Table Of Contents:

• Chapter 1: How to best use this book
• The purpose of this book
• Who is this book for?
• Pre-requisites
• Book outline
• Route planner - suggested journeys through Bayesland
• Video
• Problem sets
• Code
• R and Stan
• 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
• Independence
• Central Limit Theorems
• A derivation of Bayes’ rule
• The Bayesian inference process from the Bayesian formula
• Part II: Understanding the Bayesian formula
• Chapter 4: Likelihoods
• What is a likelihood?
• 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
• Constructing priors
• 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
• Intervals of uncertainty
• 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
• Prior distributions
• 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?
• Gamma-poisson example
• 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?
• Sensitivity analysis
• Chapter 11: Making Bayesian analysis objective?
• The illusion of the ’uninformative’ uniform prior
• Jeffreys’ priors
• Reference priors
• Empirical Bayes
• 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
• Sustainable fishing
• Prospecting for gold
• Defining the Metropolis algorithm
• When does Metropolis work?
• Efficiency of convergence: the importance of choosing the right proposal scale
• Metropolis-Hastings
• Judging convergence
• 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
• NLP space
• Solving for the sledge motion over NLP space
• How to shove the sledge
• 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
• Our first words in Stan
• Essential Stan reading
• What to do when things go wrong
• How to get further help
• 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
• Pooled model
• Interactions
• Heterogeneous coefficient model
• Hierarchical model
• Incorporating LEA-level data
• Chapter 19: Generalised linear models and other animals
• Example: electoral participation in European countries
• Discrete parameter models in Stan