Bayesian Computational Methods

If, in the mid 1980's, one had asked the average statistician about the di-culties of using Bayesian Statistics, the most likely answer would have been\Well, there is this problem of selecting a prior distribution and then, evenif one agrees on the prior, the whole Bayesian inference is simply impossibleto implement in practice!" The same question asked in the 21th Centurydoes not produce the same reply, but rather a much less aggressive complaintabout the lack of generic software (besides winBUGS), along withthe renewed worry of subjectively selecting a prior! The last 20 years haveindeed witnessed a tremendous change in the way Bayesian Statistics areperceived, both by mathematical statisticians and by applied statisticiansand the impetus behind this change has been a prodigious leap-forward inthe computational abilities. The availability of very powerful approximationmethods has correlatively freed Bayesian modelling, in terms of both modelscope and prior modelling. This opening has induced many more scientistsfrom outside the statistics community to opt for a Bayesian perspective asthey can now handle those tools on their own. As discussed below, a mostsuccessful illustration of this gained freedom can be seen in Bayesian modelchoice, which was only emerging at the beginning of the MCMC era, for lackof appropriate computational tools.In this chapter, we will rst present the most standard computational challengesmet in Bayesian Statistics (Section 2), and then relate these problemswith computational solutions. Of course, this chapter is only a terse introductionto the problems and solutions related to Bayesian computations. Formore complete references, see Robert and Casella (2004), Marin and Robert(2007a), Robert and Casella (2004) and Liu (2001), among others. We alsorestrain from providing an introduction to Bayesian Statistics per se and forcomprehensive coverage, address the reader to Marin and Robert (2007a) andRobert (2007), (again) among others.