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

Preface......Page 3
Contents......Page 5
Intro......Page 8
1.1 Introduction......Page 13
1.2 Information, Complexity and Entropy......Page 15
1.2.1 Entropic Decompositions......Page 18
1.3 Partition Entropy......Page 19
1.3.1 Complexity and the Psychology of Investment Decisions......Page 22
1.3.2 An Extension: The Frequency Domain......Page 24
1.4 Partition Entropy and the Log Odds Function......Page 25
1.5 Discrete Valued Random Variables and Histograms......Page 27
1.6 Resolving Grade Uncertainty......Page 29
1.7 Tail Complexity and Market Clearing Prices......Page 31
1.8 Literature Notes......Page 34
References......Page 35
2.1 Introduction......Page 37
2.2 Left and Right Entropic Shifts......Page 38
2.3 Shift Kernels: Concentrators and Spreaders......Page 42
2.3.1 Some Extensions......Page 46
2.4 Perspectives on Mixture Distributions......Page 47
2.5 The Entropic Kernel: Data Smoothing and End Correction......Page 50
2.5.1 End Correction with Kernel Compression......Page 54
2.6 The Social Dynamics of Opinion......Page 58
2.6.1 The Momentum of Opinion......Page 61
2.7 Application of Mixture Theory to Value at Risk......Page 64
2.8 Literature Notes......Page 66
References......Page 67
3.1 Introduction......Page 69
3.2 Moments for the Entropic Shifts......Page 70
3.3 The Double Smoothing Property......Page 72
3.4 Asymmetry and Spread Functions......Page 73
3.5 Summary Metrics for Asymmetry and Spread......Page 75
3.6 Gini’s Mean Absolute Difference and Welfare Variants......Page 77
3.7 The Economic Dynamics of Remuneration Relativities......Page 79
3.8 Relationship with Stochastic Dominance......Page 83
3.9 Literature Notes......Page 85
References......Page 86
4.1 Introduction......Page 87
4.2 Income Distribution......Page 88
4.3 Implementation as Entropic Asymmetry and Spread Metrics......Page 91
4.3.1 Social Welfare Aspects......Page 93
4.3.2 Dynamics: The v-d Phase Plane......Page 94
4.4 Application to Stock Market Performance......Page 95
4.5 Actuarial Life Expectancy......Page 99
4.6 Literature Notes......Page 104
References......Page 105
5.1 Introduction......Page 107
5.2 A First Approach Based on Spanning......Page 108
5.3 Bipolarity: The Entropic Centre and Equivalent Width......Page 110
5.4 Polar Asymmetry and Spread Metrics......Page 114
5.5 Fund Performance Measures......Page 116
5.6 Actuarial Uncertainty Revisited......Page 121
References......Page 122
6.1 Introduction......Page 123
6.2 Higher Dimensions: Directional Shifting Perspectives......Page 124
6.2.1 Left and Right Hand Smoothing Moments......Page 126
6.2.2 Co-smoothing......Page 128
6.3 Index Combinations......Page 129
6.4 Co-smoothing and the Ordered Mean Difference......Page 130
6.4.1 The OMD in Context: Some Financial Decision Theory......Page 132
6.4.2 Do Hedge Funds Really Add Value?......Page 134
6.5 Literature Notes......Page 135
References......Page 136
7.1 Introduction......Page 137
7.2 Tail Probabilities and Informational Measures......Page 138
7.3 The Conditional Value at Risk......Page 143
7.4.1 Utility Theory: A Short Review......Page 145
7.4.2 Entropic Asymmetry and Social Utility......Page 146
7.5 Information Based Rescaling in Subjectivist Probability......Page 148
7.5.1 Formalising as a Change of Measure......Page 149
7.5.2 Information Based Rescaling......Page 152
7.6 Concluding Remarks: The Scope of Informative Comparisons......Page 154
References......Page 156
Excel VBA Code for the Compressed Smoothing Kernel......Page 158
Glossary......Page 163
Index......Page 167