersection of halfspaces yields $\Theta_I$ (Fan et al., 20 Mar 2025).
• Conditional Linear Programs (CLPs): For parameters identified via covariate-varying linear constraints, bounds are defined by expectations of CLP solutions. De-biasing strategies using influence functions or entropic regularization yield consistent estimators and confidence intervals (Ben-Michael, 13 Jun 2025).
• Random Set Averaging: For cases where identified sets can be directly computed from observations, empirical Aumann support-function estimators consist of Minkowski averages or empirical maxima over selections (Molinari, 2020).
• Minimax/Regret Decision Rules: In policy learning and individualized decision-making under partial identification, optimality criteria such as maximin, minimax-regret, and Hurwicz rules are unified and computable via sharp bounds on utilities over the identified set (Cui, 2021, Ben-Michael, 13 Jun 2025).

Empirical examples demonstrate computational tractability in high-dimensional, multivariate, and functionally constrained settings.

4. Inference and Statistical Properties

A central theme in unified partial identification is the rigorous construction of confidence sets and inference procedures for identified sets and scalar functionals. Key results include:

• Set- and Pointwise Coverage: Confidence sets for the identified region are designed to guarantee either set-level ($C \subset CS_n$ with high probability) or pointwise coverage ($\theta \in CS_n$ for every $\theta \in C$), with uniformity across data-generating processes (Molinari, 2020).
• Asymptotic Normality: Under smoothness and margin conditions, plug-in/de-biased estimators for functionals of the identified set converge at rate $\sqrt{n}$, with the influence function determined by the extremal point or basis of the active CLP/OT solution (Ben-Michael, 13 Jun 2025, Fan et al., 20 Mar 2025).
• Bootstrap Procedures: Empirical support-function processes can be bootstrapped to determine critical values for the construction of set-valued confidence regions (Molinari, 2020).
• Uniform Validity: Covariate-assisted inference procedures using optimal transport duality theory yield confidence intervals that are uniformly valid, even when nuisance parameters (e.g., estimated conditional distributions) are mis-specified or selected by model selection (Ji et al., 2023).

5. Generality: Structural Assumptions and Model Classes

The unified framework encompasses a wide spectrum of statistical and econometric models:

• Moment Models with Incomplete Data: Covers classical missing data, causal inference with unobserved counterfactuals, and fairness metrics in algorithmic settings; identified sets described by OT inequalities (Fan et al., 20 Mar 2025).
• Nonparametric and Semiparametric Panel Models: Average partial effects are partially or point identified depending on index sufficiency and support conditions, with sharp bounds derived when support conditions are violated (Liu et al., 2021).
• Factor-Confounding Models: For causal inference with multiple treatments/outcomes and unmeasured confounding, joint partial identification regions are characterized by nonconvex image sets parameterized over rotations in the confounder space, and can be substantially sharpened by incorporating negative controls or domain knowledge constraints (Kang et al., 2023).
• Decision Theory under Partial Identification: All standard Bayesian and frequentist decision criteria (maximin, minimax-regret, Hurwicz) are unified by considering the family of states (compatible data-generating distributions), with optimal rules computed by direct maximization/minimization over the identified set (Cui, 2021).

6. Comparative Perspective: Relation to Classical Approaches

Compared to traditional methods (e.g., copula bounds or large linear programs for data fusion), unified frameworks leveraging convex geometry, support functions, and optimal transport achieve broader generality, computational efficiency, and tractability for high-dimensional or conditionally specified models:

Approach	Scope	Computational Complexity
Copula/Fréchet–Hoeffding bounds	Univariate, smooth copulas	Closed-form, restrictive
Linear-programming/data fusion	Multivariate, LPs	Scaling issues in high-d
OT-based support functions (unified)	Fully general, high-d, multi	OT algorithms, closed-form
Random-set/Aumann geometry	Universal	Convex programming

A principal distinction is the explicit isolation of "coupling uncertainty" (in the dependence structure of unobserved quantities), which is embedded in an inner optimal transport or convex program, exploiting the full mathematical apparatus of OT, random set theory, and duality (Fan et al., 20 Mar 2025, Molinari, 2020, Ji et al., 2023).

7. Applications and Illustrations

Unified frameworks are now standard in microeconometrics, causal inference with missing data or unmeasured confounding, fairness evaluation in algorithmic systems, and weakly supervised learning. Recent applications include:

• Demographic Disparities and TPR Disparities: OT-based support-function characterization yields closed-form or efficiently computable bounds for group disparities in algorithmic fairness, with specialized algorithms exploiting partial transport submodularity (Fan et al., 20 Mar 2025).
• Covariate-Assisted Causal Inference: Model-agnostic dual OT-based inference with a split-sample scheme provides uniformly valid confidence intervals for partially identified average treatment effects, outperforming naive modeling (Ji et al., 2023).
• Panel Data APEs: Semiparametric three-stage estimators deliver sharp bounds and consistent estimation for average partial effects under minimal conditions, generalizing linear and non-linear panel specifications (Liu et al., 2021).
• Partial Identification with Stratified Randomization: Unified GMM and design-consistent variance estimation procedures for Lee bounds are developed for finely stratified, potentially unbalanced experiments (Ferman et al., 18 Jan 2026).
• Policy Learning under Ambiguous Identification: Conditional linear program formulations allow inference and policy optimization when the value of decision rules is only partially identified, with de-biased and regularized estimators for valid regret-minimization (Ben-Michael, 13 Jun 2025, Cui, 2021).

These applications demonstrate the practical versatility and scalability of the unified partial identification paradigm.

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References:

• "Partial Identification in Moment Models with Incomplete Data via Optimal Transport" (Fan et al., 20 Mar 2025)
• "Microeconometrics with Partial Identification" (Molinari, 2020)
• "A general theory of identification" (Basse et al., 2020)
• "Partial identification via conditional linear programs: estimation and policy learning" (Ben-Michael, 13 Jun 2025)
• "Individualized Decision-Making Under Partial Identification" (Cui, 2021)
• "Model-Agnostic Covariate-Assisted Inference on Partially Identified Causal Effects" (Ji et al., 2023)
• "Partial identification and unmeasured confounding with multiple treatments and multiple outcomes" (Kang et al., 2023)
• "Partial Identification under Stratified Randomization" (Ferman et al., 18 Jan 2026)
• "Identification and Estimation of Partial Effects in Nonlinear Semiparametric Panel Models" (Liu et al., 2021)