Bayesian statistics

Objectifs

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

Compute a posteriori laws (only for SSV)

Perform Bayesian inference on 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

Plan

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)

Prérequis

Probability, inferential statistics, SAS, R (1A)

Regression, GLM, Markov chain, Bayesian computation (2A)