6. CONCLUSIONS

• First, it is not difficult to get a consensus on the observation that subjective probability is the natural concept developed by the human mind to quantify the plausibility of events in conditions of uncertainty.

• Second, one should insist on the fact that Bayes' theorem is in fact a natural way of reasoning in updating probability, and not a philosophical point of view that somebody tries to apply to data analysis ⁽²³⁾ (see [7] for details).

• Bayes' theorem is not ``all''. It only works in situations where the nice scheme of prior and likelihood is applicable. In many circumstances one can assess a subjective probability directly (try asking a carpenter how much he believes the result of his measurement!).

• The coherent bet (à la de Finetti [27]) forces people to be honest and to make the best (i.e. ``most objective'') assessments of probability.

• It is preferable not to mix up probability evaluation with decision problems ⁽²⁴⁾ . In other words, the coherent bet should be considered hypothetical. This makes a clear distinction between our beliefs and our wishes (the example in section 5.1 should teach something in this respect).

• One may think, naïvely, that the ``objective Bayesian theory'' is more suited for science than the ``subjective one''. Instead, it seems to me easier to convince experienced physicists that ``there is nothing really that is objective'', than it is to accept an objective theory containing ingredients which appear dogmatic <sup>(25)</sup> from the physicist point of view [28, 7]. Any experienced physicist knows already that the only ``objective'' thing in science is the reading of digital scales. When we want to transform this information into scientific knowledge we have to make use of many implicit and explicit beliefs (see section 3.2). Nevertheless, the ``honest'' (but naïve) ideal of objectivity can be recovered if scientific knowledge is considered as a kind of very solid networks of beliefs, based on centuries of experimentation, with fuzzy borders which correspond to the areas of present research. My preferred motto is that ``no one should be allowed to talk about objectivity, unless he has 10 or 20 years of experience in frontier science, economics, or any other real world application''. In particular, mathematicians should refrain from using the word objectivity when talking about the physical world.

• It is very important to work on applications: the simplicity and the naturalness of the Bayesian reasoning will certainly attract people.

• Many conventional methods can be easily recovered as limit cases of the Bayesian ones, if some well defined restricting conditions are valid, as already discussed in section 3.4. For example, when I make a ² fit I consider myself to be using a Bayesian method, although in a simplified form. This attitude contrasts to that of practitioners who use methods in which the Bayes' theorem is explicitly applied, but as if it were one of the many frequentist cooking recipes.

• It is important to make efforts to introduce Bayesian thinking in teaching, starting from the basic courses. I am not the first to have realized that the Bayesian approach is simple for students. The resistance comes from our colleagues, who are unwilling to renew the contents of their lectures, and who have developed a distorted way of thinking.

Finally, I would like to give a last recommendation. Don't try to convince a physicist that he already is Bayesian, or that you want to convert him to become Bayesian. A physicist feels offended if you call him ``X-ian'', be it Newtonian, Fermian, or Einsteinian. But, being human, he has a natural feel for probability, just like everybody else. I would like to generalize this idea and propose reducing the use of the adjective ``Bayesian''. I think that the important thing is to have a theory of uncertainty in which ``probability'' has the same meaning for everybody, precisely that meaning which the human mind has naturally developed and that frequentists have tried to kill. Therefore I would rather call these methods probabilistic methods. And I conclude saying that, obviously, ``I am not a Bayesian''.

Acknowledgements

It is a pleasure to thank the organizers of Maxent98 for the warm hospitality in Garching, which favoured friendly and fruitful discussions among the participants.

²³ For example frequentists completely misunderstand this points, when they state, e.g., that ``Bayesian methods proceed by invoking an interpretation of Bayes' theorem, in which one deems it sensible to consider a p.d.f. for the unknown true value m<sub>t</sub>'', or that ``a pragmatist can consider the utility of equations generated by the two approaches while skirting the issue of buying a whole philosophy of science' [6].

I find that also the Zellner's paper [26] demonstrating that Bayes' theorem makes the best use of the available information can help a lot to convince people. Back.

²⁴ Although it may seem absurd, the Bayesian approach is recognized by ``frequentists'' to be ``well adapted to decision-making situations'' [3] (see also [6, 18]). I wonder what then probability is for these authors. Back.

²⁵ Dogmatism is never desirable. It can be easily turned against the theory. For example, one criticism of [18] says, more or less, that Bayesian theory supports Jeffreys' priors, and not uniform priors, but, since Jeffreys' priors give unreasonable results in their application, then one should mistrust Bayesian methods! (see also [28].) One may object that the meaning and the role of Jeffreys' priors was misunderstood, but it seems to me difficult to control the use of objective priors or of reference analysis once they have left the community of experts aware of the ``rather special nature and role of the concept of a `minimally informative' prior specification - appropriately defined!'' [29]. Back.