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A Student’s Guide to Bayesian Statistics

A Student’s Guide to Bayesian Statistics

April 2018 | 512 pages | SAGE Publications Ltd

Supported by a wealth of learning features, exercises, and visual elements as well as online video tutorials and interactive simulations, this book is the first student-focused introduction to Bayesian statistics.

Without sacrificing technical integrity for the sake of simplicity, the author draws upon accessible, student-friendly language to provide approachable instruction perfectly aimed at statistics and Bayesian newcomers. Through a logical structure that introduces and builds upon key concepts in a gradual way and slowly acclimatizes students to using R and Stan software, the book covers:

  • An introduction to probability and Bayesian inference
  • Understanding Bayes' rule 
  • Nuts and bolts of Bayesian analytic methods
  • Computational Bayes and real-world Bayesian analysis
  • Regression analysis and hierarchical methods

This unique guide will help students develop the statistical confidence and skills to put the Bayesian formula into practice, from the basic concepts of statistical inference to complex applications of analyses.

Chapter 1: How to best use this book
The purpose of this book  
Who is this book for?  
Book outline  
Route planner - suggested journeys through Bayesland  
Problem sets  
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  
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  
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  
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
College of Business and Economics, Australian National University

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
Information Technology, Monash University

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
Psychology, Rice University

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
Senior Lecturer in Statistics, School of Science & Technology, Nottingham Trent University

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
Professor Emeritus of Statistics, Swarthmore College

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