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The Mathematics of Behavior

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  • 14 tables
  • Page extent: 356 pages
  • Size: 228 x 152 mm
  • Weight: 0.491 kg

Paperback

 (ISBN-13: 9780521615228 | ISBN-10: 0521615224)

The Mathematics of Behavior
Cambridge University Press
0521850126 - THE MATHEMATICS OF BEHAVIOR - by Earl Hunt
Table of Contents





Contents

Prefacepage ix
1INTRODUCTION1
1.1. What’s in the Book?1
1.2. Some Examples of Formal and Informal Thinking2
1.3. A Bit of History4
1.4. How Big Is the Earth? Eratosthenes’ Solution5
1.5. A Critique of Eratosthenes12
1.6. Applications of Mathematics to Social and Behavioral Issues14
1.7. Statistics16
2APPLYING PROBABILITY THEORY TO PROBLEMS IN SOCIOLOGY AND PSYCHOLOGY18
2.1. Introduction18
2.2. Defining Probability and Probability Measures19
2.3. How Closely Connected Are We?23
2.4. Conscious and Unconscious Memories27
2.5. Some Final Comments32
Appendix 2A. The Basis for Kolmogorov’s Axioms32
Appendix 2B. Some Important Properties of Probability Measures33
3FROM PHYSICS TO PERCEPTION42
3.1. The Psychophysical Problem42
3.2. Weber’s Law44
3.3. Fechner’s Law47
3.4. Stevens’s Scaling Technique: Deriving the Psychophysical Function from Magnitude Estimation53
3.5. Judging Complex Objects61
3.6. A Comment on Measurement65
4WHEN SYSTEMS EVOLVE OVER TIME67
4.1. Systems of Variables67
4.2. Differences and Differentiation68
4.3. Exponential Growth and Decay70
4.4. Numerical Analysis: The Transmission of Jokes and Colds76
4.5. Questions about Modeling81
4.6. Graphical Analysis: The Evolution of War and Peace86
4.7. Making Love, Not War: The Gottman-Murray Model of Marital Interactions96
4.8. Concluding Comments on Modeling Simple Systems101
Appendix 4A. A Proof of the Exponential Growth Equation103
5NON-LINEAR AND CHAOTIC SYSTEMS104
5.1. Continuous Change and Sudden Jumps104
5.2. The Lotka-Volterra Model of Predator and Prey Interactions106
5.3. The Logistic Equation: Introduction and Behavior When k < 1111
5.4. Non-zero Asymptotes and Cycles as k Increases116
5.5. Chaos121
5.6. Chaos and Network Models123
5.7. Closing Comments on Chaos130
6DEFINING RATIONALITY132
6.1. Axiomatic Reasoning132
6.2. Decision Making under Risk133
6.3. The Concept of Utility135
6.4. Von Neumann and Morgenstern’s Axiomatic Approach to Decision Making139
6.5. The Utility of Money143
6.6. A Summary of the Argument148
6.7. Psychological Research on Decision Making151
6.8. The Problem of Voting158
6.9. Definition and Notation161
6.10. Arrow’s Axioms: The Restrictions on Social Welfare Functions162
6.11. Illustration of the Definitions and Concepts for the Three-Person Society164
6.12. A Proof of Arrow’s Theorem166
6.13. Commentary on the Implications of Arrow’s Theorem173
6.14. Summary Comments and Questions About Axiomatic Reasoning174
7HOW TO EVALUATE EVIDENCE176
7.1. The Legacy of Reverend Bayes176
7.2. Bayes’ Theorem178
7.3. Some Numerical Examples180
7.4. Calculating the Odds184
7.5. Some Examples of Signal Detection185
7.6. A Mathematical Formulation of the Signal Detection Problem187
7.7. The Decision Analyst’s Problem191
7.8. A Numerical Example of ROC Analysis199
7.9. Establishing a Criterion203
7.10. Examples207
7.11. Four Challenge Problems213
8MULTIDIMENSIONAL SCALING216
8.1. The Basic Idea216
8.2. Steps and Technique219
8.3. Extensions to Non-geometric Data222
8.4. Extending the Idea to Conceptual Classes223
8.5. Generalizations of Semantic Space Models227
8.6. Qualifications on the Semantic Space Model229
9THE MATHEMATICAL MODELS BEHIND PSYCHOLOGICAL TESTING231
9.1. Introduction231
9.2. A Brief Review of Correlation and Covariance234
9.3. Predicting One Variable from Another: Linear Regression240
9.4. The Single Factor Model: The Case of General Intelligence244
9.5. Multifactor Theories of Intelligence and Personality249
9.6. Geometric and Graphic Interpretations254
9.7. What Sort of Results Are Obtained?255
Appendix 9A. A Matrix Algebra Presentation of Factor Analysis256
10HOW TO KNOW YOU ASKED A GOOD QUESTION259
10.1. The Problem259
10.2. An Illustrative Case: Vocabulary Testing260
10.3. The Basics of Item Response Theory262
10.4. Standardization: Estimating Item and Person Parameters Simultaneously265
10.5. The Application Phase: Adaptive Testing267
10.6. More Complicated IRT Models269
10.7. Mathematics Meets the Social World: Mathematical Issues and Social Relevance272
Appendix 10A. The Adaptive Testing Algorithm274
Appendix 10B. An Exercise in Adaptive Testing275
11THE CONSTRUCTION OF COMPLEXITY277
11.1. Some Grand Themes277
11.2. The Problem of Complexity278
11.3. Cellular Automata Can Create Complicated Constructions281
11.4. Is Capitalism Inherently Unfair? Reconstructing a Simple Market Economy283
11.5. Residential Segregation, Genocide, and the Usefulness of the Police289
11.6. Is This a New Kind of Science?294
12CONNECTIONISM297
12.1. The Brain and the Mind297
12.2. Computation at the Neural Level299
12.3. Computations at the Network Level303
12.4. A Philosophical Aside307
12.5. Connectionist Architectures309
12.6. Simulating a Phenomenon in Visual Recognition: The Interactive Activation Model311
12.7. An Artificial Intelligence Approach to Learning313
12.8. A Biological Approach to Learning: The Hebbian Algorithm319
12.9. The Auto-associator321
12.10. A Final Word324
13L’ENVOI325
References328
Index of Names333
Index of Subjects337


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