Further Reading

This book covers the foundations. The references below point to authoritative texts and papers for readers who want to go deeper. The historical sources named in the chapters (Kolmogorov, Bernoulli, de Moivre, Laplace, Markov, and others) are not repeated here; this list gathers modern texts and the specific papers behind the methods in Chapter 11.

• William Feller, An Introduction to Probability Theory and Its Applications, Volumes I and II (Wiley). The classic two-volume treatment.
• Sheldon Ross, A First Course in Probability (Pearson). A standard undergraduate text.
• Geoffrey Grimmett and David Stirzaker, Probability and Random Processes (Oxford). Strong on random processes.
• Joseph Blitzstein and Jessica Hwang, Introduction to Probability (CRC). Clear and example-driven, with free online lectures.
• Rick Durrett, Probability: Theory and Examples (Cambridge). A graduate measure-theoretic text.
• Patrick Billingsley, Probability and Measure (Wiley). The measure-theoretic foundations sketched in Chapter 1.
• David Williams, Probability with Martingales (Cambridge). A short, elegant route through measure theory and martingales.

• Larry Wasserman, All of Statistics (Springer). A fast, broad bridge from probability to statistics and machine learning.
• Andrew Gelman, John Carlin, Hal Stern, David Dunson, Aki Vehtari, and Donald Rubin, Bayesian Data Analysis, 3rd edition (CRC). The standard reference for the Bayesian material in Chapters 9 and 11.
• Christopher Bishop, Pattern Recognition and Machine Learning (Springer, 2006).
• Kevin Murphy, Probabilistic Machine Learning: An Introduction (MIT, 2022).
• David MacKay, Information Theory, Inference, and Learning Algorithms (Cambridge, 2003). Free online, and excellent on the link between probability and information.

• David Levin and Yuval Peres, Markov Chains and Mixing Times (AMS). The eigenvalue and mixing-time material from Chapter 10.
• James Norris, Markov Chains (Cambridge).
• Christian Robert and George Casella, Monte Carlo Statistical Methods (Springer).
• Steve Brooks, Andrew Gelman, Galin Jones, and Xiao-Li Meng, editors, Handbook of Markov Chain Monte Carlo (CRC, 2011).

• Nicholas Metropolis, Arianna Rosenbluth, Marshall Rosenbluth, Augusta Teller, and Edward Teller (1953), “Equation of State Calculations by Fast Computing Machines,” Journal of Chemical Physics.
• W. K. Hastings (1970), “Monte Carlo Sampling Methods Using Markov Chains and Their Applications,” Biometrika.
• Simon Duane, Anthony Kennedy, Brian Pendleton, and Duncan Roweth (1987), “Hybrid Monte Carlo,” Physics Letters B. The origin of Hamiltonian Monte Carlo.
• Radford Neal (2011), “MCMC Using Hamiltonian Dynamics,” in the Handbook of Markov Chain Monte Carlo.
• Michael Betancourt (2017), “A Conceptual Introduction to Hamiltonian Monte Carlo,” arXiv:1701.02434.
• Matthew Hoffman and Andrew Gelman (2014), “The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo,” Journal of Machine Learning Research. The step-size adaptation used by the HMC engine.
• Alp Kucukelbir, Dustin Tran, Rajesh Ranganath, Andrew Gelman, and David Blei (2017), “Automatic Differentiation Variational Inference,” Journal of Machine Learning Research. The recipe the variational engine follows.
• David Blei, Alp Kucukelbir, and Jon McAuliffe (2017), “Variational Inference: A Review for Statisticians,” Journal of the American Statistical Association.
• Diederik Kingma and Jimmy Ba (2015), “Adam: A Method for Stochastic Optimization,” International Conference on Learning Representations. The optimizer used by the variational engine.
• Charles Geyer (1992), “Practical Markov Chain Monte Carlo,” Statistical Science. The initial positive sequence estimator behind the ESS diagnostic.
• Andrew Gelman and Donald Rubin (1992), “Inference from Iterative Simulation Using Multiple Sequences,” Statistical Science. The original R-hat statistic.
• Aki Vehtari, Andrew Gelman, Daniel Simpson, Bob Carpenter, and Paul-Christian Bürkner (2021), “Rank-normalization, Folding, and Localization: An Improved R-hat for Assessing Convergence of MCMC,” Bayesian Analysis.
• George Box and Mervin Muller (1958), “A Note on the Generation of Random Normal Deviates,” Annals of Mathematical Statistics. The transform used to sample normals.
• Atılım Güneş Baydin, Barak Pearlmutter, Alexey Radul, and Jeffrey Siskind (2018), “Automatic Differentiation in Machine Learning: A Survey,” Journal of Machine Learning Research. Background for the dual-number differentiation.

• The NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov, is the modern successor to Abramowitz and Stegun and gives the error-function and Gamma-function approximations used in Chapters 5, 9, and 11.