Bayes' Rule: A Tutorial Introduction to Bayesian Analysis

by James V Stone

Overview

What does a medical test tell us about the chances of having a particular disease? How can we tell if a spoken phrase is, 'four candles' or 'fork handles'? How do we a perceive a three-dimensional world from from the two-dimensional images on our retinas? The short answer is Bayes' rule, which transforms meaningless statistics and raw data into useful information. Discovered by an 18th century mathematician and preacher, Bayes' rule is a cornerstone of modern probability theory. In this richly illustrated book, intuitive visual representations of real-world examples are used to show how Bayes' rule is actually a form of commonsense reasoning. The tutorial style of writing, combined with a comprehensive glossary, makes this an ideal primer for novices who wish to gain an intuitive understanding of Bayesian analysis. As an aid to understanding, online computer code (in MatLab, Python and R) reproduces key numerical results and diagrams.

Product Details

ISBN-13:	9780956372895
Publisher:	Sebtel Press
Publication date:	03/01/2022
Series:	Tutorial Introductions
Pages:	184
Product dimensions:	6.00(w) x 9.00(h) x 0.39(d)

About the Author

James V Stone is an Honorary Associate Professor at the University of Sheffiled, England.

Table of Contents

Preface

1 An Introduction to Bayes' Rule

1.1 Example 1: Poxy Diseases

1.2 Example 2: Forkandles

1.3 Example 3: Flipping Coins

1.4 Example 4: Light Craters

1.5 Forward and Inverse Probability

2 Bayes' Rule in Pictures

2.1 Random Variables

2.2 The Rules of Probability

2.3 Random Variables and Coin Flips

2.4 Joint Probability and Coin Flips

2.5 Probability As Geometric Area

2.6 Bayes' Rule From Venn Diagrams

2.7 Bayes' Rule and the Medical Test

3 Discrete Parameter Values

3.1 Joint Probability Functions

3.2 Patient Questions

3.3 Deriving Bayes' Rule

3.4 Using Bayes' Rule

3.5 Bayes' Rule and the Joint Distribution

4 Continuous Parameter Values

4.1 A Continuous Likelihood Function

4.2 A Binomial Prior Probability Density Function

4.3 A Posterior Probability Density Function

4.4 A Uniform Prior Probability Density Function

4.5 MAP Estimates Are Not Affected By Constants

4.6 Finding the MAP Estimate Analytically

4.7 Evolution of the Posterior

4.8 Reference Priors

4.9 Loss Functions

5 Gaussian Parameter Estimation

5.1 The Gaussian Distribution

5.2 Estimating the Population Mean

5.3 Error Bars for Gaussian Distributions

5.4 Regression as Parameter Estimation

6 A Bird's-Eye View of Bayes' Rule

6.1 Joint Gaussian Distributions

6.2 A Bird's-Eye View of Joint Distributions

6.3 A Bird's-Eye View of Bayes' Rule

6.4 Slicing Through Joint Distributions

6.5 Statistical Independence

7 Bayesian Wars

7.1 The Nature of Probability

7.2 Subjective Probability

7.3 Bayesian Wars

7.4 A Very Short History of Bayes' Rule

Further Reading

Appendices

A Glossary

B Mathematical Symbols

C The Rules of Probability

D Probability Density Functions

E The Binomial

F The Gaussian

G Least-Squares Estimation

H Reference Priors

I MatLab Code

References

Index