Steps towards a unified basis for scientific models and methods /
Culture, in fact, also plays an important role in science which is, per se, a multitude of different cultures. The book attempts to build a bridge across three cultures: mathematical statistics, quantum theory and chemometrical methods. Of course, these three domains should not be taken as equals in...
Singapore ; Hackensack, N.J. :
World Scientific Pub. Co.,
Table of Contents:
- 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
- 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 Schrödinger 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. Briefly 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.