Causation Prediction and Search.pdf

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Table of Contents
Preface ................................................................................................................................vii
Acknowledgments..............................................................................................................xiii
Notational Conventions.....................................................................................................xxi
1. Introduction and Advertisement ..................................................................................1
1.1 The Issue............................................................................................................1
1.2 Advertisements ..................................................................................................10
1.2.1 Bayes Networks from the Data...........................................................11
1.2.2 Structural Equation Models from the Data.........................................13
1.2.3 Selection of Regressors.......................................................................14
1.2.4 Causal Inference without Experiment ................................................17
1.2.5 The Structure of the Unobserved........................................................19
1.3 Themes...............................................................................................................21
2. Formal Preliminaries.....................................................................................................25
2.1 Graphs................................................................................................................25
2.2 Probability..........................................................................................................31
2.3 Graphs and Probability Distributions ................................................................32
2.3.1 Directed Acyclic Graphs.....................................................................32
2.3.2 Directed Independence Graphs...........................................................34
2.3.3 Faithfulness.........................................................................................35
2.3.4 d-separation.........................................................................................36
2.3.5 Linear Structures.................................................................................36
2.4 Undirected Independence Graphs......................................................................37
2.5 Deterministic and Pseudo-Indeterministic Systems ..........................................38
2.6 Background Notes .............................................................................................39
3. Causation and Prediction: Axioms and Explications.................................................41
3.1 Conditionals.......................................................................................................41
3.2 Causation ...........................................................................................................42
3.2.1 Direct vs. Indirect Causation ..............................................................42
3.2.2 Events and Variables ..........................................................................43
3.2.3 Examples.............................................................................................45
3.2.4 Representing Causal Relations with Directed Graphs........................47
3.3 Causality and Probability...................................................................................49
3.3.1 Deterministic Causal Structures .........................................................49
3.3.2 Pseudo-Indeterministic and Indeterministic Causal Structures ..........51
3.4 The Axioms .......................................................................................................53
3.4.1 The Causal Markov Condition............................................................53
3.4.2 The Causal Minimality Condition ......................................................55
3.4.3 The Faithfulness Condition.................................................................56
3.5 Discussion of the Conditions.............................................................................57
3.5.1 The Causal Markov and Minimality Conditions ................................57
3.5.2 Faithfulness and Simpson's Paradox...................................................64
3.6 Bayesian Interpretations ....................................................................................70
3.7 Consequences of The Axioms ...........................................................................71
3.7.1 d-Separation........................................................................................71
3.7.2 The Manipulation Theorem ................................................................75
3.8 Determinism ......................................................................................................81
3.9 Background Notes .............................................................................................86
4. Statistical Indistinguishability ......................................................................................87
4.1 Strong Statistical Indistinguishability................................................................88
4.2 Faithful Indistinguishability...............................................................................89
4.3 Weak Statistical Indistinguishability .................................................................90
4.4 Rigid Indistinguishability ..................................................................................93
4.5 The Linear Case.................................................................................................94
4.6 Redefining Variables .........................................................................................99
4.7 Background Notes .............................................................................................101
5. Discovery Algorithms for Causally Sufficient Structures..........................................103
5.1 Discovery Problems...........................................................................................103
5.2 Search Strategies in Statistics ............................................................................104
5.2.1 The Wrong Hypothesis Space ............................................................105
5.2.2 Computational and Statistical Limitations..........................................107
5.2.3 Generating a Single Hypothesis..........................................................108
5.2.4 Other Approaches ...............................................................................109
5.2.5 Bayesian Methods...............................................................................109
5.3 The Wermuth-Lauritzen Algorithm...................................................................111
5.4 New Algorithms.................................................................................................112
5.4.1 The SGS Algorithm ............................................................................114
5.4.2 The PC Algorithm...............................................................................116
5.4.3 The IG (Independence Graph) Algorithm ..........................................124
5.4.4 Variable Selection...............................................................................125
5.4.5 Incorporating Background Knowledge...............................................127
5.5 Statistical Decisions...........................................................................................128
5.6 Reliability and Probabilities of Error.................................................................130
5.7 Estimation ..........................................................................................................132
5.8 Examples and Applications ...............................................................................132
5.8.1 The Causes of Publishing Productivity...............................................133
5.8.2 Education and Fertility .......................................................................139
5.8.3 The Female Orgasm............................................................................140
5.8.4 The American Occupational Structure ...............................................142
5.8.5 The ALARM Network........................................................................145
5.8.6 Virginity..............................................................................................147
5.8.7 The Leading Crowd ............................................................................147
5.8.8 Influences on College Plans................................................................149
5.8.9 Abortion Opinions ..............................................................................150
5.8.10 Simulation Tests with Random Graphs...........................................152
5.9 Conclusion .........................................................................................................161
5.10 Background Notes ...........................................................................................162
6. Discovery Algorithms without Causal Sufficiency .....................................................163
6.1 Introduction........................................................................................................163
6.2 The PC Algorithm and Latent Variables ...........................................................165
6.3 Mistakes.............................................................................................................168
6.4 Inducing Paths ...................................................................................................173
6.5 Inducing Path Graphs ........................................................................................174
6.6 Partially Oriented Inducing Path Graphs...........................................................177
6.7 Algorithms for Causal Inference with Latent Common Causes........................181
6.8 Theorems on Detectable Causal Influence ........................................................190
6.9 Non-Independence Constraints..........................................................................191
6.10 Generalized Statistical Indistinguishability and Linearity...............................193
6.11 The Tetrad Representation Theorem ...............................................................196
6.12 An Example: Math Marks and Causal Interpretation ......................................197
6.13 Background Notes ...........................................................................................200
7. Prediction........................................................................................................................201
7.1 Introduction........................................................................................................201
7.2 Prediction Problems...........................................................................................202
7.3 Rubin-Holland-Pratt-Schlaifer Theory ..............................................................203
7.4 Prediction with Causal Sufficiency ...................................................................213
7.5 Prediction without Causal Sufficiency ..............................................................216
7.6 Examples............................................................................................................227
7.7 Conclusion .........................................................................................................237
7.8 Background Notes ............................................................................................237
8. Regression, Causation and Prediction .........................................................................238
8.1 When Regression Fails to Measure Influence ...................................................238
8.2 A Solution and Its Application ..........................................................................242
8.2.1 Components of the Armed Forces Qualification Test ........................243
8.2.2 The Causes of Spartina Biomass ........................................................244
8.2.3 The Effects of Foreign Investment on Political Repression ...............248
8.2.4 More Simulation Studies ....................................................................250
8.3 Error Probabilities for Specification Searches...................................................252
8.4 Conclusion .........................................................................................................257
9. The Design of Empirical Studies ..................................................................................259
9.1 Observational or Experimental Study?..............................................................259
9.2 Selecting Variables ............................................................................................271
9.3 Sampling ............................................................................................................272
9.4 Ethical Issues in Experimental Design ..............................................................276
9.4.1 The Kadane/Sedransk/Seidenfeld Design...........................................277
9.4.2 Causal Reasoning in the Experimental Design...................................280
9.4.3 Towards Ethical Trials........................................................................286
9.5 An Example: Smoking and Lung Cancer ..........................................................291
9.6 Appendix............................................................................................................302
10. The Structure of the Unobserved ...............................................................................306
10.1 Introduction......................................................................................................306
10.2 An Outline of the Algorithm............................................................................307
10.3 Finding Almost Pure Measurement Models....................................................310
10.3.1 Intra-Construct Foursomes ...............................................................310
10.3.2 Cross-Construct Foursomes..............................................................311
10.4 Facts about the Unobserved Determined by the Observed..............................315
10.5 Unifying the Pieces..........................................................................................316
10.6 Simulation Tests ..............................................................................................320
10.7 Conclusion .......................................................................................................322
11. Elaborating Linear Theories with Unmeasured Variables......................................323
11.1 Introduction......................................................................................................323
11.2. The Procedure.................................................................................................324
11.2.1 Scoring..............................................................................................324
11.2.2 Search ...............................................................................................327
11.3. The LISREL and EQS Procedures .................................................................329
11.3.1 Input and Output ...............................................................................329
11.3.2 Scoring..............................................................................................330
11.3.3 The LISREL VI Search ....................................................................331
11.3.4 The EQS Search................................................................................331
11.4. The Primary Study..........................................................................................332
11.4.1 The Design of Comparative Simulation Studies ..............................332
11.4.2 Study Design.....................................................................................333
11.5 Results..............................................................................................................343
11.6 Reliability and Informativeness.......................................................................346
11.7 Using LISREL and EQS as Adjuncts to Search ..............................................349
11.8 Limitations of the TETRAD II Elaboration Search.........................................351
11.9 Some Morals for Statistical Search..................................................................352
12. Open Problems.............................................................................................................354
12.1 Feedback, Reciprocal Causation, and Cyclic Graphs......................................354
12.1.1 Mason's Theorem..............................................................................355
12.1.2 Time Series and Cyclic Graphs ........................................................356
12.1.3 The Markov Condition, Factorizability and Faithfulness.................359
12.1.4 Discovery Procedures .......................................................................360
12.2 Indistinguishability Relations..........................................................................361
12.3 Time series and Granger Causality..................................................................363
12.4 Model Specification and Parameter Estimation from the Same Data Base.....365
12.5 Conditional Independence Tests......................................................................366
13. Proofs of Theorems......................................................................................................367
13.1 Theorem 2.1.....................................................................................................367
13.2 Theorem 3.1.....................................................................................................367
13.3 Theorem 3.2.....................................................................................................374
13.4 Theorem 3.3.....................................................................................................376
13.5 Theorem 3.4.....................................................................................................385
13.6 Theorem 3.5.....................................................................................................386
13.7 Theorem 3.6 (Manipulation Theorem) ............................................................395
13.8 Theorem 3.7.....................................................................................................398
13.9 Theorem 4.1.....................................................................................................401
13.10 Theorem 4.2...................................................................................................403
13.11 Theorem 4.3...................................................................................................403
13.12 Theorem 4.4...................................................................................................404
13.13 Theorem 4.5...................................................................................................404
13.14 Theorem 4.6...................................................................................................405
13.15 Theorem 5.1...................................................................................................405
13.16 Theorem 6.1...................................................................................................408
13.17 Theorem 6.2...................................................................................................411
13.18 Theorem 6.3...................................................................................................414
13.19 Theorem 6.4...................................................................................................417
13.20 Theorem 6.5...................................................................................................418
13.21 Theorem 6.6...................................................................................................419
13.22 Theorem 6.7...................................................................................................424
13.23 Theorem 6.8...................................................................................................425
13.24 Theorem 6.9...................................................................................................425
13.25 Theorem 6.10 (Tetrad Representation Theorem) ..........................................426
13.26 Theorem 6.11.................................................................................................460
13.27 Theorem 7.1...................................................................................................460
13.28 Theorem 7.2...................................................................................................462
13.29 Theorem 7.3...................................................................................................463
13.30 Theorem 7.4...................................................................................................470
13.31 Theorem 7.5...................................................................................................471
13.32 Theorem 9.1...................................................................................................472
13.33 Theorem 9.2...................................................................................................472
13.34 Theorem 10.1.................................................................................................473
13.35 Theorem 10.2.................................................................................................476
13.36 Theorem 11.1.................................................................................................479
Glossary ..............................................................................................................................481
Bibliography.......................................................................................................................495
Index....................................................................................................................................517