This short overview of the problems and solutions considered for Bayesian Statistics is nothing but an introduction to the game: there are much more complex problems than those illustrated above and much more advanced techniques than those presented in these pages. The reader is then encouraged to enter the literature on the topic, maybe with other introductory surveys like [7] and [3], but mostly through books like [10], [13], [27], [21] and [37].
We have not mentioned so far entries to Bayesian softwares like winBUGS, developed by the MRC Unit in Cambridge ([18,41]), Ox ([12]), BATS ([33]), BACC ([15]) and the Minitab package of [2], which all cover some aspects of Bayesian computing. Obviously, these packages require some expertise from the user and are thus more difficult of use than the classical open source or commercial softwares like R, Splus, Statgraphics, StatXact, SPSS or SAS. In other words, they are not black boxes that could be used by laymen with no statistical background. But this entrance fee to the use of Bayesian softwares is inevitable, given the versatile nature of Bayesian analysis: since it offers much more variability than standard inferential procedures, through the choice of prior distributions and loss functions for instance, it also requires more input from the user! And, once these preliminary steps have been overcome, the programming involved in a software like winBUGS is rather limited and certainly not harder than writing a code in R or Matlab.
As stressed in this chapter, computational issues are central to the design and implementation of Bayesian analysis. The new era opened by the MCMC methodology has brought much more freedom in the use of Bayesian methods, as reflected by the increase of Bayesian studies in applied Statistics. As usually the case, a strong increase in the use of a methodology also sees a corresponding increase in its misuse! Inconsistent data-dependent priors and improper posteriors are sometimes appearing in studies and, more generally, the assessment of prior modelling (or even of MCMC convergence) are rarely conducted with sufficient care. This is somehow a price to pay for the wider range of Bayesian studies, while the improvement of corresponding software should bring more guidelines and warnings about these misuses of Bayesian analysis.