# Rational Descriptions, Decisions and Designs

## 1st Edition

### Pergamon Unified Engineering Series

**Authors:**Myron Tribus

**Editors:**Thomas Irvine James P. Hartnett

**eBook ISBN:**9781483146232

**Imprint:**Pergamon

**Published Date:**1st January 1969

**Page Count:**498

## 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

## Details

- No. of pages:
- 498

- Language:
- English

- Copyright:
- © Pergamon 1969

- Published:
- 1st January 1969

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483146232

## About the Author

### Myron Tribus

## About the Editor

### Thomas Irvine

### Affiliations and Expertise

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