CAUSALITY
CAUSALITY by Judea Pearl
PREFACE (updated 9/99)
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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
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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
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1.3 Causal Bayesian Networks
- 1.3.1 Causal networks as oracles for interventions
- 1.3.2 Causal relationships and their stability
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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
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1.5 Causal vs. Statistical Terminology
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2.1 Introduction
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2.2 The Causal Modeling Framework
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2.3 Model Preference (Occam's razor)
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2.4 Stable Distributions
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2.5 Recovering DAG Structures
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2.6 Recovering Latent Structures
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2.7 Local Criteria for Causal Relations
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2.8 Non-Temporal Causation and Statistical Time
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2.9 Conclusions
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- 2.9.1 On minimality, Markov, and stability
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3.1 Introduction
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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
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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
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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
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3.5 Graphical Tests of Identifiability
- 3.5.1 Identifying models
- 3.5.2 Nonidentifying models
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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
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4.1 Introduction
- 4.1.1 Actions, acts, and probabilities
- 4.1.2 Actions in decision analysis
- 4.1.3 Actions and counterfactuals
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4.2 Conditional Actions and Stochastic Policies
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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
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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
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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
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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
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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
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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
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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
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5.5 Conclusion
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6.1 Simpson's Paradox: An Anatomy
- 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)
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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
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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
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6.4 Stable vs. Incidental Unbiasedness
- 6.4.1 Motivation
- 6.4.2 Formal definitions
- 6.4.3 Operational test for stable no-confounding
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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
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6.6 Conclusions
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6.7 Acknowledgment
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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
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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
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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
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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
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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
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8.1 Introduction
- 8.1.1 Imperfect and indirect experiments
- 8.1.2 Noncompliance and intent-to-treat
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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
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8.3 Counterfactuals and Legal Responsibility}
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8.4 A Test for Instruments
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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
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8.6 Conclusion
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9.1 Introduction
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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
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9.3 Examples and Applications
- 9.3.1 Example-1: Betting against a fair coin
- 9.3.2 Example-2: The firing squad
- 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
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9.4 Identification in Non-monotonic Models
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9.5 Conclusions
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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
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10.2 Production, Dependence, and Sustenance
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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
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10.4 Conclusions