Steps towards a unified basis for scientific models and methods - Helland, Inge S. (World Scientific Pub. Co., 2010)
Includes bibliographical references (p. 245-253) and index
1. 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 significance. 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 Cramér-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. 3.2. Group actions and statistical models. 3.3. Invariant measures on the parameter space. 3.4. Subparameters, inference and orbits. 3.5. Estimation under symmetry. 3.6. Estimation under symmetry. 3.6. Credibility sets and confidence sets. 3.7. Examples. Orbits and model reduction. 3.8. Model reduction for subparameter estimation and prediction. 3.9. Estimation of the maximally invariant parameter : REML. 3.10. Design of experiments situations. 3.11. Group actions defined 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 Gödel 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 --