LIMITED OFFER
Save 50% on book bundles
Immediately download your ebook while waiting for your print delivery. No promo code is needed.
Rational Descriptions, Decisions and Designs
Pergamon Unified Engineering Series
- 1st Edition - January 1, 1969
- Author: Myron Tribus
- Editors: Thomas F. Irvine, James P. Hartnett
- Language: English
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 4 6 2 3 - 2
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… Read more
Purchase options
Institutional subscription on ScienceDirect
Request a sales quoteRational 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.
Foreword
Preface
Concerning the Use of the Book as a Text
Chapter 1. What Do We Mean by Rational
What Has the Concern for Rationality to Do with Design
Inductive Logic
Desiderata for an Inductive Logic Computer
A First Consequence of the Desiderata
A Second Consequence of the Desiderata
The Functional Equations
A Solution to the Functional Equation
A Second Functional Relation
A Particular Choice of Variable
Concerning Allowable Transformations
The Problem of Inference
Appendix A—The Solution to the First Functional Equation
Appendix B—The Solution to the Second Functional Equation
Chapter 2. Rational Descriptions, some Mathematical Preliminaries
Probability of an "OR" Statement
A Geometrical Interpretation of the Additive Rules
Contingency Table
The Encoding of Symmetrical Knowledge
Unsymmetrical Consequences of Symmetrical Knowledge
The Concept of a Bernoulli Trial
The Multinomial Distribution
Stirling's Approximation for n! and Log n!
The Probability of a Particular Frequency
The Concept of Expectation
Expectations and Mathematical Expectations
The Variance and Higher Moments
Additional Remarks on the Difference between Frequency and Probability
Continuous Distributions
Deterministic Knowledge about a Discrete or Continuous Variable
Chapter 3. Bayes' Equation and Rational Inference
Bayes' Equation
Bayes' Equation and the Role of Prior Probabilities—The Extension Rule
On the Futility of Arguments over the Need for Prior Probabilities
The Concept of Statistical Dependence
Hypothesis Testing and the Evidence Form of Bayes' Equation
Example Problem: The Spark Plug Manufacturer
Multiple Outcomes and Non-Bernoulli Processes
Is There a Hypothesis We Had Not Considered
The Chi-Square Test of an Hypothesis
Test of a Random Number Generator
Chapter 4. A Rational Measure of Uncertainty
Entropy as a Measure of Uncertainty
The Use of Other than Binary Questions
Chapter 5. The Principle of Maximum Entropy
The Maximum Entropy Formalism
Proof that S is at a Maximum and Not a Local Saddle Point
Some Properties of the Maximum Entropy Distribution
Maximum Entropy Probability Distributions
The Uniform Distribution
The Exponential Distribution
The Truncated and the Normal Gaussian Distributions
The Gamma Distribution
The Incomplete Gamma Function
The Beta Distribution
Some Other Probability Distributions
Entropy and Hypothesis Testing
Some Sample Calculations
Appendix C—The Error Function and its Approximation
Appendix D—Using the Digamma Function to Fit the Beta
Distribution
Chapter 6. Contingency Tables
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 IB)
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 X2A)
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
Chapter 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
Queuing Problems
The Poisson Process from Maximum Entropy Considerations
The Poisson Distribution as the Limit of a Binomial Distribution
Application to the Queuing Problem
Rare Occurrences
An Approximate Equation for Small Probabilities (The Weakest Link)
The Largest Member of a Set
Inferring the Parameter a
Return Period
An Alternative Method (Method of Thomas)
Appendix G—The Z-Transform Pair
Appendix H—The Ramp, Step and Delta Functions
Appendix I—The Sum of Variables, Each of which is Described by a Gaussian
Appendix J—Simulation via High Speed Digital Computer
Appendix K—The Use of Jacobians in Change of Variables
Chapter 8. Decision Theory
What are the Elements of a Decision
Decision Trees
Strategies and Values
Utility or Value Functions
The Making of Strategies
More About Utilities
The Utility of Money
A Utility Curve with Constant Behavior
Utilities and Prior Probabilities
The Value of Perfect Information
The Design of an Experiment
Sequential Testing
An Alternative Formulation for Decision Analysis
A Competitive Bid Under Uncertainty
Appendix L —Finding the Optimum Location for Terminating the Wald Sequential Procedure
Chapter 9. Engineering Design
Towards a Theory of the Design Process
The Utility
Generating Alternatives
The Design Process
Why Decision Theory at All
Decision Theory Solution to the Widget Problem
The Design of a Transmitter
Design of a Voltmeter
Appendix M—Relating System Performance to Performance of Components
Chapter 10. Reliability Engineering
Rational Descriptions of Reliability
The Concept of Level of Complexity
Some Basic Concepts of Reliability Engineering
The Force of Mortality
Wearing Out Or Wearing In
Updating Reliability Data
Using Life Test Data to Establish an Exponential Distribution
The Effect of Failures
The Probability of Failure
The Probability of Success
Asymptotic Behavior as n Gets Very Large
A Different State of Prior Knowledge
Bayesian Inference and Classical Procedures Contrasted
Failure of a System of Series Connected Components
The Tyranny of Numbers
Parallel or Redundant Systems
Series and Parallel Systems
The Probability of Failure under Load
Approximate Probability Distributions from Data on Tolerances
Matrix Methods
Example: The Wheatstone Bridge
Example: An Application to a Dynamic System
Example: The Simple Transistor Amplifier
Appendix N—Evaluation of a Definite Integral
Index
- No. of pages: 498
- Language: English
- Edition: 1
- Published: January 1, 1969
- Imprint: Pergamon
- eBook ISBN: 9781483146232
TI
Thomas F. Irvine
Affiliations and expertise
Department of Mechanical Engineering
State University of New York at Stony Brook
Stony Brook, New YorkRead Rational Descriptions, Decisions and Designs on ScienceDirect