Rational Descriptions, Decisions and Designs
Pergamon Unified Engineering Series
Published on 22. October 2013
498 pages
978-1-4831-4623-2 (ISBN)
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.
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Content
ForewordPrefaceConcerning the Use of the Book as a TextChapter 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 EquationChapter 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 VariableChapter 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 GeneratorChapter 4. A Rational Measure of Uncertainty Entropy as a Measure of Uncertainty The Use of Other than Binary QuestionsChapter 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 DistributionChapter 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 TableChapter 7.
