Bayesian Statistics

Ph.D. in Economics, Statistics, and Data Science - University of Milano-Bicocca

Author
Affiliation

Raffaele Argiento, Bernardo Nipoti, and Tommaso Rigon

DEMS

Detailed syllabus

Bayesian Statistics is a Ph.D. course organized in three modules

Principles of Bayesian statistics (Bernardo Nipoti, 15h)

  • Exchangeability and de Finetti’s theorem
  • The Bayesian framework
  • Conjugate prior distributions
  • Bayesian point estimation
  • Test and credible intervals
  • The normal model
  • The multivariate normal model
  • Introduction to hierarchical modelling

Bayesian computations (Tommaso Rigon, 15h)

  • Metropolis-Hastings and Gibbs sampling
  • Optimal scaling & adaptive Metropolis
  • MALA algorithm & Hamiltonian Monte Carlo
  • Missing data problems, Gibbs sampling and the EM algorithm
  • Laplace appr., Variational Bayes, and Expectation Propagation

Mixture models in Bayesian Statistics (Raffaele Argiento, 12h)

  • Finite and infinite mixture models, the basic concepts of kernel, mixing measure and components of a mixture.
  • The latent component allocation variables and the clustering it induces on the data by a mixture model. Difference between component and cluster.
  • The prior on the parameter “cluster” induced by the mixing measure: the exchangeable product partition function (EPPF) and its properties. The prior on the number of components and the prior on the number of clusters.
  • Linear and non-linear functionals of the posterior distributions and how to approximate them via Markov Chain Monte Carlo (MCMC) algorithms.
  • Marginal and conditional algorithms for mixture model, the Chinese restaurant process and its generalization.

Exam

The exam rules are described here. In short, you will be asked to read a paper, write a short review, and then make a presentation. Here is an example from previous years:

  1. Paper assigned to a PhD student, using the keyword system;
  2. Short review, written using the Bayesian Analysis template;
  3. Slides of the presentation.

References

Books

  • Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A. and Rubin, D.B., 2013. Bayesian data analysis. CRC press.
  • Hoff, P. D. (2009). A First Course in Bayesian Statistical Methods. Springer.
  • Robert, C. P., and Casella, G. (2009). Introducing Monte Carlo methods with R. Springer.
  • Robert, C., 2007. The Bayesian choice: from decision-theoretic foundations to computational implementation. Springer Science & Business Media.

Articles

Additional references are available online on the instructors’ websites.