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 $p$ 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.