The Information Theory of Comparisons: With Applications to Statistics and the Social Sciences

Description

This book finds a broad domain of relevance in statistics and the social sciences. Its conceptual development is supported by applications to economics and income distribution, finance, education, demographics and actuarial science, political studies, psychology, and general statistics. Fresh perspectives on directional complexity have generated an informational theory of ‘more versus less’, with representative polar outcomes as good or bad, or rich or poor. New duality metrics for spread and asymmetry have resulted, motivated by internal perspectives on the part of subjects, such as attitudes to their comparative (dis)advantage. This book is a readable review of these developments. Concepts and applications are described in tandem with each other. They consolidate recent contributions to the research literature, augmented with fresh insights and applications. Dynamic extensions include modeling shifting social attitudes, while the broader agenda encompasses topical areas such as subjectivist probability, investment decision making, and income distribution. Preface Contents Intro 1 Partition Entropy 1.1 Introduction 1.2 Information, Complexity and Entropy 1.2.1 Entropic Decompositions 1.3 Partition Entropy 1.3.1 Complexity and the Psychology of Investment Decisions 1.3.2 An Extension: The Frequency Domain 1.4 Partition Entropy and the Log Odds Function 1.5 Discrete Valued Random Variables and Histograms 1.6 Resolving Grade Uncertainty 1.7 Tail Complexity and Market Clearing Prices 1.8 Literature Notes References 2 Entropic Shifting Perspectives & Applications 2.1 Introduction 2.2 Left and Right Entropic Shifts 2.3 Shift Kernels: Concentrators and Spreaders 2.3.1 Some Extensions 2.4 Perspectives on Mixture Distributions 2.5 The Entropic Kernel: Data Smoothing and End Correction 2.5.1 End Correction with Kernel Compression 2.6 The Social Dynamics of Opinion 2.6.1 The Momentum of Opinion 2.7 Application of Mixture Theory to Value at Risk 2.8 Literature Notes References 3 Moments, Measures & Metrics 3.1 Introduction 3.2 Moments for the Entropic Shifts 3.3 The Double Smoothing Property 3.4 Asymmetry and Spread Functions 3.5 Summary Metrics for Asymmetry and Spread 3.6 Gini’s Mean Absolute Difference and Welfare Variants 3.7 The Economic Dynamics of Remuneration Relativities 3.8 Relationship with Stochastic Dominance 3.9 Literature Notes References 4 Information Comparisons in Practice 4.1 Introduction 4.2 Income Distribution 4.3 Implementation as Entropic Asymmetry and Spread Metrics 4.3.1 Social Welfare Aspects 4.3.2 Dynamics: The v-d Phase Plane 4.4 Application to Stock Market Performance 4.5 Actuarial Life Expectancy 4.6 Literature Notes References 5 Binary Perspectives for Spread & Asymmetry 5.1 Introduction 5.2 A First Approach Based on Spanning 5.3 Bipolarity: The Entropic Centre and Equivalent Width 5.4 Polar Asymmetry and Spread Metrics 5.5 Fund Performance Measures 5.6 Actuarial Uncertainty Revisited 5.7 Literature Notes References 6 Higher Dimensions 6.1 Introduction 6.2 Higher Dimensions: Directional Shifting Perspectives 6.2.1 Left and Right Hand Smoothing Moments 6.2.2 Co-smoothing 6.3 Index Combinations 6.4 Co-smoothing and the Ordered Mean Difference 6.4.1 The OMD in Context: Some Financial Decision Theory 6.4.2 Do Hedge Funds Really Add Value? 6.5 Literature Notes References 7 Entropy, Risk & Comparability 7.1 Introduction 7.2 Tail Probabilities and Informational Measures 7.3 The Conditional Value at Risk 7.4 Cardinal Versus Ordinal Comparisons 7.4.1 Utility Theory: A Short Review 7.4.2 Entropic Asymmetry and Social Utility 7.5 Information Based Rescaling in Subjectivist Probability 7.5.1 Formalising as a Change of Measure 7.5.2 Information Based Rescaling 7.6 Concluding Remarks: The Scope of Informative Comparisons 7.7 Literature Notes References Excel VBA Code for the Compressed Smoothing Kernel Glossary Index