Rational Descriptions, Decisions and Designs

Rational Descriptions, Decisions and Designs

Pergamon Unified Engineering Series

1st Edition - January 1, 1969

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  • Author: Myron Tribus
  • eBook ISBN: 9781483146232

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Description

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.

Table of Contents


  • 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

Product details

  • No. of pages: 498
  • Language: English
  • Copyright: © Pergamon 1969
  • Published: January 1, 1969
  • Imprint: Pergamon
  • eBook ISBN: 9781483146232

About the Author

Myron Tribus

About the Editors

Thomas F. Irvine

Affiliations and Expertise

Department of Mechanical Engineering State University of New York at Stony Brook Stony Brook, New York

James P. Hartnett

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