The art of causal conjecture

First Page......Page 1
Title Page......Page 3
Copyright......Page 4
Contents......Page 7
Series Foreword......Page 13
Preface......Page 15
Acknowledgments......Page 19
1 Introduction......Page 21
1.1 Probability Trees......Page 23
1.2 Many Observers, Many Stances, Many Natures......Page 28
1.3 Causal Relations as Relations in Nature\'s Tree......Page 29
1.4 Evidence......Page 33
1.5 Measuring the Average Effect of a Cause......Page 37
1.6 Causal Diagrams......Page 40
1.7 Humean Events......Page 43
1.9 An Outline of the Book......Page 47
2 Event Trees......Page 51
2.1 Situations and Events......Page 52
2.2 The Ordering of Situations and Moivrean Events......Page 55
2.3 Cuts......Page 59
2.4 Humean Events......Page 63
2.5 Moivrean Variables......Page 69
2.6 Humean Variables......Page 73
2.7 Event Trees for Stochastic Processes......Page 74
2.8 Timing in Event Trees......Page 76
2.9 Intersecting Event Trees......Page 80
2.10 Notes on the Literature......Page 81
3 Probability Trees......Page 83
3.1 Some Types of Probability Trees......Page 84
3.2 Axioms for the Probabilities of Moivrean Events......Page 88
3.3 Zero Probabilities......Page 90
3.4 A Sample-Space Analysis of the Event-Tree Axioms......Page 92
3.5 Probabilities and Expected Values for Variables......Page 94
3.6 Martingales......Page 99
3.7 The Expectation of a Variable in a Cut......Page 103
3.8 Conditional Expected Value and Conditional Expectation......Page 107
4 The Meaning of Probability......Page 111
4.1 The Interpretation of Expected Value......Page 112
4.2 The Interpretation of Expectation......Page 115
4.3 The Long Run......Page 118
4.4 Changes in Belief......Page 121
4.5 The Empirical Validation of Probability......Page 126
4.6 The Diversity of Uses of Probability......Page 128
4.7 Notes on the Literature......Page 130
5 Independent Events......Page 133
5.1 Independence......Page 134
5.2 Weak Independence......Page 138
5.3 The Principle of the Common Cause......Page 141
5.4 Conditional Independence......Page 148
5.5 Notes on the Literature......Page 153
6 Events Tracking Events......Page 155
6.1 Tracking......Page 157
6.2 Tracking and Conditional Independence......Page 162
6.3 Stochastic Subsequence......Page 163
6.4 Singular Diagrams for Stochastic Subsequence......Page 167
6.5 Conjunctive and Interactive Forks......Page 169
7 Events as Signs of Events......Page 173
7.1 Sign......Page 174
7.2 Weak Sign......Page 179
7.3 The Ethics of Causal Talk......Page 180
7.4 Screening Off......Page 182
8 Independent Variables......Page 187
8.1 Unconditional Independence......Page 190
8.2 Conditional Independence......Page 195
8.3 Independence for Partitions......Page 197
8.4 Independence for Families of Variables......Page 202
8.5 Individual Properties of the Independence Relations......Page 206
9 Variables Tracking Variables......Page 209
9.1 Tracking and Conditional Independence: A Summary......Page 210
9.2 Strong Tracking......Page 212
9.3 Strong Tracking and Conditional Independence......Page 218
9.4 Stochastic Subsequence......Page 221
9.5 Functional Dependence......Page 223
9.6 Tracking in Mean......Page 224
9.7 Linear Tracking......Page 227
9.8 Tracking by Partitions......Page 230
9.9 Tracking by Families of Variables......Page 232
10 Variables as Signs of Variables......Page 235
10.1 Sign......Page 239
10.2 Linear Sign......Page 242
10.3 Scored Sign......Page 245
10.4 Families of Variables......Page 247
11 An Abstract Theory of Event Trees......Page 249
11.1 Event Trees as Sets of Sets......Page 250
11.2 Event Trees as Partially Ordered Sets......Page 252
11.3 Regular Event Trees......Page 260
11.4 The Resolution of Moivrean Variables......Page 264
11.5 Humean Events and Variables......Page 266
12 Martingale Trees......Page 267
12.1 Examples of Decision Trees......Page 269
12.2 The Meaning of Probability in a Decision Tree......Page 273
12.3 Martingales......Page 277
12.4 The Structure of Martingale Trees......Page 281
12.5 Probability and Causality......Page 285
12.6 Lower and Upper Probability......Page 289
12.7 The Law of Large Numbers......Page 292
12.8 Notes on the Literature......Page 294
13 Refining......Page 295
13.1 Examples of Refinement......Page 297
13.2 A Constructive Definition of Finite Refinement......Page 301
13.3 Axioms for Refinement......Page 302
13.5 Refining Martingale Trees......Page 308
13.6 Grounding......Page 314
14 Principles of Causal Conjecture......Page 319
14.1 The Diversity of Causal Explanation......Page 322
14.2 The Mean Effect of the Happening of a Moivrean Event......Page 325
14.3 The Effect of a Humean Variable......Page 331
14.4 Attribution and Generality......Page 336
14.5 The Statistical Measurement of the Effect of a Cause......Page 339
14.6 Measurement by Experiment......Page 340
14.7 Using Our Knowledge of How Things Work......Page 342
14.9 The Sampling Frame......Page 349
14.10 Notes on the Literature......Page 350
15 Causal Models......Page 351
15.1 The Causal Interpretation of Statistical Prediction......Page 353
15.2 Generalizing to a Family of Exogenous Variables......Page 357
15.3 Some Joint Causal Diagrams......Page 359
15.4 Causal Path Diagrams......Page 362
15.5 Causal Relevance Diagrams......Page 366
15.6 The Meaning of Latent Variables......Page 372
15.7 Notes on the Literature......Page 377
16 Representing Probability Trees......Page 379
16.1 Three Graphical Representations......Page 381
16.2 Skeletal Simplifications......Page 388
16.3 Martingale Trees in Type Theory......Page 391
Appendix A: Huygens\'s Probability Trees......Page 399
Huygens\'s Manuscript in Translation......Page 400
B.l Undirected Graphs......Page 405
B.2 Directed Graphs......Page 406
C.l Partial and Quasi Orderings......Page 413
C.2 Singular and Joint Diagrams for Binary Relations......Page 414
C.3 Lattices......Page 415
C.4 The Lattice of Partitions of a Set......Page 416
D.l Probability Measures......Page 419
D.2 Variables......Page 420
D.3 Families of Variables......Page 421
D.4 Expected Value......Page 422
D.5 The Law of Large Numbers......Page 425
D.6 Conditional Probability......Page 426
D.7 Conditional Expected Value......Page 427
Appendix E: Prediction in Probability Spaces......Page 429
E.l Conditional Distribution......Page 431
E.2 Regression on a Single Variable......Page 432
E.3 Regression on a Partition or a Family of Variables......Page 435
E.4 Linear Regression on a Single Variable......Page 438
E.5 Linear Regression on a Family of Variables......Page 442
Appendix F: Sample-Space Concepts of Independence......Page 445
F.l Overview......Page 446
F.2 Independence Proper......Page 452
F.3 Unpredictability in Mean......Page 454
F.4 Simple Uncorrelatedness......Page 457
F.5 Mixed Uncorrelatedness......Page 458
F.6 Partial Uncorrelatedness......Page 460
F.7 Independence for Partitions......Page 462
F.8 Independence for Families of Variables......Page 465
F.9 The Basic Role of Uncorrelatedness......Page 468
F.10 Dawid\'s Axioms......Page 469
Appendix G: Prediction Diagrams......Page 473
G.l Path Diagrams......Page 474
G.2 Generalized Path Diagrams......Page 482
G.3 Relevance Diagrams......Page 486
G.4 Bubbled Relevance Diagrams......Page 495
H.l Probability Conditionals and Probability Distributions......Page 497
H.2 Abstract Stochastic Processes......Page 499
H.3 Embedding Variables and Processes in a Sample Space......Page 500
Glossary of Notation......Page 505
References......Page 511
C......Page 521
E......Page 523
F......Page 524
I......Page 525
L......Page 526
P......Page 527
R......Page 528
S......Page 529
T......Page 530
Z......Page 531