Bayes’ Rule

A Tutorial Introduction to Bayesian Analysis

James V Stone

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 Aected 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

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