Stone J.V. Bayes' Rule with R. A Tutorial Introduction to Bayesian Analysis

New York: Sebtel Press, 2016. — 187 p.Discovered by an 18th century mathematician and preacher, Bayes' rule is a cornerstone of modern probability theory. In this richly illustrated book, a range of accessible examples is used to show how Bayes' rule is actually a natural consequence of common sense reasoning. Bayes' rule is then derived using intuitive graphical representations of probability, and Bayesian analysis is applied to parameter estimation. The tutorial style of writing, combined with a comprehensive glossary, makes this an ideal primer for novices who wish to become familiar with the basic principles of Bayesian analysis. Note that this book includes R (3.2) code snippets, which reproduce key numerical results and diagrams.An Introduction to Bayes’ Rule
Example 1: Poxy Diseases
Example 2: Forkandles
Example 3: Flipping Coins
Example 4: Light Craters
Forward and Inverse Probability
Bayes’ Rule in Pictures
Random Variables
The Rules of Probability
Joint Probability and Coin Flips
Probability As Geometric Area
Bayes’ Rule From Venn Diagrams
Bayes’ Rule and the Medical Test
Discrete Parameter Values
Joint Probability Functions
Patient Questions
Deriving Bayes’ Rule
Using Bayes’ Rule
Bayes’ Rule and the Joint Distribution
Continuous Parameter Values
A Continuous Likelihood Function
A Binomial Prior
The Posterior
A Rational Basis For Bias
The Uniform Prior
Finding the MAP Analytically
Evolution of the Posterior
Reference Priors
Loss Functions
Gaussian Parameter Estimation
The Gaussian Distribution
Estimating the Population Mean
Error Bars for Gaussian Distributions
Regression as Parameter Estimation
A Bird’s Eye View of Bayes’ Rule
Joint Gaussian Distributions
A Bird’s-Eye View of the Joint Distribution
A Bird’s-Eye View of Bayes’ Rule
Slicing Through Joint Distributions
Statistical Independence
Bayesian Wars
The Nature of Probability
Bayesian Wars
A Very Short History of Bayes’ Rule
Further Reading
AppendicesMathematical Symbols
The Rules of Probability
Probability Density Functions
The Binomial Distribution
The Gaussian Distribution
Least-Squares Estimation
Reference Priors