Bayesian testing and testing bayesians

This chapter focuses on Bayesian methods and illustrates both the intrinsic unity of Bayesian thinking, and its basic flexibility to adjust to and to cope with a wide range of circumstances. Two ideas are emphasized in the chapter. Firstly hypothesis testing and model choice have been dealt with as a single class of problems met with so strikingly varied motivations that no clear distinction among them seems to be operationally fruitful. Secondly Bayesian thinking is rich enough to accommodate to that variety of situations and is much more flexible than a mechanical prior-to-posterior transformation; in particular, the predictive distributions have been shown to play an important role for this class of problems. Putting emphasis on general inference, on model label or on parameters of interest, has led to put less emphasis on solving specific decision problems. Thus strictly decision oriented procedures have at time been alluded to but not dealt with in any systematic way. This is the case of model choice procedures based on specific criteria such as Akaike information criterion (AIC) or Bayesian information criterion (BIC) with different degrees of involvement with the Bayesian idea. The chapter also provides some examples to examine how testing problems may be handled by statisticians open-minded towards Bayesian ideas