The art of causal conjecture [...] XA-GB

In The Art of Causal Conjecture , Glenn Shafer lays out a new mathematical and philosophical foundation for probability and uses it to explain concepts of causality used in statistics, artificial intelligence, and philosophy. The various disciplines that use causal reasoning differ in the relative weight they put on security and precision of knowledge as opposed to timeliness of action. The natural and social sciences seek high levels of certainty in the identification of causes and high levels of precision in the measurement of their effects. The practical sciences -- medicine, business, engineering, and artificial intelligence -- must act on causal conjectures based on more limited knowledge. Shafer's understanding of causality contributes to both of these uses of causal reasoning. His language for causal explanation can guide statistical investigation in the natural and social sciences, and it can also be used to formulate assumptions of causal uniformity needed for decision making in the practical sciences. Causal ideas permeate the use of probability and statistics in all branches of industry, commerce, government, and science. The Art of Causal Conjecture shows that causal ideas can be equally important in theory. It does not challenge the maxim that causation cannot be proven from statistics alone, but by bringing causal ideas into the foundations of probability, it allows causal conjectures to be more clearly quantified, debated, and confronted by statistical evidence. 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 1. Introduction -- 2. Event Trees -- 3. Probability Trees -- 4. Meaning Of Probability -- 5. Independent Events -- 6. Events Tracking Events -- 7. Events As Signs Of Events -- 8. Independent Variables -- 9. Variables Tracking Variables -- 10. Variables As Signs Of Variables -- 11. Abstract Theory Of Event Trees -- 12. Martingale Trees -- 13. Refining -- 14. Principles Of Causal Conjecture -- 15. Causal Models -- 16. Representing Probability Trees -- App. A. Huygens Probability Trees. App. B. Some Elements Of Graph Theory -- App. C. Some Elements Of Order Theory -- App. D. Sample-space Framework For Probability -- App. E. Prediction In Probability Spaces -- App. F. Sample-space Concepts Of Independence -- App. G. Prediction Diagrams -- App. H. Abstract Stochastic Processes. Glenn Shafer. Includes Bibliographical References (p. [491]-500) And Index.