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3.3.3 Example: Smoking and the Genotype Theory 83
 
3.3.3 Example: Smoking and the Genotype Theory 83
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3.4 A Calculus of Intervention 85
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'''3.4 A Calculus of Intervention 85'''
    
3.4.1 Preliminary Notation 85
 
3.4.1 Preliminary Notation 85
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3.4.4 Causal Inference by Surrogate Experiments 88
 
3.4.4 Causal Inference by Surrogate Experiments 88
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3.5 Graphical Tests of Identifiability 89
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'''3.5 Graphical Tests of Identifiability 89'''
    
3.5.1 Identifying Models 91
 
3.5.1 Identifying Models 91
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3.5.2 Nonidentifying Models 93
 
3.5.2 Nonidentifying Models 93
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3.6 Discussion 94
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'''3.6 Discussion 94'''
    
3.6.1 Qualifications and Extensions 94
 
3.6.1 Qualifications and Extensions 94
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3.6.4 Relations to Robins’s G-Estimation 102
 
3.6.4 Relations to Robins’s G-Estimation 102
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4 Actions, Plans, and Direct Effects 107
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'''<big>4 Actions, Plans, and Direct Effects 107</big>'''
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4.1 Introduction 108
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'''4.1 Introduction 108'''
    
4.1.1 Actions, Acts, and Probabilities 108
 
4.1.1 Actions, Acts, and Probabilities 108
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4.1.3 Actions and Counterfactuals 112
 
4.1.3 Actions and Counterfactuals 112
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4.2 Conditional Actions and Stochastic Policies 113
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'''4.2 Conditional Actions and Stochastic Policies 113'''
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4.3 When Is the Effect of an Action Identifiable? 114
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'''4.3 When Is the Effect of an Action Identifiable? 114'''
    
4.3.1 Graphical Conditions for Identification 114
 
4.3.1 Graphical Conditions for Identification 114
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4.3.2 Remarks on Efficiency 116
 
4.3.2 Remarks on Efficiency 116
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4.3.3 Deriving a Closed-Form Expression  
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4.3.3 Deriving a Closed-Form Expression for Control Queries 117  
 
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for Control Queries 117
      
4.3.4 Summary 118
 
4.3.4 Summary 118
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4.4 The Identification of Dynamic Plans 118
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'''4.4 The Identification of Dynamic Plans 118'''
    
4.4.1 Motivation 118
 
4.4.1 Motivation 118
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4.4.4 Plan Identification: A Procedure 124
 
4.4.4 Plan Identification: A Procedure 124
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4.5 Direct and Indirect Effects 126
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'''4.5 Direct and Indirect Effects 126'''
    
4.5.1 Direct versus Total Effects 126
 
4.5.1 Direct versus Total Effects 126
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4.5.5 Indirect Effects and the Mediation Formula 132
 
4.5.5 Indirect Effects and the Mediation Formula 132
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5 Causality and Structural Models in Social Science and Economics 133
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'''<big>5 Causality and Structural Models in Social Science and Economics 133</big>'''
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5.1 Introduction 134
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'''5.1 Introduction 134'''
    
5.1.1 Causality in Search of a Language 134
 
5.1.1 Causality in Search of a Language 134
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5.1.3 Graphs as a Mathematical Language 138
 
5.1.3 Graphs as a Mathematical Language 138
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5.2 Graphs and Model Testing 140
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'''5.2 Graphs and Model Testing 140'''
    
5.2.1 The Testable Implications of Structural Models 140
 
5.2.1 The Testable Implications of Structural Models 140
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5.2.3 Model Equivalence 145
 
5.2.3 Model Equivalence 145
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5.3 Graphs and Identifiability 149
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'''5.3 Graphs and Identifiability 149'''
    
5.3.1 Parameter Identification in Linear Models 149
 
5.3.1 Parameter Identification in Linear Models 149
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5.3.2 Comparison to Nonparametric Identification 154
 
5.3.2 Comparison to Nonparametric Identification 154
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5.3.3 Causal Effects: The Interventional Interpretation of
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5.3.3 Causal Effects: The Interventional Interpretation of Structural Equation Models 157
 
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Structural Equation Models 157
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5.4 Some Conceptual Underpinnings 159
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'''5.4 Some Conceptual Underpinnings 159'''
    
5.4.1 What Do Structural Parameters Really Mean? 159
 
5.4.1 What Do Structural Parameters Really Mean? 159
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5.4.3 Exogeneity, Superexogeneity, and Other Frills 165
 
5.4.3 Exogeneity, Superexogeneity, and Other Frills 165
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5.5 Conclusion 170
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'''5.5 Conclusion 170'''
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5.6 Postscript for the Second Edition 171
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'''5.6 Postscript for the Second Edition 171'''
    
5.6.1 An Econometric Awakening?   171
 
5.6.1 An Econometric Awakening?   171
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5.6.3 Robustness of Causal Claims  172
 
5.6.3 Robustness of Causal Claims  172
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6 Simpson’s Paradox, Confounding, and Collapsibility 173
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'''<big>6 Simpson’s Paradox, Confounding, and Collapsibility 173</big>'''
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6.1 Simpson’s Paradox: An Anatomy 174
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'''6.1 Simpson’s Paradox: An Anatomy 174'''
    
6.1.1 A Tale of a Non-Paradox 174
 
6.1.1 A Tale of a Non-Paradox 174
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6.1.4 A Paradox Resolved (Or: What Kind of Machine Is Man?) 180
 
6.1.4 A Paradox Resolved (Or: What Kind of Machine Is Man?) 180
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6.2 Why There Is No Statistical Test for Confounding, Why Many
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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 182'''
 
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Think There Is, and Why They Are Almost Right 182
      
6.2.1 Introduction 182
 
6.2.1 Introduction 182
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6.2.2 Causal and Associational Definitions 184
 
6.2.2 Causal and Associational Definitions 184
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6.3 How the Associational Criterion Fails 185
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'''6.3 How the Associational Criterion Fails 185'''
    
6.3.1 Failing Sufficiency via Marginality 185
 
6.3.1 Failing Sufficiency via Marginality 185
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6.3.4 Failing Necessity via Incidental Cancellations 188
 
6.3.4 Failing Necessity via Incidental Cancellations 188
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6.4 Stable versus Incidental Unbiasedness 189
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'''6.4 Stable versus Incidental Unbiasedness 189'''
    
6.4.1 Motivation 189
 
6.4.1 Motivation 189
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6.4.3 Operational Test for Stable No-Confounding 192
 
6.4.3 Operational Test for Stable No-Confounding 192
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6.5 Confounding, Collapsibility, and Exchangeability 193
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'''6.5 Confounding, Collapsibility, and Exchangeability 193'''
    
6.5.1 Confounding and Collapsibility 193
 
6.5.1 Confounding and Collapsibility 193
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6.5.3 Exchangeability versus Structural Analysis of Confounding 196
 
6.5.3 Exchangeability versus Structural Analysis of Confounding 196
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6.6 Conclusions 199
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'''6.6 Conclusions 199'''
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7 The Logic of Structure-Based Counterfactuals 201
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'''<big>7 The Logic of Structure-Based Counterfactuals 201</big>'''
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7.1 Structural Model Semantics 202
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'''7.1 Structural Model Semantics 202'''
    
7.1.1 Definitions: Causal Models, Actions, and Counterfactuals 202
 
7.1.1 Definitions: Causal Models, Actions, and Counterfactuals 202
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7.1.4 The Twin Network Method 213
 
7.1.4 The Twin Network Method 213
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7.2 Applications and Interpretation of Structural Models 215
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'''7.2 Applications and Interpretation of Structural Models 215'''
 
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7.2.1 Policy Analysis in Linear Econometric Models:
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An Example 215
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7.2.1 Policy Analysis in Linear Econometric Models: An Example 215
    
7.2.2 The Empirical Content of Counterfactuals 217
 
7.2.2 The Empirical Content of Counterfactuals 217
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7.2.5 Simon’s Causal Ordering 226
 
7.2.5 Simon’s Causal Ordering 226
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7.3 Axiomatic Characterization 228
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'''7.3 Axiomatic Characterization 228'''
    
7.3.1 The Axioms of Structural Counterfactuals 228
 
7.3.1 The Axioms of Structural Counterfactuals 228
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7.3.3 Axioms of Causal Relevance 234
 
7.3.3 Axioms of Causal Relevance 234
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7.4 Structural and Similarity-Based Counterfactuals 238
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'''7.4 Structural and Similarity-Based Counterfactuals 238'''
    
7.4.1 Relations to Lewis’s Counterfactuals 238
 
7.4.1 Relations to Lewis’s Counterfactuals 238
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7.4.4 Relations to the Neyman–Rubin Framework 243
 
7.4.4 Relations to the Neyman–Rubin Framework 243
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7.4.5 Exogeneity and Instruments: Counterfactual and  
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7.4.5 Exogeneity and Instruments: Counterfactual and Graphical Definitions 245  
 
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Graphical Definitions 245
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7.5 Structural versus Probabilistic Causality 249
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'''7.5 Structural versus Probabilistic Causality 249'''
    
7.5.1 The Reliance on Temporal Ordering 249
 
7.5.1 The Reliance on Temporal Ordering 249
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7.5.5 Summary 256
 
7.5.5 Summary 256
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8 Imperfect Experiments: Bounding Effects and Counterfactuals 259
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'''<big>8 Imperfect Experiments: Bounding Effects and Counterfactuals 259</big>'''
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8.1 Introduction 259
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'''8.1 Introduction 259'''
    
8.1.1 Imperfect and Indirect Experiments 259
 
8.1.1 Imperfect and Indirect Experiments 259
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8.1.2 Noncompliance and Intent to Treat 261
 
8.1.2 Noncompliance and Intent to Treat 261
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8.2 Bounding Causal Effects with Instrumental Variables  262
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'''8.2 Bounding Causal Effects with Instrumental Variables  262'''
    
8.2.1 Problem Formulation: Constrained Optimization 262
 
8.2.1 Problem Formulation: Constrained Optimization 262
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8.2.2 Canonical Partitions: The Evolution of  
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8.2.2 Canonical Partitions: The Evolution of Finite-Response Variables 263  
 
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Finite-Response Variables 263
      
8.2.3 Linear Programming Formulation 266
 
8.2.3 Linear Programming Formulation 266
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8.2.6 Example: The Effect of Cholestyramine 270
 
8.2.6 Example: The Effect of Cholestyramine 270
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8.3 Counterfactuals and Legal Responsibility 271
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'''8.3 Counterfactuals and Legal Responsibility 271'''
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8.4 A Test for Instruments 274
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'''8.4 A Test for Instruments 274'''
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8.5 A Bayesian Approach to Noncompliance 275
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'''8.5 A Bayesian Approach to Noncompliance 275'''
    
8.5.1 Bayesian Methods and Gibbs Sampling 275
 
8.5.1 Bayesian Methods and Gibbs Sampling 275
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8.5.2 The Effects of Sample Size and Prior Distribution 277
 
8.5.2 The Effects of Sample Size and Prior Distribution 277
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8.5.3 Causal Effects from Clinical Data with Imperfect
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8.5.3 Causal Effects from Clinical Data with Imperfect Compliance 277
 
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Compliance 277
      
8.5.4 Bayesian Estimate of Single-Event Causation 280
 
8.5.4 Bayesian Estimate of Single-Event Causation 280
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8.6 Conclusion 281
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'''8.6 Conclusion 281'''
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9 Probability of Causation: Interpretation and Identification 283
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'''<big>9 Probability of Causation: Interpretation and Identification 283</big>'''
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9.1 Introduction 283
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'''9.1 Introduction 283'''
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9.2 Necessary and Sufficient Causes: Conditions of Identification 286
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'''9.2 Necessary and Sufficient Causes: Conditions of Identification 286'''
    
9.2.1 Definitions, Notation, and Basic Relationships 286
 
9.2.1 Definitions, Notation, and Basic Relationships 286
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9.2.4 Identifiability under Monotonicity and Nonexogeneity 293
 
9.2.4 Identifiability under Monotonicity and Nonexogeneity 293
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9.3 Examples and Applications 296
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'''9.3 Examples and Applications 296'''
    
9.3.1 Example 1: Betting against a Fair Coin 296
 
9.3.1 Example 1: Betting against a Fair Coin 296
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9.3.3 Example 3: The Effect of Radiation on Leukemia 299
 
9.3.3 Example 3: The Effect of Radiation on Leukemia 299
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9.3.4 Example 4: Legal Responsibility from Experimental and
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9.3.4 Example 4: Legal Responsibility from Experimental and Nonexperimental Data 302
 
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Nonexperimental Data 302
      
9.3.5 Summary of Results 303
 
9.3.5 Summary of Results 303
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9.4 Identification in Nonmonotonic Models 304
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'''9.4 Identification in Nonmonotonic Models 304'''
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9.5 Conclusions 307
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'''9.5 Conclusions 307'''
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10 The Actual Cause 309
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'''<big>10 The Actual Cause 309</big>'''
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10.1 Introduction: The Insufficiency of Necessary Causation 309
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'''10.1 Introduction: The Insufficiency of Necessary Causation 309'''
    
10.1.1 Singular Causes Revisited 309
 
10.1.1 Singular Causes Revisited 309
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10.1.3 Overdetermination and Quasi-Dependence 313
 
10.1.3 Overdetermination and Quasi-Dependence 313
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10.1.4 Mackie’s INUS Condition 313
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10.1.4 Mackie's INUS Condition 313
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10.2 Production, Dependence, and Sustenance 316
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'''10.2 Production, Dependence, and Sustenance 316'''
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10.3 Causal Beams and Sustenance-Based Causation 318
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'''10.3 Causal Beams and Sustenance-Based Causation 318'''
    
10.3.1 Causal Beams: Definitions and Implications 318
 
10.3.1 Causal Beams: Definitions and Implications 318
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10.3.5 Temporal Preemption 325
 
10.3.5 Temporal Preemption 325
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10.4 Conclusions 327
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'''10.4 Conclusions 327'''
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11 Reflections, Elaborations, and Discussions with Readers 331
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'''<big>11 Reflections, Elaborations, and Discussions with Readers 331</big>'''
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11.1 Causal, Statistical, and Graphical Vocabulary  331
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'''11.1 Causal, Statistical, and Graphical Vocabulary  331'''
    
11.1.1 Is the Causal-Statistical Dichotomy Necessary?  331
 
11.1.1 Is the Causal-Statistical Dichotomy Necessary?  331
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11.1.2 d-Separation without Tears (Chapter 1, pp. 16–18)  335
 
11.1.2 d-Separation without Tears (Chapter 1, pp. 16–18)  335
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11.2 Reversing Statistical Time (Chapter 2, p. 58–59)  337
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'''11.2 Reversing Statistical Time (Chapter 2, p. 58–59)  337'''
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11.3 Estimating Causal Effects  338
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'''11.3 Estimating Causal Effects  338'''
    
11.3.1 The Intuition behind the Back-Door Criterion (Chapter 3, p. 79)  338
 
11.3.1 The Intuition behind the Back-Door Criterion (Chapter 3, p. 79)  338
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11.3.7 The Validity of G-Estimation  352
 
11.3.7 The Validity of G-Estimation  352
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11.4 Policy Evaluation and the do-Operator  354
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'''11.4 Policy Evaluation and the do-Operator  354'''
    
11.4.1 Identifying Conditional Plans (Section 4.2, p. 113)  354
 
11.4.1 Identifying Conditional Plans (Section 4.2, p. 113)  354
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11.4.7 The Illusion of Nonmodularity  364
 
11.4.7 The Illusion of Nonmodularity  364
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11.5 Causal Analysis in Linear Structural Models  366
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'''11.5 Causal Analysis in Linear Structural Models  366'''
    
11.5.1 General Criterion for Parameter Identification (Chapter 5, pp. 149–54)  366
 
11.5.1 General Criterion for Parameter Identification (Chapter 5, pp. 149–54)  366
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11.5.5 External Variation versus Surgery  376
 
11.5.5 External Variation versus Surgery  376
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11.6 Decisions and Confounding (Chapter 6)  380
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'''11.6 Decisions and Confounding (Chapter 6)  380'''
    
11.6.1 Simpson’s Paradox and Decision Trees  380
 
11.6.1 Simpson’s Paradox and Decision Trees  380
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11.6.4 Why Isn’t Confounding a Statistical Concept?  387
 
11.6.4 Why Isn’t Confounding a Statistical Concept?  387
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11.7 The Calculus of Counterfactuals  389
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'''11.7 The Calculus of Counterfactuals  389'''
    
11.7.1 Counterfactuals in Linear Systems  389
 
11.7.1 Counterfactuals in Linear Systems  389
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11.7.3 d-Separation of Counterfactuals  393
 
11.7.3 d-Separation of Counterfactuals  393
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11.8 Instrumental Variables and Noncompliance  395
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'''11.8 Instrumental Variables and Noncompliance  395'''
    
11.8.1 Tight Bounds under Noncompliance   395
 
11.8.1 Tight Bounds under Noncompliance   395
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11.9 More on Probabilities of Causation  396
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'''11.9 More on Probabilities of Causation  396'''
    
11.9.1 Is “Guilty with Probability One” Ever Possible?  396
 
11.9.1 Is “Guilty with Probability One” Ever Possible?  396
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