Bayesian methods

This course introduces regression models using Bayesian philosophy and software. The course commences with a brief refresher on probability defined in frequentist terms and a discussion on the limitations of this view when applied to unique events. Different schools of statistical thought, the distinction between confidence intervals and credible intervals, and the roles of generative models and of estimands in the scientific process are discussed. The course develops full-probability modelling for single- and multi-parameter models with conjugate prior distributions. Comparisons between exact solutions and numerical approximations are provided to illustrate the use of numerical algorithms. Further, Markov Chain Monte Carlo (MCMC) algorithms as a generic approaches to Bayesian modeling are introduced. The focus is on understanding the challenge of convergence in distribution, on diagnosing lack of convergence, and on remediating convergence problems. Posterior predictive checks as a systematic method to assess model fit are described. Model comparison via information criteria is justified. Finally, stacking of multiple models is illustrated. The unified workflow of a Bayesian analysis is illustrated in the case of generalised linear mixed models (GLMM). Some discussion on contemporary challenges and prior elicitation is provided.