- 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
^{(25)}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__''.

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*_{t}'', 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.