Contents -- Preface -- 1. The Basic Elements -- 1.1 Introduction: Complementarity and Its Implications -- 1.2 Conceptually Defined Variables -- 1.3 Inaccessible c-Variables -- 1.4 On Decisions from a Statistical Point of View -- 1.5 Contexts for Experiments -- 1.6 Experiments and Selected Parameters -- 1.7 Hidden Variables and c-Variables -- 1.8 Causality, Counterfactuals -- 1.9 Probability Theory -- 1.10 Probability Models for Experiments -- 1.11 Elements of Group Theory -- 2. Statistical Theory and Practice -- 2.1 Historical Development of Statistics as a Science -- 2.2 The Starting Point of Statistical Theory -- 2.3 Estimation Theory -- 2.4 Confidence Intervals, Testing and Measures of Signifficance -- 2.5 Simple Situations Where Statistics is Useful -- 2.6 Bayes' Formula and Bayesian Inference -- 2.7 Regression and Analysis of Variance -- 2.8 Model Checking in Regression -- 2.9 Factorial Models -- 2.10 Contrasts in ANOVA Models -- 2.11 Reduction of Data in Experiments: Sufficiency -- 2.12 Fisher Information and the Cram er-Rao Inequality -- 2.13 The Conditionality Principle -- 2.14 A Few Design of Experiment Issues -- 2.15 Model Reduction -- 2.16 Perfect Experiments -- 3. Statistical Inference under Symmetry -- 3.1 Introduction -- 3.1.1 Orbits -- 3.2 Group Actions and Statistical Models -- 3.3 Invariant Measures on the Parameter Space -- 3.4 Subparameters, Inference and Orbits -- 3.5 Estimation under Symmetry -- 3.5.1 The Main Result -- 3.5.2 Consequences -- 3.6 Credibility Sets and Confidence Sets -- 3.7 Examples. Orbits and Model Reduction -- 3.8 Model Reduction for Subparameter Estimation and Prediction -- 3.8.1 Estimation of Subparameters -- 3.8.2 Multiple Regression under Rotation Invariance -- 3.8.3 Towards Partial Least Squares Regression -- 3.9 Estimation of the Maximally Invariant Parameter: REML -- 3.9.1 On Orbit Indices and on REML.

3.9.2 The Model and the Group -- 3.9.3 Estimation -- 3.10 Design of Experiments Situations -- 3.11 Group Actions De ned on a c-Variable Space -- 3.12 Some Concluding Remarks -- 4. The Transition from Statistics to Quantum Theory -- 4.1 Theoretical Statistics and Applied Statistics -- 4.2 The Godel Theorem Analogy -- 4.3 Wave Mechanics -- 4.4 The Formal Axioms of Quantum Theory -- 4.5 The Historical Development of Formal Quantum Mechanics -- 4.6 A Large Scale Model -- 4.7 A General Definition -- A c-System -- 4.8 Quantum Theory Axioms under Symmetry and Complementarity -- 4.9 The Electron Spin Example -- 5. Quantum Mechanics from a Statistical Basis -- 5.1 Introduction -- 5.2 The Hilbert Spaces of a Given Experiment -- 5.3 The Common Hilbert Space -- 5.4 States and State Variables -- 5.5 The Born Formula -- 5.6 The Electron Spin Revisited -- 5.7 Statistical Inference in a Quantum Setting -- 5.8 Proof of the Quantum Rules from Our Axioms -- 5.9 The Case of Continuous Parameters -- 5.10 On the Context of a System, and on the Measurement Process -- 6. Further Development of Quantum Mechanics -- 6.1 Introduction -- 6.2 Entanglement -- 6.3 The Bell Inequality Issue -- 6.4 Statistical Models in Connection to Bell's Inequality -- 6.5 Groups Connected to Position and Momentum. Planck's Constant -- 6.6 The Schrodinger Equation -- 6.7 Classical Information and Information in Quantum Mechanics -- 6.8 Some Themes and 'Paradoxes' in Quantum Mechanics -- 6.9 Histories -- 6.10 The Many Worlds and Many Minds -- 7. Decisions in Statistics -- 7.1 Focusing in Statistics -- 7.2 Linear Models -- 7.3 Focusing in Decision Theory -- 7.4 Briey on Schools in Statistical Inference -- 7.5 Experimental Design -- 7.6 Quantum Mechanics and Testing of Hypotheses -- 7.7 Complementarity in Statistics -- 8. Multivariate Data Analysis and Statistics -- 8.1 Introduction.

8.2 The Partial Least Squares Data Algorithms -- 8.3 The Partial Least Squares Population Model -- 8.4 Theoretical Aspects of Partial Least Squares -- 8.5 The Best Equivariant Predictor -- 8.6 The Case of a Multivariate Dependent Variable -- 8.7 The Two Cultures in Statistical Modelling -- 8.8 Model Reduction and PLS -- 8.9 A Multivariate Example Resembling Quantum Mechanics -- 9. Quantum Mechanics and the Diversity of Concepts -- 9.1 Introduction -- 9.2 Daily Life Complementarity -- 9.3 From Learning Parameter Values to Learning to Make Other Decisions -- 9.4 Basic Learning: With and Without a Teacher -- 9.5 On Psychology -- 9.6 On Social Sciences -- 10. Epilogue -- Appendix -- A.1 Mathematical Aspects of Basic Statistics -- A.1.1 Kolmogorov's Axioms -- A.1.2 The Derivation of the Binomial Distribution -- A.1.3 The Normal Distribution and Series of Observations -- A.1.4 Some Results for Linear Models -- A.1.5 On the Fisher Information -- A.1.6 Prediction Errors in Example 2.15.1. -- A.2 Transformation Groups and Group Representation -- A.2.1 Further Properties of Group Actions -- A.2.2 Haar Measure and the Modular Function -- A.2.3 Proofs Concerning Orbits, Model Reduction and Estimation of Orbit Indices -- A.2.4 On Group Representation Theory -- A.3 Technical Aspects of Quantum Mechanics -- A.3.1 Parameters of Several Statistical Experiments -- A.3.2 Proofs from Section 5.4 -- A.4 Some Aspects of Partial Least Squares Regression -- Bibliography -- Index.