Index: Causal Inference Foundations

Causal Inference Foundations

Routing Summary

This folder covers foundational causal inference concepts — potential outcomes, estimands, frequentist estimators — plus causal DAG graph theory and summarization. Contains 9 notes.

• Need the potential outcomes setup, SUTVA, ignorability, overlap? → Potential Outcomes Framework
• Need formal definitions of ITE, SATE, CATE, PATE, MATE? → Causal Estimands
• Need Frequentist estimators (IPW, outcome modeling, doubly-robust)? → Frequentist Causal Estimation
• Need overview of causal DAG summarization (problem formulation + CaGReS results)? → Zeng 2025 - Overview
• Need the formal definition of summary causal DAGs + NP-hardness proof? → Summary Causal DAGs
• Need node contraction = edge addition (Theorem 4.1, canonical DAG)? → Canonical Causal DAGs
• Need the CaGReS greedy algorithm (GetCost, optimizations, complexity)? → CaGReS Algorithm
• Need s-separation for CI identification in summary DAGs? → s-Separation in Summary DAGs
• Need do-calculus soundness/completeness for summary DAGs + ATE computation? → Do-Calculus in Summary Causal DAGs

Concept Map

Concept	Note	Type	Depends On	Key Result
SUTVA	Potential Outcomes Framework	definition	—	No interference, no multiple versions
Ignorability	Potential Outcomes Framework	definition	—	Unconfoundedness + overlap → identification
Propensity score	Potential Outcomes Framework	definition	SUTVA	$e(x)=\mathrm{Pr}(Z=1\mid X=x)$; balancing score
ITE	Causal Estimands	definition	Potential Outcomes Framework	${\tau }_i=Y_i(1)-Y_i(0)$
CATE	Causal Estimands	definition	ITE	$\tau (x)={\mu }_1(x)-{\mu }_0(x)$
PATE / SATE / MATE	Causal Estimands	definition	ITE	Population vs. sample vs. empirical-$X$ average
IPW estimator	Frequentist Causal Estimation	theorem	Ignorability	Consistent if propensity score model correct
Doubly-robust estimator	Frequentist Causal Estimation	theorem	IPW + outcome model	Consistent if either model correct
Summary causal DAG	Summary Causal DAGs	definition	Directed Acyclic Graphs	Pair $(\mathcal{H},f)$; node contraction; NP-hard to optimize (Theorem 3.2)
Canonical causal DAG	Canonical Causal DAGs	theorem	Summary Causal DAGs	$\text{RB}(\mathcal{H})\equiv \text{RB}({\mathcal{G}}_{\mathcal{H}})$ (Theorem 4.1); contraction = edge addition
CaGReS algorithm	CaGReS Algorithm	concept	Canonical Causal DAGs	Greedy; $O((n-k)\cdot n^2)$; minimizes added edges; 4 optimizations
s-Separation	s-Separation in Summary DAGs	theorem	Canonical Causal DAGs	Sound + complete for CI in summary DAGs (Theorem 4.2)
Do-calculus on summary DAGs	Do-Calculus in Summary Causal DAGs	theorem	s-Separation in Summary DAGs	Soundness (Thm 6.1) + completeness (Thm 6.2); ATE valid on summary DAG

Notes

• Potential Outcomes Framework — CONTAINS: SUTVA (Assumption), Ignorability/Overlap (Assumption 2.1), identification equation, design vs. analysis stage distinction
• Causal Estimands — CONTAINS: ITE, SATE, CATE, PATE, MATE formal definitions with LaTeX; principal causal effects (IV preview)
• Frequentist Causal Estimation — CONTAINS: outcome modeling estimator, IPW (definition), Hájek IPW, doubly-robust estimator (definition + double robustness theorem), matching/weighting overview
• Zeng 2025 - Overview — CONTAINS: paper overview (arXiv 2504.14937); 4 contributions; experimental summary over 6 datasets; comparison to baselines
• Summary Causal DAGs — CONTAINS: Def 1 (summary DAG), Def 2 (compatibility), node contraction operation, 3 summarization constraints, Theorem 3.2 (NP-hardness), REDSHIFT example
• Canonical Causal DAGs — CONTAINS: Def 5 (canonical causal DAG), Def CI Sets equivalence, Theorem 4.1 (RB equivalence = contraction ↔ edge addition), why adding edges preserves validity
• CaGReS Algorithm — CONTAINS: Algorithm 1 (CaGReS pseudocode), Algorithm 2 (GetCost), 4 optimizations (semantic constraint, caching, low-cost merges, $O((n-k)\cdot n^2)$ complexity), FLIGHTS example
• s-Separation in Summary DAGs — CONTAINS: Def 6 (valid CI), Def 7 (s-separation), s-separation algorithm, Theorem 4.2 (soundness + completeness), example with ${\mathcal{H}}_1$
• Do-Calculus in Summary Causal DAGs — CONTAINS: Theorem 6.1 (soundness), Theorem 6.2 (completeness), ATE formula, REDSHIFT ATE example with robustness illustration

Sources

• Li et al. - 2022 - Bayesian causal inference a critical review.pdf — §2, pp. 2–5
• Zeng et al. - 2025 - Causal DAG Summarization (Full Version).pdf — Zeng, Cafarella, Kenig, Markakis, Youngmann & Salimi. arXiv 2504.14937v1. Causal DAG summarization via node contraction; CaGReS algorithm.

See Also

• Bayesian Inference — how Bayesian CI builds on these foundations
• Directed Acyclic Graphs — prerequisite DAG theory (d-separation, do-calculus, RB)
• Code Prompts for Causal Structure — how DAG structure maps to code if statements for LLM reasoning