Bayesian statistics

Objectives

Explain the general principle of the Bayesian statistical approach
Make an appropriate choice of a priori laws
Formulate the probabilistic writing of a latent variable model in hierarchical form and give its representation in the form of a directed acyclic graph.
Calculation of a posteriori laws (only for SSV)
Bayesian inference of classical models (e.g., linear regression, GLM) and latent variable models (e.g., mixed models) using the R packages "rjags" and "rstan".
Conduct a convergence study of an MCMC algorithm
Compare different models using Bayesian selection criteria
Validate a model – from a predictive point of view – under the Bayesian paradigm

Course outline

Bayes formula / A priori law / A posteriori law / A posteriori predictive law / Epistemic uncertainty
Bayesian estimators / Credibility intervals
Conjugate a priori laws / Jeffreys a priori laws
Latent variable models and hierarchical representation / Oriented acyclic graph
Deterministic a posteriori approximations
MCMC algorithms (Principle, Gibbs/Metropolis-Hastings/Hamiltonian dynamics, convergence diagnostics)
Bayesian predictive validation (posterior predictive check, cross validation)
Bayesian model selection (Bayes factor, Deviance Information Criterion, Widely Applicable Information Criterion)

Prerequisites

Probability, inferential statistics, SAS, R (1A)
Linear regression, GLM, Markov chain, Bayesian computation (2A)