Probability Basic Calculus Basic Algebra Knowledge of R

a) Introduce the basic concepts of Bayesian inference and the similarities and differences with frequentist inference. b) Show when and how conjugate prior distributions can be applied. c) Illustrate how to use Monte Carlo and MCMC methods for implementing Bayesian inference in situations where a conjugate prior distribution is not available. d) Demonstrate how inference can be made for linear models and generalized linear models using Bayesian techniques, as well as Bayesian techniques for model selection. e) Show how advanced software such as R, Stan, or Python can be used to implement Bayesian methods. 1) Analytical and synthesis skills. 2) Modeling and problem-solving. 3) Oral and written communication.

1. Review of basic concepts: a) Law of total probability; Bayes theorem; the law of total expectation. b) Prior distribution; likelihood function; posterior distribution; marginal likelihood; predictive distribution. c) Differences between frequentist probability and subjective probability. d) Introduction to naïve Bayes classification. 2. Models with conjugate prior distributions. a) The beta-binomial model. b) Conjugate prior distributions. c) Poisson-gamma and / or Poisson-exponential models. d) Conjugate priors for exponential families. e) Interpretation of parameters. f) Limits of conjugate prior distributions and improper priors. 3. Bayesian inference for the Gaussian distribution. a) The normal-gamma distribution. b) The Behrens and Fisher problem. c) Semi-conjugate inference: an introduction to Gibbs sampling. 4. Implementation of Bayesian inference. a) Monte Carlo: rejection sampling, importance sampling. b) Markov chain Monte Carlo (MCMC): convergence of MCMC algorithms; Gibbs; Metropolis; Hamiltonian MCMC; other methods. 5. Linear models and Bayesian classification. a) Regresssion with an improper prior b) Conjugate regression and the relationship to ridge regression c) Bayesian lasso. d) Generalized linear models. e) Classification. 6. Model selection. a) Bayes factors. b) Information criteria (BIC, DIC, WAIC). c) Examples in regression and linear models.

Theory (4 ECTS). Theoretical classes with support material available on the Web. Practice (2 ECTS) problem-solving classes. Computing practices in computer labs. Presentations and debates.