Potential Outcomes Paradigm

• The potential outcomes paradigm is a causal inference framework defining unobservable counterfactual outcomes for each treatment, enabling estimation of effects like ATE and CATE.
• It incorporates diverse methodologies—Bayesian imputation, design-based inference, and generative models—to robustly estimate causal effects even under conditions of selection bias and overdispersion.
• Applications span experimental design, personalized decision-making, and dynamic causal analysis, though challenges remain regarding untestable assumptions and dynamic complexities.

The potential outcomes paradigm is a foundational formalism for causal inference, defining causal effects by positing, for each unit, a set of latent variables—the “potential outcomes”—describing the value that would be observed for each possible treatment regime. Only one potential outcome is observed per unit, the others are counterfactual, which motivates the problem of causal identification and estimation. Known variously as the Neyman–Rubin, Rubin Causal Model, or counterfactual approach, it has seen extensive development in experimental design, decision theory, machine learning, structural epidemiology, and generative modeling.

1. Formal Structure and Fundamental Assumptions

The basic formulation treats each unit $i=1,\dots,N$ as possessing a vector of potential outcomes, usually denoted $\{Y_i(z): z\in\mathcal Z\}$, where $\mathcal Z$ is the set of possible treatment assignments. For binary treatment $W_i$, each unit has two potential outcomes $Y_i(0)$ and $Y_i(1)$, with the observed outcome given by $Y_i^{\mathrm{obs}} = W_i Y_i(1) + (1-W_i) Y_i(0)$ (Lee et al., 2020).

The paradigm relies on several key assumptions:

• Consistency: The observed outcome equals the potential outcome under the administered treatment.
• SUTVA (Stable-Unit-Treatment-Value Assumption): No interference and well-defined treatments.
• Ignorability/Unconfoundedness: $(Y_i(0), Y_i(1)) \perp W_i \mid X_i$ for observed covariates (Jun et al., 2020).
• Positivity/Overlap: All treatments have positive probability for all covariate strata.

These properties enable identification of key causal estimands such as average treatment effect (ATE), conditional average treatment effect (CATE), and related functionals.

2. Bayesian, Design-Based, and Machine Learning Methodologies

Numerous inferential protocols have been developed within the potential outcomes framework:

• Bayesian Imputation for Count Data: Lee et al. propose a fully Bayesian machinery for count potential outcomes, incorporating mixed-Poisson models with latent frailties, conditionally conjugate priors, Gibbs sampling, and asymptotic MCMC acceleration via normal approximation. The average treatment effect is estimated as the posterior mean and credible interval of imputed POs, supporting overdispersed or zero-inflated counts (Lee et al., 2020).
• Design-Based Inference and Riesz Representation: Yang (2024) augments the Neyman–Rubin model by modeling potential outcomes as random functions, embedding them in a $L^2$ Hilbert space, and applying Riesz representation to define causal estimands as bounded linear functionals. The technique supports random outcome-level noise and generalizes to dependent designs, yielding unbiased estimators and robust large-sample properties under local dependence (Yang, 2 May 2025).
• Distributional and Generative Models: Recent advances employ conditional generative models, e.g., normalizing flows (PO-Flow), diffusion models (DiffPO), generative adversarial networks (GDR-learners), and variational autoencoders to estimate full distributions or samples of $Y(a) \mid X$, enabling uncertainty quantification and rich counterfactual analysis (Wu et al., 21 May 2025, Ma et al., 2024, Melnychuk et al., 26 Sep 2025). These methods are constructed with orthogonal (doubly robust) loss functions for quasi-oracle efficiency (Melnychuk et al., 26 Sep 2025).
• Bounds and Decision Analysis: Sharper understanding emerges from bounding the unidentifiable components of potential outcomes. Min-max and covering-number bounds on interval estimators of $Y(a)$, risk-difference or relative risk bounds under monotonicity, and estimation of lower and upper confidence limits provide robust causal inference in finite samples. Decision-theoretic extensions allow for additive counterfactual loss functions, supporting identification and practical evaluation of treatment regimes in multi-arm settings (Makar et al., 2019, Jun et al., 2020, Lu et al., 2015, Koch et al., 13 May 2025, Kawakami et al., 13 Nov 2025).

3. Extensions: Ordinal, Multivariate, and Joint Potential Outcomes

When outcomes do not have a numeric scale, as in ordinal non-numeric settings, the potential outcomes framework shifts focus to estimands defined via the conditional or joint distribution of potential outcomes:

• Ordinal Outcomes: Causal estimands such as the probability that treatment is beneficial ($\tau = \Pr\{Y_i(1) \ge Y_i(0)\}$), strictly beneficial, or more refined quantiles are defined. These are partially identified via marginal empirical distributions, with sharp bounds derivable via linear programming formulations (Lu et al., 2015).
• Conditional and Joint Distributions: Multi-experimental and multi-arm settings allow for identification of the joint distribution $P(Y^0, Y^1)$ via assumptions of transportability and full-rank moment conditions across several randomized trials. This machinery underpins the estimation of principal causal effects and enables robust surrogate evaluation (Wu et al., 29 Apr 2025).
• Personalized Temporal Counterfactuals: In high-dimensional, longitudinal data, e.g., wearable sensors, the paradigm supports dynamic counterfactual engines wherein individual-specific potential outcomes are generated through causal graph discovery and conditional boosting models, projecting future physiological trajectories under hypothetical interventions (Subramanian et al., 20 Aug 2025).

4. Applications: Experimental Design, Selection Bias, and Decision Making

• Experimental Controls: The potential outcomes framework is used to precisely define null/non-null treatment controls, outcome controls, and contrast controls, which serve both as assumptions and as diagnostics for measuring and mitigating unwanted variation in experimental studies. Null control arms and contrasts make the detection of systematic errors mathematically explicit and promote reproducibility (Hunter et al., 2021).
• Selection Bias Analysis: Kenah (2023) reformulates selection bias nonparametrically in terms of independence of sample selection indicators under interventions, enabling graphical analyses via SWIGs and recognition of bias both under and away from the null. This formalism unifies structural DAG and classical parameter-based definitions (Kenah, 2020).
• Risk Assessment Instrument Pitfalls: RAIs trained on observable outcomes rather than potential outcomes can actually worsen the outcomes they are intended to prevent. Even under perfect prediction and no unmeasured confounding, thresholding observed-outcome models can move mass in the wrong direction. Optimal causal decision rules require explicit estimation of potential outcome regressions (Mishler et al., 2021).
• Additive Counterfactual Loss in Decision Theory: Koch and Imai show that counterfactual risk is identifiable from data if and only if the loss function is additive in potential outcomes. This enables principled policy design in multi-arm choice problems, supporting ethical, fairness, or overtreatment-penalty goals (Koch et al., 13 May 2025).

5. Limitations, Controversies, and Alternative Frameworks

Despite its deep influence, criticisms of the potential outcomes paradigm persist:

• Abstraction and Testability: The reliance on multiple unobservable counterfactuals per unit, and on independence statements over abstract joint distributions, is fundamentally metaphysical and not empirically testable (Höltgen et al., 2024).
• Dynamic Causality and Stochastic Processes: Situations involving feedback, mediation (e.g., truncation by death), and continuous-time processes can be more coherently represented using dynamic causal models and change-of-measure techniques from stochastic calculus, rather than by extending potential outcomes or principal stratification (Commenges, 2019).
• Alternative Finite-Population Frameworks: Approaches based on treatment-wise prediction for finite populations eschew unobservable counterfactuals, instead leveraging testable calibration conditions and direct prediction functions to evaluate the average causal effect in the target population. This sharpens the distinction between statistical and scientific inference and highlights the model-dependence inherent in causal claims (Höltgen et al., 2024).

6. Impact and Ongoing Directions

The potential outcomes paradigm continues to serve as the backbone for rigorous causal analysis in randomized experiments, observational studies, machine learning, and personalized medicine. Its recent intersection with modern generative models, doubly robust estimation, decision-theoretic advancements, and finite-sample nonparametric bounds signals an ongoing expansion. Recent work on joint potential outcomes across multiple experiments, distributional and ranking-based estimands for heterogeneous treatment effects, and connections to dynamical causal systems reflect an active research frontier.

Researchers increasingly face practical questions about transportability, personalized counterfactual prediction, principled policy learning under complex loss functions, and robust inference under selection and sample bias. The paradigm’s capacity for formalization, adaptation, and extension continues to motivate technical innovation and critical debate in the foundations of causality.