Causal Inference: A Tale of Three Frameworks (2511.21516v1)

Abstract: Causal inference is a central goal across many scientific disciplines. Over the past several decades, three major frameworks have emerged to formalize causal questions and guide their analysis: the potential outcomes framework, structural equation models, and directed acyclic graphs. Although these frameworks differ in language, assumptions, and philosophical orientation, they often lead to compatible or complementary insights. This paper provides a comparative introduction to the three frameworks, clarifying their connections, highlighting their distinct strengths and limitations, and illustrating how they can be used together in practice. The discussion is aimed at researchers and graduate students with some background in statistics or causal inference who are seeking a conceptual foundation for applying causal methods across a range of substantive domains.

• The paper develops formal translation mechanisms between potential outcomes, NPSEMs, and DAGs to clarify identification and inference in causal analysis.
• It compares key assumptions and methodologies, highlighting how each framework handles counterfactual reasoning and interventional effects.
• The study prescribes an integrated workflow that enhances causal inference practices and informs robust applications in statistical and AI systems.

Comparative Analysis of Three Causal Inference Frameworks

Introduction

The manuscript "Causal Inference: A Tale of Three Frameworks" (2511.21516) presents a rigorous comparative paper of the potential outcomes (PO) framework, nonparametric structural equation models (NPSEMs), and directed acyclic graphs (DAGs). Each formalism underpins different facets of causal reasoning, ranging from counterfactual estimation to graphical identification of interventional effects. The text systematically analyzes their conceptual foundations, identifies key translation mechanisms among frameworks, and delineates implications for identification, statistical inference, and practical causal workflow development.

Foundational Frameworks

Potential Outcomes

The PO framework formalizes causal questions via random variables that encode hypothetical outcomes under alternative interventions. Its centerpiece is the distinction between factual and counterfactual quantities, with average causal effects (ACE), conditional average treatment effects (CATE), and other population-level estimands derived from the distributional properties of $Y(a)$ for a treatment $A$.

Assumptions central to identification include SUTVA (no interference or multiple treatment versions), consistency ($Y=Y(A)$ for units receiving treatment $A$), and no unmeasured confounding (conditional ignorability $A \perp Y(a) \mid L$). Under these conditions, the g-formula ensures nonparametric identification of ACE:

$$\text{ACE} = \mathbb{E}_L [\mathbb{E}(Y|A=1,L) - \mathbb{E}(Y|A=0,L)],$$

where integration over $L$ accounts for confounder adjustment.

Structural Equation Models

NPSEMs model each variable as a deterministic function of its parents and exogenous noise, $V_j = f_j(U_j, \varepsilon_j)$. The autonomy (modularity) assumption permits interventional analysis via replacement of structural equations (do-operator) and preservation of other mechanisms. If exogenous errors are independent (NPSEM-IE), cross-world independence between collections of potential outcomes is implied, strengthening identification conditions beyond PO.

Interventions transform equations, e.g., fixing $A=a$ in a system yields predicted outcomes $Y=f_Y(L,a,\varepsilon_Y)$. The model formalizes counterfactual dependence and mechanism invariance, and explicit error independence grounds stronger identification (see below).

Directed Acyclic Graphs

DAGs provide a graphical encoding of conditional independence and causal dependencies. Their utility lies in reading off identification criteria such as the backdoor path criterion, applying d-separation, and translating graph structure to factorization:

$$P(V_1,\ldots,V_p) = \prod_{j=1}^{p} P(V_j|\text{Pa}(V_j)).$$

Causal DAG models introduce interventional semantics (modularity) via truncated factorization, enabling computation of effects under intervention:

$$P_{\text{do}(v_S)}(V_{[p]\setminus S}) = \prod_{k\notin S} P(V_k|\text{Pa}(V_k\setminus S), V_{\text{Pa}(V_k)\cap S}=v_{\text{Pa}(V_k)\cap S}).$$

Figure 1: A complex causal DAG model illustrating edge structure among observed and latent variables.

Graphical tools such as do-calculus, SWIGs, and the ID algorithm enable algebraic and algorithmic reduction of interventional queries to observational distributions, subject to graphical and structural assumptions.

Translation Mechanisms and Interconnections

The manuscript establishes formal methods for mapping between frameworks:

• NPSEM-IE to PO: Structural equations recursively encode potential outcomes, e.g., $Y(l,a) = f_Y(l, a, \varepsilon_Y)$, making the mechanistic basis explicit.
• PO to NPSEM-IE: Canonical representation via one-step-ahead counterfactuals reconstructs NPSEMs from systems of potential outcomes.
• NPSEM-IE to DAG: DAG structure is induced by parent sets in structural equations, and independent errors guarantee the global Markov and modularity properties.
• DAG to NPSEM-IE: Functional representation lemma ensures existence of measurable functions representing conditional distributions, compatible with the DAG structure.

A particular focus is placed on SWIGs, which bridge PO and DAG approaches, enabling conditional independence checks directly involving interventions ($Y(a)$) and graphical assessment under modified node semantics.

Comparative Analysis and Identification Strength

Minimalist (PO) vs. Mechanistic (NPSEM-IE) and Graphical (DAG) Orientations

• PO supports succinct specification of estimands under minimal assumptions but lacks explicit mechanism encoding.
• NPSEM-IE provides a generative account, incorporating structural mechanisms and counterfactual cross-world constraints.
• DAGs facilitate visualization and adjustment set selection, clarifying the structure of confounding and mediation.

Single-World vs. Cross-World Independence

• PO and DAG settings articulate only single-world independence, e.g., ignorability ($A \perp Y(a) \mid L$).
• NPSEM-IE, via error independence, imposes strong cross-world independence (e.g., $A \perp (Y(1), Y(0)) \mid L$), which reduces the model space dramatically but remains untestable in randomized designs.
• NPSEM-IE may thus enable identification of quantities (mediation, cross-world counterfactuals) that remain unidentifiable under standard DAG semantics.

Scenarios of Enhanced Identification

• Mediation formula: Identification of natural direct and indirect effects via mediation formulas exploits cross-world independence, only available under NPSEM-IE.
• Monotonicity assumptions: Structural or functional assumptions (unexpressed in DAGs) can yield identification or sharp bounds, e.g., local average treatment effects in IV models.
• Additive noise models: Linear, non-Gaussian, or post-nonlinear NPSEMs ensure graph identifiability, distinguishing among Markov equivalent DAGs; pure DAG approaches cannot.

Practical Recommendations and Implications

The article advocates an integrated workflow, combining strengths:

1. Define estimands in potential outcomes notation (clarifies the scientific question).
2. Articulate substantive and identification assumptions using DAGs, SWIGs, or NPSEMs (structural transparency).
3. Establish identification via graphical criteria (d-separation, backdoor), algebraic tools (g-formula, do-/po-calculus), or algorithmic approaches (ID algorithm).
4. Apply appropriate estimation methodology (regression, matching, weighting, machine learning) consonant with model structure and data availability.

This workflow improves transparency of causal inference, enables identification checklists, and reduces dependence on superfluous cross-world assumptions. The implications for AI include enhanced robustness to distributional shift, interpretability, and trustworthy design of decision systems that account for genuine interventions.

Conclusion

This comprehensive review systematically juxtaposes potential outcomes, structural equation models, and DAGs, highlighting their unique capabilities, assumptions, and areas of application. The manuscript demonstrates formal translation mechanisms among frameworks, analyzes when identification is strengthened by mechanistic or graphical modeling, and prescribes an actionable workflow for practical causal analysis. These insights facilitate principled causal reasoning in statistical science and machine learning applications, underpinning both foundational theoretical advances and robust practical deployment of AI systems.