Besides the intuitive use of Bayesian reasoning, there are, in fact, some applications in which the Bayes' theorem is explicitly applied. This happens when frequentist methods ``do not work'', i.e. they give manifestly absurd results, or in solving more complicated problems than just inferring the value of a quantity, like, for example, the deconvolution of a spectrum (``unfolding''). Nevertheless, these methods are mostly used with a utilitarian spirit, without having really understood the meaning of subjective probability, or even remaining skeptical about it. They are used as one uses one of the many frequentist ``ad hoc-eries'' (14) , after it has been ``proved'' that they work by MC simulation (15) .
Some of the cases in which the conventional methods do not work have even induced the PDG [3] to present Bayesian methods. But, according to the PDG, a paper published this year [18] finally gives a frequentist solution to the problems, and this solution is recommended for publishing the results. Let us review the situation citing directly [18]: ``Classical confidence intervals are the traditional way in which high energy physicists report errors on results of experiments. ... In recent years, there has been considerable dissatisfaction ...for upper confidence limits... This dissatisfaction led the PDG to describe procedures for Bayesian interval construction in the troublesome cases: Poisson processes with background and Gaussian errors with a bounded physical region. ... In this paper, we...use (...) to obtain a unified set of classical confidence intervals for setting upper limits and quoting two-sided confidence intervals. ...We then obtain confidence intervals which are never unphysical or empty. Thus they remove an original motivation for the description of Bayesian intervals by the PDG.'' In fact, the 1998 issue of the Review of Particle Physics still exhibits the Bayesian approach (with the typical misconceptions that frequentists have about it), but then it suggests two papers by frequentists [6, 2] (``a balanced discussion'' [3]) to help practitioners to form their own idea on the subject, and, finally, it warmly recommends the new frequentist approach. It is easy to imagine what the reaction of the average HEP physicist will be when confronted by the authority of the PDG, unaware that ``the PDG'' which rules analysis methods is in reality constituted of no more than one or two persons who recommend a paper written by their friends (as is clear from the references and the cross acknowledgements). One should also notice that this paper claims important progress in statistics, but was in fact published in a physics journal (I wonder what the reaction of a referee of a statistics journal would have been...).
In conclusion, there is still a large gap between good sense and the dominating statistical culture. For this reason we must still be very careful in interpreting published results and in evaluating whether or not the conventional methods used lead to correct scientific conclusions ``by chance''. Some cases of misleading results will be described in the next section.
14 For example, this was exactly the attitude which I had some years ago, when I wrote a Bayesian unfolding program [17], and that of the large majority my colleagues who still use the program. Now, after having attended the 1998 Valencia Meeting on Bayesian Statistics, I have realized that this pragmatic frequentist-like use of Bayesian methods is rather common. Back.
15 I would like to point out that sometimes the conclusions derived from MC checks of Bayesian procedures may be misleading, as discussed in detail in [7]. Back.