Steps Towards a Unified Basis for Scientific Models and Methods

Contents......Page 16
Preface......Page 8
1.1 Introduction: Complementarity and Its Implications......Page 20
1.2 Conceptually Defined Variables......Page 24
1.3 Inaccessible c-Variables......Page 26
1.4 On Decisions from a Statistical Point of View......Page 29
1.5 Contexts for Experiments......Page 31
1.6 Experiments and Selected Parameters......Page 32
1.7 Hidden Variables and c-Variables......Page 33
1.8 Causality, Counterfactuals......Page 36
1.9 Probability Theory......Page 38
1.10 Probability Models for Experiments......Page 42
1.11 Elements of Group Theory......Page 45
2.1 Historical Development of Statistics as a Science......Page 48
2.2 The Starting Point of Statistical Theory......Page 50
2.3 Estimation Theory......Page 54
2.4 Confidence Intervals, Testing and Measures of Signifficance......Page 57
2.5 Simple Situations Where Statistics is Useful......Page 62
2.6 Bayes\' Formula and Bayesian Inference......Page 66
2.7 Regression and Analysis of Variance......Page 70
2.8 Model Checking in Regression......Page 73
2.9 Factorial Models......Page 74
2.10 Contrasts in ANOVA Models......Page 75
2.11 Reduction of Data in Experiments: Sufficiency......Page 77
2.12 Fisher Information and the Cram er-Rao Inequality......Page 79
2.13 The Conditionality Principle......Page 80
2.14 A Few Design of Experiment Issues......Page 86
2.15 Model Reduction......Page 87
2.16 Perfect Experiments......Page 89
3.1 Introduction......Page 90
3.1.1 Orbits......Page 92
3.2 Group Actions and Statistical Models......Page 95
3.3 Invariant Measures on the Parameter Space......Page 99
3.4 Subparameters, Inference and Orbits......Page 102
3.5 Estimation under Symmetry......Page 106
3.5.1 The Main Result......Page 107
3.5.2 Consequences......Page 108
3.6 Credibility Sets and Confidence Sets......Page 111
3.7 Examples. Orbits and Model Reduction......Page 113
3.8 Model Reduction for Subparameter Estimation and Prediction......Page 115
3.8.1 Estimation of Subparameters......Page 116
3.8.2 Multiple Regression under Rotation Invariance......Page 117
3.8.3 Towards Partial Least Squares Regression......Page 118
3.9.1 On Orbit Indices and on REML......Page 120
3.9.2 The Model and the Group......Page 121
3.9.3 Estimation......Page 122
3.10 Design of Experiments Situations......Page 124
3.11 Group Actions De ned on a c-Variable Space......Page 125
3.12 Some Concluding Remarks......Page 126
4.1 Theoretical Statistics and Applied Statistics......Page 128
4.2 The Godel Theorem Analogy......Page 130
4.3 Wave Mechanics......Page 131
4.4 The Formal Axioms of Quantum Theory......Page 133
4.5 The Historical Development of Formal Quantum Mechanics......Page 135
4.6 A Large Scale Model......Page 137
4.7 A General Definition; A c-System......Page 139
4.8 Quantum Theory Axioms under Symmetry and Complementarity......Page 140
4.9 The Electron Spin Example......Page 143
5.1 Introduction......Page 146
5.2 The Hilbert Spaces of a Given Experiment......Page 149
5.3 The Common Hilbert Space......Page 153
5.4 States and State Variables......Page 158
5.5 The Born Formula......Page 163
5.6 The Electron Spin Revisited......Page 168
5.7 Statistical Inference in a Quantum Setting......Page 170
5.8 Proof of the Quantum Rules from Our Axioms......Page 175
5.9 The Case of Continuous Parameters......Page 176
5.10 On the Context of a System, and on the Measurement Process......Page 178
6.1 Introduction......Page 182
6.2 Entanglement......Page 183
6.3 The Bell Inequality Issue......Page 184
6.4 Statistical Models in Connection to Bell\'s Inequality......Page 186
6.5 Groups Connected to Position and Momentum. Planck\'s Constant......Page 188
6.6 The Schrodinger Equation......Page 191
6.7 Classical Information and Information in Quantum Mechanics......Page 192
6.8 Some Themes and \'Paradoxes\' in Quantum Mechanics......Page 195
6.9 Histories......Page 197
6.10 The Many Worlds and Many Minds......Page 198
7.1 Focusing in Statistics......Page 202
7.2 Linear Models......Page 204
7.3 Focusing in Decision Theory......Page 206
7.4 Briey on Schools in Statistical Inference......Page 208
7.5 Experimental Design......Page 210
7.6 Quantum Mechanics and Testing of Hypotheses......Page 211
7.7 Complementarity in Statistics......Page 215
8.1 Introduction......Page 218
8.2 The Partial Least Squares Data Algorithms......Page 219
8.3 The Partial Least Squares Population Model......Page 222
8.4 Theoretical Aspects of Partial Least Squares......Page 223
8.5 The Best Equivariant Predictor......Page 227
8.7 The Two Cultures in Statistical Modelling......Page 228
8.8 Model Reduction and PLS......Page 230
8.9 A Multivariate Example Resembling Quantum Mechanics......Page 232
9.1 Introduction......Page 234
9.2 Daily Life Complementarity......Page 235
9.3 From Learning Parameter Values to Learning to Make Other Decisions......Page 236
9.4 Basic Learning: With and Without a Teacher......Page 237
9.5 On Psychology......Page 238
9.6 On Social Sciences......Page 239
10. Epilogue......Page 242
A.1.2 The Derivation of the Binomial Distribution......Page 246
A.1.3 The Normal Distribution and Series of Observations......Page 247
A.1.4 Some Results for Linear Models......Page 248
A.1.6 Prediction Errors in Example 2.15.1.......Page 249
A.2.1 Further Properties of Group Actions......Page 250
A.2.2 Haar Measure and the Modular Function......Page 251
A.2.3 Proofs Concerning Orbits, Model Reduction and Estimation of Orbit Indices......Page 253
A.2.4 On Group Representation Theory......Page 255
A.3.1 Parameters of Several Statistical Experiments......Page 256
A.3.2 Proofs from Section 5.4......Page 258
A.4 Some Aspects of Partial Least Squares Regression......Page 259
Bibliography......Page 264
Index......Page 274