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Bayesian Statistics

A PhD-level course at EMAp.

To compile the slides, you'll need to do

pdflatex -interaction=nonstopmode --shell-escape bayes_stats

a few times to get it right.

Pre-requisites

Books

Resources

  • An overview of computing techniques for Bayesian inference can be found here.
  • See Esteves, Stern and Izbicki's course notes.
  • Rafael Stern's excellent course.
  • Principles of Uncertainty by the inimitable J. Kadane is a book about avoiding being a sure loser. See this review by Christian Robert.
  • Bayesian Learning by Mattias Vilani is a book for a computation-minded audience.
  • Michael Betancourt's website is a treasure trove of rigorous, modern and insightful applied Bayesian statistics. See this as a gateway drug.
  • Awesome Bayes is a curated list of bayesian resources, including blog posts and podcasts.

Acknowledgements

Guido Moreira and Isaque Pim suggested topics, exercises and exam questions. Lucas Moschen made many good suggestions.

Exercises

We keep a list here. I recommend you check back every so often because this is likely to be updated (if infrequently).

News

  • Papers for the assignment are here. A bib file is also made available. Turn in your choice by 18h (Brasília time) on 2024-06-19.
  • Discussion guide is now available. Hand-in deadlineis 2024-07-05 at 16h Brasília time.

Syllabus

Lecture 0: Overview

Lecture 1: Principled Inference, decision-theoretical foundations

  • Berger and Wolpert's 1988 monograph is the definitive text on the Likelihood Principle (LP).
  • See this paper By Franklin and Bambirra for a generalised version of the LP.
  • As advanced reading, one can consider Birnbaum (1962) and a helpful review paper published 30 years later by Bjornstad.
  • Michael Evans has a few papers on the LP. See Evans, Fraser & Monette (1986) for an argument using a stronger version of CP and Evans, 2013 for a flaw with the original 1962 paper by Birnbaum.
  • Deborah G. Mayo challenged Birnbaum's argument on the LP. But Mayo implicitly changed the statement of the SP, nullifing her point. This Cross-Validate post adds more details to the story and to the relevance of the LP.

Lecture 2: Belief functions, coherence, exchangeability

Lecture 3: Priors I: rationale and construction; conjugate analysis

Lecture 4: Priors II: types of priors; implementation

Required reading

Optional reading

Lecture 5: Bayesian point estimation

  • The paper The Federalist Papers As a Case Study by Mosteller and Wallace (1984) is a very nice example of capture-recapture models. It is cited in Sharon McGrayne's book "The Theory That Would Not Die" as triumph of Bayesian inference. It is also a serious contender for coolest paper abstract ever.
  • This post in the Andrew Gelman blog discusses how to deal with the sample size (n) in a Bayesian problem: either write out a full model that specifies a probabilistic model for n or write an approximate prior pi(theta|n).

Lecture 6: Bayesian Testing I

Required reading

Optional reading

Lecture 7: Bayesian Testing II

  • This paper by Lavine and Schervish provides a nice "disambiguation" for what Bayes factors can and cannot do inferentially.
  • Yao et al. (2018) along with ensuing discussion is a must-read for an understanding of modern prediction-based Bayesian model comparison.

Lecture 8: Asymptotics

  • The entry on the Encyclopedia of Mathematics on the Bernstein-von Mises theorem is nicely written.
  • The integrated nested Laplace approximation (INLA) methodology leverages Laplace approximations to provide accurate approximations to the posterior in latent Gaussian models, which cover a huge class of models used in applied modelling. This by Thiago G. Martins and others, specially section 2, is a good introduction.

Lecture 9: Applications I

  • Ever wondered what to do when both the number of trials and success probability are unknown in a binomial model? Well, this paper by Adrian Raftery has an answer. See also this discussion with JAGS and Stan implementations.
  • This case study shows how to create a model from first (physical) principles.

Lecture 10: Applications II

Lecture 11: Discussion Bayes vs Frequentism

Disclaimer: everything in this section needs to be read with care so one does not become a zealot!

Computational resources