Analogy and Structure

Royal Skousen
1992
Kluwer Academic Publishers
Dordrecht
ISBN 0-7923-1935-4

TABLE OF CONTENTS

INTRODUCTION 3
PART I:
STRUCTURALIST DESCRIPTIONS
Chapter 1: MEASURING THE CERTAINTY OF PROBABILISTIC RULES
1.0 Synopsis 13
1.1 Discrete Probabilistic Rules 16
1.2 Certainty 17
1.3 Unbiased and Biased Rules 19
1.4 Deterministic and Non-Deterministic Rules 20
1.5 Minimizing the Certainty of a Rule 20
1.6 Further Differentiation of Outcomes 23
1.7 Two Measures of Uncertainty 25
1.8 The Certainty of Non-Finite Rules 25
1.9 Comparison of C, U, and I (to the order Ξ±) 27
1.10 Why the Order of Certainty is Positive 28
1.11 Natural Motivations of Certainty 30
    1.11.1 Interpreting C1 30
    1.11.2 Interpreting C2 37
    1.11.3 A Conceptual Difference Between Q and H 38
1.12 Quadratic Q Versus Linear H 39
1.13 Axiomatic Differences Between Q and H 40
Chapter 2: SYSTEMS OF RULES
2.0 Synopsis 41
2.1 Subrules and Subcontexts 45
2.2 Constructing a System of Rules 45
2.3 The Certainty of a System of Rules 47
2.4 Deterministic and Non-Deterministic Systems 49
2.5 Biased and Unbiased Systems 49
2.6 The Minimal Value for the Certainty of a System 50
2.7 Derived Systems of Rules 53
2.8 The Effect on Certainty of Splitting up a Rule 54
2.9 Rule Homogeneity 55
2.10 Randomness 56
2.11 Measuring the Correctness of a System 56
2.12 The Degree of Correctness 59
2.13 The Uncertainty of a System 60
2.14 Natural Interpretations of the Uncertainty of a System 61
2.15 The Difference in Uncertainty 62
    2.15.1 Properties of βˆ†H 62
    2.15.2 Properties of βˆ†Q 65
Chapter 3: THE AGREEMENT DENSITY FOR CONTINUOUS RULES
3.0 Synopsis 71
3.1 Definition of Agreement Density 73
3.2 Agreement Density for Univariate Continuous Distributions 74
3.3 Agreement Density for Multivariate Continuous Distributions 82
3.4 The Agreement Density as a Measure of Concentration 83
3.5 Differences between Z and Z' 84
3.6 Maximizing Z' as a Measure of Correctness 85
3.7 An Example 87
3.8 Entropy Density 89
Chapter 4: MAXIMUM LIKELIHOOD STATISTICS
4.0 Synopsis 92
4.1 Probabilities Versus Statistics 96
4.2 A Maximum Likelihood Estimator 97
4.3 Estimators of Certainty and Uncertainty for a Rule 98
4.4 Biasedness of the Estimated Certainty for a Rule 99
4.5 Estimating the Certainty and Uncertainty for a System 100
4.6 Statistics for the Change in Uncertainty 102
4.7 Chi-Square Tests Based on the Change in Uncertainty 104
    4.7.1 G2: 2nβˆ†H (to the base e) 105
    4.7.2 U2: (n-1)(J-1)βˆ†Q/Q(R) 110
4.8 Pearson's Chi-Square Statistic 112
4.9 A Warning 115
Chapter 5: OPTIMAL DESCRIPTIONS
5.0 Synopsis 118
5.1 Defining Optimality 119
5.2 Asymptotic Values for the Chi-Square Distribution 121
5.3 An Optimal Description is a Correct Description 124
5.4 An Optimal Description Minimizes the Number of Rules 125
5.5 Minimizing the Number of Outcomes 132
Chapter 6: SIMPLEST DESCRIPTIONS
6.0 Synopsis 136
6.1 Well-Formed Contexts 139
6.2 The Complexity of a Contextual Specification 140
6.3 Logical Simplicity 140
6.4 Conjuncts of Positive Contexts 144
6.5 Fundamental Types of Behavior 146
    6.5.1 Categorical 146
    6.5.2 Exceptional/Regular 147
    6.5.3 Idiosyncratic 147
    6.5.4 A Preference Scale 148
6.6 Examples 149
6.7 Rule Ordering and Exceptionality 154
6.8 A Problem with Ordering 157
6.9 Redundancy 158
Chapter 7: PREFERRED DERIVATIONS
7.0 Synopsis 161
7.1 Binary Structures 166
7.2 Step-by-Step Procedures 169
7.3 Basic Behavior 172
7.4 Analysis Versus Synthesis 173
7.5 The Learning Factor 179
7.6 Minimizing Excess Questions and Guesses 191
Chapter 8: ANALYZING THE EFFECT OF A VARIABLE
8.0 Synopsis 194
8.1 Variable Analysis of Contextual Specifications 196
8.2 The Effect of a Variable 198
8.3 A Statistical Example 202
8.4 Defining the Overall Effect for a Set of Variables 207
 
PART II:
ANALOGICAL DESCRIPTIONS
Chapter 9: PROBLEMS WITH STRUCTURALIST DESCRIPTIONS
9.1 Statistical Problems 211
9.2 Problems with Rule Usage 214
    9.2.1 Non-Deterministic Behavior 215
    9.2.2 Partitioning 215
    9.2.3 Non-Unique Rule Application 217
Chapter 10: AN ANALOGICAL APPROACH
10.1 Local Homogeneity 219
10.2 Idiosyncratic Behavior 224
    10.2.1 Frequency and Contextual Specification 227
10.3 Exceptional/Regular Behavior 229
    10.3.1 Properties of Exceptional/Regular Behavior 232
    10.3.2 The Exponential Effect 234
    10.3.3 The Effect of Frequency 236
10.4 Categorical Behavior 237
10.5 Deviant Forms and Missing Information 242
Chapter 11: A NATURAL TEST FOR HOMOGENEITY
11.1 A Natural Estimator of Z 246
11.2 The Rate of Agreement 249
11.3 A Natural Statistical Test 251
11.4 Some Examples 254
11.5 Non-Symmetry of βˆ†ΞΆ 257
11.6 Advantages 258
11.7 Some Theorems 259
Chapter 12: STATISTICAL ANALOGY
12.1 An Algorithm 266
    12.1.1 Categorical Behavior 267
    12.1.2 Exceptional/Regular Behavior 273
    12.1.3 Idiosyncratic Behavior 276
12.2 Homogeneous Non-Deterministic Behavior 278
12.3 Bounds on Statistical Homogeneity 283
Chapter 13: DEFINING OTHER LEVELS OF SIGNIFICANCE
13.1 Heterogeneity at Smaller Levels of Significance 286
13.2 Eliminating Statistics Altogether 295
Chapter 14: ACTUAL EXAMPLES
14.1 Infant Survival 301
14.2 Food Poisoning 303
14.3 Final-Stop Deletion 305
14.4 Past-Tense Forms in Finnish 310
Chapter 15: ANALOGICAL ANALYSES OF CONTINUOUS VARIABLES
15.1 Continuous Contexts 323
15.2 A Non-Parametric Approach 329
15.3 Ties 339
15.4 Continuous Outcomes 342
15.5 Continuous Contexts and Outcomes 346
Chapter 16: BEHAVIORAL FACTORS
16.1 Imperfect Memory 349
16.2 The Instability of Non-Deterministic Behavior 353
16.3 Another Rule of Usage: Selection by Plurality 357
16.4 Efficiency and Processing Time 359
Concluding Remarks: A FINAL ANALOGY 364
REFERENCES 366
INDEX 371