Rational Descriptions, Decisions and Designs is a reference for understanding the aspects of rational decision theory in terms of the basic formalism of information theory. The text provides ways to achieve correct engineering design decisions.
The book starts with an understanding for the need to apply rationality, as opposed to uncertainty, in design decision making. Inductive logic in computers is explained where the design of the machine and the accompanying software are considered. The text then explains the functional equations and the problems of arriving at a rational description through some mathematical preliminaries. Bayes' equation and rational inference as tools for adjusting probabilities when something new is encountered in earlier probability distributions are explained. The book presents as well a case study concerning the error made in following specifications of spark plugs. The author also explains the Bernoulli trials, where a probability that a better hypothesis than that already adopted may exist. The rational measure of uncertainty and the principle of maximum entropy with sample calculations are included in the text. After considering the probabilities, the decision theory is taken up where engineering design follows. Examples regarding transmitter and voltmeter designs are presented. The book ends by explaining probabilities of success and failure as applied to reliability engineering, that it is a state of knowledge rather than the state of a thing.
The text can serve as a textbook for students in technology engineering and design, and as a useful reference for mathematicians, statisticians, and fabrication engineers.
Author(s): Tribus, Myron
Publisher: Pergamon Press
Year: 1969
Language: English
Pages: 489
City: New York
Foreword xi
Preface xv
Concerning the Use of the Book as a Text xix
1. What Do We Mean by Rational? 1
What Has the Concern for Rationality to Do With Design? 2
Inductive Logic 3
Desiderata for an Inductive Logic Computer 4
A First Consequence of the Desiderata 6
A Second Consequence of the Desiderata 13
The Functional Equations 14
A Solution to the Functional Equation 18
A Second Functional Relation 21
A Particular Choice of Variable 24
Concerning Allowable Transformations 26
The Problem of Inference 29
Appendix A-The Solution to the First Functional Equation 31
Appendix B-The Solution to the Second Functional Equation 34
2. Rational Descriptions, Some Mathematical Preliminaries 37
Probability of an "OR" Statement 37
A Geometrical Interpretation of the Additive Rules 38
Contingency Table 39
The Encoding of Symmetrical Knowledge 48
Unsymmetrical Consequences of Symmetrical Knowledge 50
The Concept of a Bernoulli Trial 51
The Multinomial Distribution 53
Stirling's Approximation for n! and log n! 54
The Probability of a Particular Frequency 54 The Concept of Expectation 60
Expectations and Mathematical Expectations 62
The Variance and Higher Moments 63
Additional Remarks on the Difference Between Frequency and Probability 65
Continuous Distributions 67
Deterministic Knowledge About a Discrete or Continuous Variable 71
3. Bayes' Equation and Rational Inference 73
Bayes' Equation 74
Bayes' Equation and the Role of Prior Probabilities-The Extension Rule 78
On the Futility of Arguments over the Need for Prior Probabilities 80
The Concept of Statistical Dependence 82
Hypothesis Testing and the Evidence Form of Bayes' Equation 83
Example Problem: The Spark Plug Manufacturer 86
Multiple Outcomes and Non-Bernoulli Processes 93
Is There a Hypothesis We Had Not Considered? 96
The Chi-Square Test of an Hypothesis 101
Test of a Random Number Generator 104
4. A Rational Measure of Uncertainty 107
Entropy as a Measure of Uncertainty 111
The Use of Other than Binary Questions 114
5. The Principle of Maximum Entropy 119
The Maximum Entropy Formalism 120
Proof that S is at a Maximum and not a Local Saddle Point 123
Some Properties of the Maximum Entropy Distribution 124
Maximum Entropy Probability Distributions 127
The Uniform Distribution 128
The Exponential Distribution 129
The Truncated and the Normal Gaussian Distributions 131
The Gamma Distribution 144
The Incomplete Gamma Function 148
The Beta Distribution 150
Some Other Probability Distributions 154
Entropy and Hypothesis Testing 156
Some Sample Calculations 159
Appendix C-The Error Function and its Approximation 167
Appendix D-Using the Digamma Function to Fit the Beta Distribution 175
6. Contingency Tables 181
Some Examples of Contingency Tables
Prior Information
The Summation Convention
An Important Integral
The 2 x 2 Table with State of Knowledge 1A
The r x s Table with State of Knowledge 1A
The Entropy Test and the Chi Square Test
The Effect of Controls During Experimentation (Knowledge 1B)
Testing the Analysis by Simulation
Describing the Statistical Dependence
The Effect of Prior Knowledge of the System (State of Knowledge X3A)
The Relation Between Priors for the Center and the Margin of the Table
The Effect of Knowing Precisely the Probability of One Attribute (State of Knowledge X2.J
Combining the Results from Two Tables
On the Consistency of the Method
The Three Level Table (State of Knowledge X1A)
Simulation Runs to Test the Accuracy of the Analysis of Three Level Tables
Treating Hypotheses H8 to H11
Appendix E-A Definite Multiple Integral
Appendix F-A Computer Program for the Three Level Table
7. Rational Descriptions
Estimating a Competitor's Production
Number of Parts in Service
Estimating the Number of Defects
Probability Distribution for Sums-The "Stack-Up" Problem
An Assembly of Two Components (Rectangular Distributions)
A Three Component Assembly (Rectangular Distributions)
The Central Limit Theorem and the Principle of Maximum Entropy
The Sum of Variables, Each of Which is Described by a Gaussian
Successive Observations of a Rate Constant for Mass Transfer
Estimating the Rate of Arrival of Orders
Proof of the Recursion Formula
Queueing Problems
The Poisson Process from Maximum Entropy Considerations
The Poisson Distribution as the Limit of a Binomial Distribution
Application to the Queueing Problem
Rare Occurrences
An Approximate Equation for Small Probabilities (The Weakest Link) 295
The Largest Member of a Set 298
Inferring the Parameter a 303
Return Period 305
An Alternative Method (Method of Thomas) 306
Appendix G-The Z-Transform Pair 309
Appendix H-The Ramp, Step and Delta Functions 312
Appendix I-The Sum of Variables, Each of which is Described by a Gaussian 315
Appendix J-Simulation Via High Speed Digital Computer 317
Appendix K-The Use of Jacobians in Change of Variables 325
8. Decision Theory 329
What are the Elements of a Decision? 329
Decision Trees 332
Strategies and Values 335
Utility or Value Functions 336
The Making of Strategies 337
More About Utilities 344
The Utility of Money 346
A Utility Curve with Constant Behavior 352
Utilities and Prior Probabilities 354
The Value of Perfect Information 358
The Design of an Experiment 362
Sequential Testing 371
An Alternative Formulation for Decision Analysis 380
A Competitive Bid Under Uncertainty 381
Appendix L -Finding the Optimum Location for Terminating
the Wald Sequential Procedure 384
9. Engineering Design 389
Towards a Theory of the Design Process 389
The Utility 392
Generating Alternatives 393
The Design Process 394
Why Decision Theory at All? 396
Decision Theory Solution to the Widget Problem 398
The Design of a Transmitter 404
Design of a Voltmeter 410
Appendix M-Relating System Performance to Performance of Components 417
10. Reliability Engineering 421
Rational Descriptions of Reliability 421
The Concept of Level of Complexity 424
Some Basic Concepts of Reliability Engineering 425
The Force of Mortality 427
Wearing Out Or Wearing In? 428
Updating Reliability Data 430
Using Life Test Data to Establish an Exponential Distribution 431
The Effect of Failures 437
The Probability of Failure 438
The Probability of Success 438
Asymptotic Behavior as n Gets Very Large 439
A Different State of Prior Knowledge 440
Bayesian Inference and Classical Procedures Contrasted 442
Failure of a System of Series Connected Components 448
The Tyranny of Numbers 450
Parallel or Redundant Systems 451
Series and Parallel Systems 453
The Probability of Failure Under Load 454
Approximate Probability Distributions from Data on Tolerances 456
Matrix Methods 458
Example: The Wheatstone Bridge 459
Example: An Application to a Dynamic System 461
Example: The Simple Transistor Amplifier 467
Appendix N-Evaluation of a Definite Integral 469
Index 473
