The basis of Bayesian statistics

In this short introduction to Bayesian statistics, we aim to demonstrate that it is a coherent and, above all, practical approach to solving statistical inference problems. The historical foundations of this discipline, as well as its theoretical and philosophical justifications, will not be presented here, the reader being referred for that purpose to the reference works cited in [Doc AF 605] which are Bernardo and Smith (1994); Carlin and Louis (2001); Gelman et al. (2001) and Robert (2007) (or Robert (2006) for the French version). On the contrary, our aim is to demonstrate that this approach to statistical inference is modern, adapted to computer simulation tools and capable of responding to the most advanced modeling problems in all disciplines, rather than anchoring it to its quarrels of the past. In the first part, we present the foundations of Bayesian inference, emphasizing the specificities of a priori modeling and test construction. Then, we explicitly implement the concepts previously introduced in the practical case of a linear regression model.