CAUSALITY

## CAUSALITY by Judea Pearl

### 1 INTRODUCTION TO PROBABILITIES, GRAPHS, AND CAUSAL MODELS (updated 1/2000)

1.1 Introduction to Probability Theory
1.1.1 Why probabilities
1.1.2 Basic concepts in probability theory
1.1.3 Combining predictive and diagnostic supports
1.1.4 Random Variables and Expectations
1.1.5 Conditional independence and graphoids
1.2 Graphs and Probabilities
1.2.1 Graphical Notation and Terminology
1.2.2 Bayesian Networks
1.2.3 The d-separation criterion
1.2.4 Inference with Bayesian networks
1.3 Causal Bayesian Networks
1.3.1 Causal networks as oracles for interventions
1.3.2 Causal relationships and their stability
1.4 Functional Causal Models
1.4.1 Structural Equations
1.4.2 Probabilistic predictions in causal models
1.4.3 Interventions and causal effects in functional models
1.4.4 Counterfactuals in functional models
1.5 Causal vs. Statistical Terminology

### 2 A THEORY OF INFERRED CAUSATION (updated 1/2000)

2.1 Introduction
2.2 The Causal Modeling Framework
2.3 Model Preference (Occam's razor)
2.4 Stable Distributions
2.5 Recovering DAG Structures
2.6 Recovering Latent Structures
2.7 Local Criteria for Causal Relations
2.8 Non-Temporal Causation and Statistical Time
2.9 Conclusions
2.9.1 On minimality, Markov, and stability

### 3 CAUSAL DIAGRAMS AND THE IDENTIFICATION OF CAUSAL EFFECTS (updated 2/2000)

3.1 Introduction
3.2 Intervention in Markovian Models
3.2.1 Graphs as models of interventions
3.2.2 Interventions as variables
3.2.3 Computing the effect of interventions
3.2.4 Identification in causal models
3.3 Controlling Confounding Bias
3.3.1 The back-door criterion
3.3.2 The front-door criterion
3.3.3 Example: Smoking and the genotype theory
3.4 A Calculus of Intervention
3.4.1 Preliminary notation
3.4.2 Inference rules
3.4.3 Symbolic derivation of causal effects: An example
3.4.4 Causal inference by surrogate experiments
3.5 Graphical Tests of Identifiability
3.5.1 Identifying models
3.5.2 Nonidentifying models
3.6 Discussion
3.6.1 Qualifications and extensions
3.6.2 Diagrams as a mathematical language
3.6.3 Translation from Graphs to Potential Outcomes
3.6.4 Relations to Robin's G-estimation

### 4 ACTIONS, PLANS, AND DIRECT EFFECTS (updated 2/2000)

4.1 Introduction
4.1.1 Actions, acts, and probabilities
4.1.2 Actions in decision analysis
4.1.3 Actions and counterfactuals
4.2 Conditional Actions and Stochastic Policies
4.3 When is the Effect of an Action Identifiable?
4.3.1 Graphical conditions for identification
4.3.2 Remarks on efficiency
4.3.3 Deriving a closed-form expression for control queries
4.3.4 Summary
4.4 The Identification of Plans
4.4.1 Motivation
4.4.2 Plan identification: Notation and assumptions
4.4.3 Plan identification: A general criterion
4.4.4 Plan identification: A procedure
4.5 Direct Effects and their Identification
4.5.1 Direct vs. Total Effects:
4.5.2 Direct effects, definition and identification
4.5.3 Example: Sex Discrimination in College Admission
4.5.4 Average direct effects

### 5 CAUSALITY AND STRUCTURAL MODELS IN THE SOCIAL SCIENCES (updated 3/2000)

5.1 Introduction
5.1.1 Causality in search of a language
5.1.2 SEM: How its meaning got obscured
5.1.3 Graphs as a mathematical language: An example
5.2 Graphs and Model Testing
5.2.1 The testable implications of structural models
5.2.2 Testing the testable
5.2.3 Model equivalence
5.3 Graphs and Identifiability
5.3.1 Parameter identification in linear models
5.3.2 Comparison to nonparametric identification
5.3.3 Causal effects: The interventional interpretation of structural equation models
5.4 Some Conceptual Underpinnings
5.4.1 What do structural parameters really mean?
5.4.2 Interpretation of effect decomposition
5.4.3 Exogeneity, superexogeneity and other frills
5.5 Conclusion

### 6 SIMPSON'S PARADOX, CONFOUNDING, AND COLLAPSIBILITY (updated 2/2000 (final corrections inserted but not proofed))

6.1.1 A tale of a non-paradox
6.1.2 A tale of statistical agony
6.1.3 Causality vs. enchangeability
6.1.4 A paradox resolved (or what kind of machine is man)
6.2 Why there is no statistical test for confounding, why many think there is, and why they are almost right
6.2.1 Introduction
6.2.2 Causal and associational definitions
6.3 How the Statistical Criterion Fails
6.3.1 Failing sufficiency due to marginality
6.3.2 Failing sufficiency due to closed-world assumptions
6.3.3 Failing necessity due to barren proxies
6.3.4 Failing necessity due to incidental cancelations
6.4 Stable vs. Incidental Unbiasedness
6.4.1 Motivation
6.4.2 Formal definitions
6.4.3 Operational test for stable no-confounding
6.5 Confounding, Collapsibility, and Exchangeability
6.5.1 Confounding and collapsibility
6.5.2 Confounding vs. confounders
6.5.3 Exchangeability vs. structural analysis of confounding
6.6 Conclusions
6.7 Acknowledgment

### 7 THE LOGIC OF COUNTERFACTUALS AND STRUCTURAL MODELS (updated 2/2000 (final corrections inserted but not proofed))

7.1 Structural Model Semantics
7.1.1 Definitions: Causal models, actions and counterfactuals
7.1.2 Evaluating counterfactuals: Deterministic analysis
7.1.3 Evaluating counterfactuals: Probabilistic analysis
7.1.4 The twin-networks method
7.2 Applications and Interpretation of Structural Models
7.2.1 Policy analysis in linear econometric models: An example
7.2.2 The empirical content of counterfactuals
7.2.3 Causal explanations, utterances, and their interpretation
7.2.4 From mechanisms to actions to causation
7.2.5 Simon's Causal ordering
7.3 Axiomatic Characterization
7.3.1 The axioms of structural counterfactuals
7.3.2 Causal effects from counterfactual logic: An example
7.3.3 Axioms of causal relevance
7.4 Structural and Similarity-based Counterfactuals
7.4.1 Relations to Lewis' counterfactuals
7.4.2 Imaging vs.conditioning
7.4.3 Relation to Neyman-Rubin framework
7.4.4 Exogeneity revisited with counterfactuals or On Errors, graphs and counterfactuals
7.5 Structural vs. Probabilistic Causality
7.5.1 The reliance on temporal ordering
7.5.2 The perils of circularity
7.5.3 The closed-world assumption
7.5.4 Singular vs.general causes
7.5.5 Summary

### 8 BOUNDS FOR NONCOMPLIANCE AND OTHER EXPERIMENTAL IMPERFECTIONS (updated 3/2000 (final corrections inserted but not proofed))

8.1 Introduction
8.1.1 Imperfect and indirect experiments
8.1.2 Noncompliance and intent-to-treat
8.2 Bounding Causal Effects
8.2.1 Problem formulation
8.2.2 The evolution of potential-pesponse variables
8.2.3 Linear programming formulation
8.2.4 The natural bounds
8.2.5 Effect of treatment on the treated
8.2.6 Example: The effect of cholestyramine
8.3 Counterfactuals and Legal Responsibility}
8.4 A Test for Instruments
8.5 Causal Inference From Finite Sample
8.5.1 Gibbs sampling
8.5.2 The effects of sample size and prior distribution
8.5.3 Causal effects from clinical data with imperfect compliance
8.5.4 Bayesian estimate of single-event causation
8.6 Conclusion

### 9 PROBABILITY OF CAUSATION: INTERPRETATION AND IDENTIFICATION (updated 3/2000 (final corrections inserted but not proofed))

9.1 Introduction
9.2 Necessary and Sufficient Causes: Conditions of Identification
9.2.1 Definitions, notation, and basic relationships
9.2.2 Bounds and basic relationships under exogeneity
9.2.3 Identifiability under monotonicity and exogeneity
9.2.4 Identifiability under monotonicity and non-exogeneity
9.3 Examples and Applications
9.3.1 Example-1: Betting against a fair coin
9.3.3 Example-3: The effect of radiation on leukemia
9.3.4 Example-4: Legal responsibility from experimental and non-experimental data
9.3.5 Summary of results
9.4 Identification in Non-monotonic Models
9.5 Conclusions

### 10 THE ACTUAL CAUSE(updated 1/2000)

10.1 Introduction: The Insufficiency of Necessary Causation
10.1.1 Singular causes revisited
10.1.2 Preemption and the role of structural information
10.1.3 Over-determination and quasi-dependence
10.1.4 Mackie's INUS condition
10.2 Production, Dependence, and Sustenance
10.3 Causal Beams and Sustenance-Based Causation
10.3.1 Causal beams: Definitions and implications
10.3.2 Examples: From disjunction to general formulas
10.3.3 Beams, preemption, and the probability of single-event causation
10.3.4 Path-switching causation
10.3.5 Temporal preemption
10.4 Conclusions