Chapter 1: How to best use this book

Route planner - suggested journeys through Bayesland

Why don’t more people use Bayesian statistics?

What are the tangible (non-academic) benefits of Bayesian statistics?

Part I: An introduction to Bayesian inference

Chapter 2: The subjective worlds of Frequentist and Bayesian statistics

Bayes’ rule - allowing us to go from the effect back to its cause

The purpose of statistical inference

The world according to Frequentists

The world according to Bayesians

Do parameters actually exist and have a point value?

Frequentist and Bayesian inference

Bayesian inference via Bayes’ rule

Implicit versus Explicit subjectivity

Chapter 3: Probability - the nuts and bolts of Bayesian inference

Probability distributions: helping us explicitly state our ignorance

A derivation of Bayes’ rule

The Bayesian inference process from the Bayesian formula

Part II: Understanding the Bayesian formula

Chapter 4: Likelihoods

Why use ‘likelihood’ rather than ‘probability’?

What are models and why do we need them?

How to choose an appropriate likelihood?

Exchangeability vs random sampling

Maximum likelihood - a short introduction

Chapter 5: Priors

What are priors, and what do they represent?

The explicit subjectivity of priors

Combining a prior and likelihood to form a posterior

A strong model is less sensitive to prior choice

Chapter 6: The devil’s in the denominator

An introduction to the denominator

The difficulty with the denominator

How to dispense with the difficulty: Bayesian computation

Chapter 7: The posterior - the goal of Bayesian inference

Expressing parameter uncertainty in posteriors

Bayesian statistics: updating our pre-data uncertainty

The intuition behind Bayes’ rule for inference

Point parameter estimates

From posterior to predictions by sampling

Part III: Analytic Bayesian methods

Chapter 8: An introduction to distributions for the mathematically-un-inclined

The interrelation among distributions

Sampling distributions for likelihoods

How to choose a likelihood

Table of common likelihoods, their uses, and reasonable priors

Distributions of distributions, and mixtures - link to website, and relevance

Chapter 9: Conjugate priors and their place in Bayesian analysis

What is a conjugate prior and why are they useful?

Normal example: giraffe height

Table of conjugate priors

The lessons and limits of a conjugate analysis

Chapter 10: Evaluation of model fit and hypothesis testing

Posterior predictive checks

Why do we call it a p value?

Statistics measuring predictive accuracy: AIC, Deviance, WAIC and LOO-CV

Marginal likelihoods and Bayes factors

Choosing one model, or a number?

Chapter 11: Making Bayesian analysis objective?

The illusion of the ’uninformative’ uniform prior

A move towards weakly informative priors

Part IV: A practical guide to doing real life Bayesian analysis: Computational Bayes

Chapter 12: Leaving conjugates behind: Markov Chain Monte Carlo

The difficulty with real life Bayesian inference

Discrete approximation to continuous posteriors

The posterior through quadrature

Integrating using independent samples: an introduction to Monte Carlo

Why is independent sampling easier said than done?

Ideal sampling from a posterior using only the un-normalised posterior

Moving from independent to dependent sampling

What’s the catch with dependent samplers?

Chapter 13: Random Walk Metropolis

Defining the Metropolis algorithm

When does Metropolis work?

Efficiency of convergence: the importance of choosing the right proposal scale

Effective sample size revisited

Chapter 14: Gibbs sampling

Back to prospecting for gold

Defining the Gibbs algorithm

Gibbs’ earth: the intuition behind the Gibbs algorithm

The benefits and problems with Gibbs and Random Walk Metropolis

A change of parameters to speed up exploration

Chapter 15: Hamiltonian Monte Carlo

Hamiltonian Monte Carlo as a sledge

Solving for the sledge motion over NLP space

The acceptance probability of HMC

The complete Hamiltonian Monte Carlo algorithm

The performance of HMC versus Random Walk Metropolis and Gibbs

Optimal step length of HMC: introducing the “No U-Turn Sampler”

Chapter 16: Stan

Why Stan, and how to get it

Getting setup with Stan using RStan

What to do when things go wrong

Part V: Hierarchical models and regression

Chapter 17: Hierarchical models

The spectrum from fully-pooled to heterogeneous

Non-centered parameterisations in hierarchical models

Case study: Forecasting the EU referendum result

The importance of fake data simulation for complex models

Chapter 18: Linear regression models

Example: high school test scores in England

Heterogeneous coefficient model

Incorporating LEA-level data

Chapter 19: Generalised linear models and other animals

Example: electoral participation in European countries

Discrete parameter models in Stan