Partial Identification in Causal Inference

• The paper introduces a framework that replaces single causal estimates with sharp, assumption-dependent interval bounds when standard identification conditions are unmet.
• It employs techniques like instrumental variables, linear programming, and robust optimization to derive precise bounds on causal estimands across complex settings.
• The framework provides transparent diagnostics on assumption strength, enhancing decision-making in policy, medicine, and social sciences.

Partial identification is a framework in causal inference that replaces point estimation of causal effects with informative, assumption-dependent bounds when key identifiability conditions—such as no unmeasured confounding—are violated or untestable. Instead of delivering a single value for a causal estimand, the partial-identification approach characterizes an interval of values consistent with observed data and a given set of assumptions. This paradigm is central whenever causal identification is fundamentally limited by unmeasured variables, noncompliance, selection bias, or measurement error. The framework is highly expressive, unifying disparate domains such as instrumental variables, principal stratification, proxy variables, multi-marginal causal inference, generalizability from non-representative samples, and individualized policy assignment under uncertainty.

1. Fundamental Concepts and Problem Set-Up

The core object of interest in partial identification is a causal estimand—such as the average treatment effect (ATE), conditional average treatment effect (CATE), principal causal effects, or other structured functionals of the potential outcomes—whose exact value cannot be uniquely determined from the available data and assumptions. In a canonical setup, let $X$ denote pre-treatment covariates, $A \in \{-1,1\}$ a binary action or treatment, and $Y$ the observed outcome. Each unit is associated with potential outcomes $\{Y_{-1}, Y_{1}\}$. Under unmeasured confounding, the law of $(Y_1, Y_{-1})$ is not point-identified from observable data, and the analyst can only infer bounds $L_a(x) \leq \mu_a(x) \leq U_a(x)$ for $\mu_a(x) = E[Y_a|X=x]$. This bracketing extends to any estimand expressible in terms of latent counterfactual or principal-stratum means, encompassing individual-level and population-level causal quantities (Cui, 2021).

The implied identified set is the collection

$$\Theta = \left\{\theta(\cdot): \; L_a(x) \leq \mu_a(x) \leq U_a(x)\;\;\forall x,a \right\}$$

from which functional bounds on causal estimands are derived by optimizing the expected value (or another loss) with respect to the extremal elements of $\Theta$ (Wu et al., 2024, Cui, 2021). This set can be constructed from various sources, including instrumental variable inequalities (Balke–Pearl bounds), sensitivity analyses, principal stratification constraints, or robust optimization over measurement-error neighborhoods (Cui, 2021, Guo et al., 2022).

2. Bounding Strategies: Constructing and Interpreting Sharp Bounds

Bounding of causal effects proceeds by characterizing the sharp (i.e., tightest possible) value range compatible with observed data and imposed assumptions. Several key bounding strategies arise:

• Instrumental Variables (IV) and Linear Programming: For binary or categorical variables under IV assumptions, the joint probability constraints on observables translate into a linear (or polynomial) program over principal strata (deterministic functions translating unobservable factors into observed outcomes), yielding the Balke–Pearl bounds and their many generalizations for both average effects and more complex functionals (Cui, 2021, Boussim, 13 Oct 2025, Duarte et al., 2021).
• Partial Identification Without Completeness/Bridge Functions: Proxy variable settings, where proxies for unmeasured confounders are available but the completeness condition fails, rely on non-smooth functional envelopes, such as extremal ratios of proxy densities or conditional expectations, and their smooth approximations for statistical inference (Ghassami et al., 2023).
• Measurement Error and Robust Optimization: If covariates are measured with noise, robust optimization techniques generate identification intervals for the ATE. The uncertainty set on the joint distribution is typically defined via an $\ell_1$ or total-variation ball around observed distributions, leading to a tractable dual problem for the functional of interest (Guo et al., 2022).
• Marginal Optimal Transport and Fréchet Bounds: In settings where only marginal distributions of potential outcomes are identified (e.g., under randomization or with incomplete compliance), sharp bounds correspond to the solution of a multi-marginal optimal transport (MOT) problem, with closed-form or efficiently computable solutions for a broad class of quadratic or linear estimands (Gao et al., 2024, Ji et al., 2023).
• Data Fusion and Sensitivity Parameters: When fusing multiple sources (e.g., randomized and observational), interpretable sensitivity parameters for confounding and exchangeability violations provide a family of bounds indexed by violation severity, supporting robust qualitative conclusions under realistic assumption relaxation (Lanners et al., 30 May 2025).
• Information-Theoretic Methods: Data-driven $f$-divergence bounds on interventional vs. observational distributions allow partial identification of treatment effects under arbitrary unmeasured confounding, requiring only the observed propensity score and not outcome boundedness, auxiliary variables, or structural model specification (Jung et al., 23 Jan 2026).

Sharpness is crucial: a bound is sharp if it is both sound (contains all values attainable by any data-generating process consistent with the assumptions) and complete (contains only such values) (Duarte et al., 2021).

3. Decision Theory and Optimality under Partial Identification

Any rule or decision based on partially identified causal effects must acknowledge the bounded informativeness of the identified set. Three decision-theoretic perspectives emerge (Cui, 2021):

• Worst-Case (Maximin) Utility: The most conservative rule $d^L$ maximizes the worst-case (minimum) expected utility over the feasible $\Theta$, leading to the "lower-bound rule" $d^L(x) = \arg\max_d L_{d(x)}(x)$ pointwise. This rule guarantees the highest minimum reward.
• Minimax-Regret: The minimax-regret rule minimizes the maximal gap ("regret") compared to the unknown optimal policy $d^*$ across adverse instantiations of $\Theta$. The solution has a closed-form: at each $x$, if $U(x)<0$ select $-1$, if $L(x)>0$ select $+1$, else choose the action with the smaller absolute lower or upper bound. Allowing for randomization (mixed strategies) further improves the worst-case regret by solving a constrained minimization, with explicit expressions for the optimal randomization probability (Cui, 2021).
• Paradox of Individualization: When seeking individualized (covariate-dependent) policies under partial identification, it may occur that subgroup-level bounds are too wide for meaningful distinction, yet the aggregate (population) bounds suffice for a population-level optimal action, exposing a potential paradox in combining individualized targeting with partial identification (Cui, 2021).

4. Major Applications and Extensions

Partial identification permeates a broad array of causal settings, often defining the standard for valid inference under limited information:

• Principal Stratification: Causal effects in post-treatment-defined strata—such as always-survivor or complier groups—are typically only partially identified absent principal ignorability. Explicit two-dimensional regions (in linear models, for coefficients relating potential intermediates to outcomes) characterize sharp bounds, tightened via additional sign or dominant-effect constraints, and extend seamlessly to semiparametric settings (Wu et al., 2024, Chen et al., 2024).
• Generalizability and External Validity: When the study sample is not representative relative to the population, population average treatment effects are bounded via various intervals depending on the strength of assumptions—ranging from full-interval bounds, to bounds with population-frame information, to bounds under bounded sample variation and monotonic response (Chan, 2016).
• Instrumental Variables with Invalid Instruments: By relaxing IV exclusion or independence conditions using interpretable "budgets" capturing the number or extent of invalid instruments, the feasible set for causal parameters is a finite union of convex sets, leading to strictly tighter and more informative bounds than $L_p$-ball relaxations, especially in Mendelian randomization (Penn et al., 2024).
• Causal Effects with Categorical Outcomes: For IV models with categorical outcomes, the impact of selection bias is encoded via association parameters. Point identification arises from a strong invariance condition (association similarity), while weaker monotonicity or bounded-association assumptions yield linear-programming-based sharp partial identification intervals (Boussim, 13 Oct 2025).

5. Inference, Computation, and Practical Implementation

Efficient inference under partial identification involves both computational optimization and valid statistical uncertainty quantification:

• Optimization Algorithms: The translation of discrete or continuous partial identification problems into polynomial programs (via functional parameterization of principal strata) enables automated bounding via primal–dual branch-and-bound techniques with $\epsilon$-sharpness guarantees (Duarte et al., 2021). Similar LP, convex, or robust optimization formulations underlie most modern pipelines (Guo et al., 2022, Ji et al., 2023).
• Empirical Inference: Valid confidence intervals can be constructed for bound endpoints, typically via asymptotics (for linear or differentiable functionals) or multiplier bootstrap procedures suited for selection among multiple models or covariates (Ji et al., 2023). For bounds defined as non-smooth functionals, approximation via smooth surrogates (e.g., log-sum-exp) is combined with resampling schemes (Ghassami et al., 2023).
• Robustness Analysis: Practitioners are encouraged to report the width of bounds as an interpretation of sensitivity to assumption violations and data informativeness, and to use "breakdown frontiers" or sensitivity paths tracing how qualitative conclusions change as critical assumption-violation parameters are relaxed (Lanners et al., 30 May 2025, Cui, 2021).
• Automated and Unified Software: Decision trees or meta-learners trained on data features can assist in selecting the bounding method most appropriate for a given data scenario. Packages such as "CausalBoundingEngine" provide a unified interface to symbolic, optimization-based, and information-theoretic bounding tools, facilitating compatibility, extensibility, and transparency in applied research (Maringgele, 19 Aug 2025).

6. Theoretical and Practical Significance

The partial identification framework enables meaningful causal inference in settings where ignorability, monotonicity, or other identifying assumptions are contentious or untestable. Its key virtues include:

• Transparency: Explicit visibility into the dependency of results on the maintained assumption strength—bounds widen, sometimes dramatically, as assumptions weaken.
• Robustness: Even loose bounds may be sufficient to determine qualitative features (such as the sign of an effect) under only modest added structure.
• Assumption Diagnostic: Narrowness or width of bounds directly reflects the combined informativeness of data and assumptions, guiding the design of further data collection or justifying the need for additional (e.g., instrumental) variables.
• Generalizability and Flexibility: The approach seamlessly accommodates continuous/discrete data, multivariate treatments or outcomes, network interference, and complex missingness, with general frameworks available for both classical and modern high-dimensional settings (Cui, 2021, Wu et al., 2024, Maringgele, 19 Aug 2025, Jung et al., 23 Jan 2026).

In high-stakes policy, medicine, and social science applications, partial identification offers a principled path to honest reporting and robust individualized decision-making under realistic constraints. The ongoing development of computationally efficient, user-accessible tools positions the framework as a cornerstone of modern causal inference.