##

Beliefs versus frequencies

At this point a remark on the important (and often misunderstood)
issue of the relation between degrees of beliefs and relative
frequencies is in order.
The concept of subjective probability does not
preclude the use of
*relative frequencies* in the reasonings.
In particular, beliefs can be evaluated from the relative
frequencies of other events, analogous to the
one of interest, have occurred in the past.
This can be done roughly (see Hume's quote in Appendix B)
or in a rigorous way,
using probability theory
under well defined assumptions
(Bayes' theorem applied
to the inference of
the parameter of a binomial distribution).

Similarly, if we believe that a given event will
occur with 80% probability, it is absolutely correct
to say that, if we think at a large number of analogous
independent events that we consider equally probable,
we expect that in about 80% of the cases the event
will occur. This also comes from probability theory
(Bernoulli theorem).

This means that, contrary to what one reads often
in the literature and on the web,
evaluating probabilities from past frequencies
and expressing beliefs by expected frequencies does
not imply to adhere to *frequentism*.^{21} The importance
of this remark in the context of this paper is that
people might find natural, for their reasons,
to evaluate and to express beliefs this way,
although they are perfectly aware that the event
about they are reasoning is unique. For further comments see
Appendix B.

Giulio D'Agostini
2010-09-30