Concepts of probability

We have arrived at the point where it is necessary to define better what probability is. This is done in Section . As a general comment on the different approaches to probability, I would like, following Ref. [19], to cite de Finetti[11]:

``The only relevant thing is uncertainty - the extent of our knowledge and ignorance. The actual fact of whether or not the events considered are in some sense determined, or known by other people, and so on, is of no consequence.

The numerous, different opposed attempts to put forward particular points of view which, in the opinion of their supporters, would endow Probability Theory with a `nobler status', or a `more scientific' character, or `firmer' philosophical or logical foundations, have only served to generate confusion and obscurity, and to provoke well-known polemics and disagreements - even between supporters of essentially the same framework.

The main points of view that have been put forward are as follows.

The classical view is based on physical considerations of symmetry, in which one should be obliged to give the same probability to such `symmetric' cases. But which `symmetry'? And, in any case, why? The original sentence becomes meaningful if reversed: the symmetry is probabilistically significant, in someone's opinion, if it leads him to assign the same probabilities to such events.

The logical view is similar, but much more superficial and irresponsible inasmuch as it is based on similarities or symmetries which no longer derive from the facts and their actual properties, but merely from sentences which describe them, and their formal structure or language.

The frequentistic (or statistical) view presupposes that one accepts the classical view, in that it considers an event as a class of individual events, the latter being `trials' of the former. The individual events not only have to be `equally probable', but also `stochastically independent' ...(these notions when applied to individual events are virtually impossible to define or explain in terms of the frequentistic interpretation). In this case, also, it is straightforward, by means of the subjective approach, to obtain, under the appropriate conditions, in perfectly valid manner, the result aimed at (but unattainable) in the statistical formulation. It suffices to make use of the notion of exchangeability. The result, which acts as a bridge connecting the new approach to the old, has often been referred to by the objectivists as ``de Finetti's representation theorem'.

It follows that all the three proposed definitions of `objective' probability, although useless per se, turn out to be useful and good as valid auxiliary devices when included as such in the subjectivist theory.''

Also interesting is Hume's point of view on probability, where concept and evaluations are neatly separated. Note that these words were written in the middle of the 18th century [20].

``Though there be no such thing as Chance in the world; our ignorance of the real cause of any event has the same influence on the understanding, and begets a like species of belief or opinion.

There is certainly a probability, which arises from a superiority of chances on any side; and according as this superiority increases, and surpasses the opposite chances, the probability receives a proportionable increase, and begets still a higher degree of belief or assent to that side, in which we discover the superiority. If a dye were marked with one figure or number of spots on four sides, and with another figure or number of spots on the two remaining sides, it would be more probable, that the former would turn up than the latter; though, if it had a thousand sides marked in the same manner, and only one side different, the probability would be much higher, and our belief or expectation of the event more steady and secure. This process of the thought or reasoning may seem trivial and obvious; but to those who consider it more narrowly, it may, perhaps, afford matter for curious speculation.

...

Being determined by custom to transfer the past to the future, in all our inferences; where the past has been entirely regular and uniform, we expect the event with the greatest assurance, and leave no room for any contrary supposition. But where different effects have been found to follow from causes, which are to appearance exactly similar, all these various effects must occur to the mind in transferring the past to the future, and enter into our consideration, when we determine the probability of the event. Though we give the preference to that which has been found most usual, and believe that this effect will exist, we must not overlook the other effects, but must assign to each of them a particular weight and authority, in proportion as we have found it to be more or less frequent.''