A Student's Guide to Bayesian Statistics

Ben Lambert is a researcher at Imperial College London where he works on the epidemiology of malaria. He has worked in applied statistical inference for about a decade, formerly at the University of Oxford, and is the author of over 500 online lectures on econometrics and statistics. He also somewhat strangely went to school in Thomas Bayes' home town for many years, Tunbridge Wells.

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

An excellent resource on Bayesian analysis accessible to students from a diverse range of statistical backgrounds and interests. Easy to follow with well documented examples to illustrate key concepts. -- Bronwyn Loong When I was a grad student, Bayesian statistics was restricted to those with the mathematical fortitude to plough through source literature. Thanks to Lambert, we now have something we can give to the modern generation of nascent data scientists as a first course. Love the supporting videos, too! -- Wray Buntine Written in highly accessible language, this book is the gateway for students to gain a deep understanding of the logic of Bayesian analysis and to apply that logic with numerous carefully selected hands-on examples. Lambert moves seamlessly from a traditional Bayesian approach (using analytic methods) that serves to solidify fundamental concepts, to a modern Bayesian approach (using computational sampling methods) that endows students with the powerful and practical powers of application. I would recommend this book and its accompanying materials to any students or researchers who wish to learn and actually do Bayesian modeling. -- Fred Oswald A balanced combination of theory, application and implementation of Bayesian statistics in a not very technical language. A tangible introduction to intangible concepts of Bayesian statistics for beginners. -- Golnaz Shahtahmassebi The late, famous statistician Jimmie Savage would have taken great pleasure in this book based on his work in the 1960s on Bayesian statistics. He would have marveled at the presentations in the book of many new and strong statistical and computer analyses. -- Gudmund R. Iversen While there is increasing interest in Bayesian statistics among scholars of different social science disciplines, I always looked for a text book which is accessible to a wide range of students who do not necessarily have extended knowledge of statistics. Now, I believe that this is the first textbook of Bayesian statistics, which can also be used for social science undergraduate students. Ben Lambert begins with a general introduction to statistical inference and successfully brings the readers to more specific and practical aspects of Bayesian inference. In addition to its well-considered structure, many graphical presentations and reasonable examples contribute for a broader audience to obtain well-founded understanding of Bayesian statistics. -- Susumu Shikano This book offers a path to get into the field of Bayesian statistics with no previous knowledge. Building from elementary to advanced topics, including theoretic and computational aspects, and focusing on the application, it is an excellent read for newcomers to the Bayesian world. -- Panagiotis Tsiamyrtzis