Bayes for beginners



Even for the most currmudgeonly frequentist it is sometimes refreshing to move your effort from deriving frequentist estimators for intractable models, to using the damn Bayesian ones, which fail in different and interesting ways than you are used to. If it works and you are feeling fancy you might then justify your Bayesian method on frequentist grounds, which washes away the sin.

Here are some scattered tidbits about getting into it. No attempt is made to be comprehensive, novel, or to even expert.

Teaching

Course material

So many! Too many. Actually I kinda like McElreath’s stuff to teach from; You get practical quite quickly.

Linear regression

This workhorse pops up everywhere.

Deisenroth and Zafeiriou, Mathematics for Inference and Machine Learning give an ML perspective.

Workflow

The visualisation howto from, basically, the Stan team, is a deeper than it sounds and highly recommended (Gabry et al. 2019).

Michael Betancourt’s examples, for example his workflow tips, are a good start for practical work, incorporating the inevitable collision of statistical and computational difficulties.

See also BAT the Bayesian Analysis Toolkit, which does sophisticated Bayes modelling although AFAICT uses a fairly basic Sampler?

Notes on Rao-Blackwellisation for doing faster MCMC inference, and even handling discrete parameters in Stan.

Nonparametrics

Dirichlet processes, Gaussian Process regression etc. πŸ—

As a methodology of science

Not quite.

References

Alquier, Pierre. 2021. β€œUser-Friendly Introduction to PAC-Bayes Bounds.” arXiv:2110.11216 [Cs, Math, Stat], October.
Bacchus, F, H E Kyburg, and M Thalos. 1990. β€œAgainst Conditionalization.” Synthese 85 (3): 475–506.
Barbier, Jean, and Nicolas Macris. 2017. β€œThe Stochastic Interpolation Method: A Simple Scheme to Prove Replica Formulas in Bayesian Inference.” arXiv:1705.02780 [Cond-Mat], May.
Bernardo, JosΓ© M., and Adrian F. M. Smith. 2000. Bayesian Theory. 1 edition. Chichester: Wiley.
Carpenter, Bob, Matthew D. Hoffman, Marcus Brubaker, Daniel Lee, Peter Li, and Michael Betancourt. 2015. β€œThe Stan Math Library: Reverse-Mode Automatic Differentiation in C++.” arXiv Preprint arXiv:1509.07164.
Caruana, Rich. 1998. β€œMultitask Learning.” In Learning to Learn, 95–133. Springer, Boston, MA.
Deisenroth, Marc, and Stefanos Zafeiriou. 2017. β€œMathematics for Inference and Machine Learning.” Dept. Comput., Imperial College London, London, UK, Tech. Rep., Accessed on Jul, 126.
Diaconis, Persi, and Donald Ylvisaker. 1979. β€œConjugate Priors for Exponential Families.” The Annals of Statistics 7 (2): 269–81.
Domingos, Pedro. 2020. β€œEvery Model Learned by Gradient Descent Is Approximately a Kernel Machine.” arXiv:2012.00152 [Cs, Stat], November.
Fink, Daniel. 1997. β€œA Compendium of Conjugate Priors,” 46.
Gabry, Jonah, Daniel Simpson, Aki Vehtari, Michael Betancourt, and Andrew Gelman. 2019. β€œVisualization in Bayesian Workflow.” Journal of the Royal Statistical Society: Series A (Statistics in Society) 182 (2): 389–402.
Gelman, Andrew. 2006. β€œPrior Distributions for Variance Parameters in Hierarchical Models (Comment on Article by Browne and Draper).” Bayesian Analysis 1 (3): 515–34.
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Gelman, Andrew, Jennifer Hill, and Aki Vehtari. 2021. Regression and other stories. Cambridge, UK: Cambridge University Press.
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Gelman, Andrew, and Yuling Yao. 2021. β€œHoles in Bayesian Statistics.” Journal of Physics G: Nuclear and Particle Physics 48 (1): 014002.
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Li, Meng, and David B. Dunson. 2016. β€œA Framework for Probabilistic Inferences from Imperfect Models.” arXiv:1611.01241 [Stat], November.
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Mandt, Stephan, Matthew D. Hoffman, and David M. Blei. 2017. β€œStochastic Gradient Descent as Approximate Bayesian Inference.” JMLR, April.
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