Subjective probability aims to solve a big problem: how to calculate probability when none of the definitions examined in the previous chapters is applicable. The probability that it will rain tomorrow, that a newly graduated student will find a job, or that a basketball player will pass the basket record are examples of the subjective conception of probability, as they require neither the knowledge of the mechanism that regulates the phenomenon nor the repeatability of the phenomenon. The value attributed depends on the state of information of the person making the assessment. The already mentioned Bayes' Theorem shows how to make probability evaluations and subsequently modify them in the presence of new data. Faced with the vast field of application, there is a very scarce (one would have to say nothing) attention on the part of the teachers who prefer to concentrate the little time dedicated to the subject to more standardized situations.
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Chance is the pseudonym of God when he did not want to sign. ~ Anatole France
One of the perhaps lesser-known aspects is that Greek philosophers already had a notion of probability, but in a very particular sense. In ancient times “probable” was the attribute of an opinion guaranteed by a recognized and as such credible authority. Today one would say that probability was purely epistemic (a Greek word derived from the verb epistanai, which means to know or understand) and did not have a random aspect. This epistemic probability, according to Robert Ineichen (1999), was qualitative, not a quantitative one.
Only in more recent times have quantitative aspects been systematically taken into account, but with various theoretical-practical problems.
There are events for which none of the definitions discussed in the previous chapters is applicable because it is not possible to define the probability a priori and it is not even possible to calculate the frequency, as they are often unique events. If we ask ourselves the question “what is the probability that tomorrow it will rain?” it would be absurd to take for true the classical conception because every day the probability would be 1/2 (rain or no rain) and also it would be impossible to apply the frequentist probability because it is a unique phenomenon while this approach requires the repeatability of the test under the same conditions. At the most, we could take into account the historical series related to tomorrow's day and rely on a statistical evaluation of the event.
In this context there is, at a certain point, a change of perspective: the probability does not lie in the event, but lives in the observer: probability is subjective, probability is not a property of reality, it is a property of the subject who evaluates it according to the information in his possession; it can change from individual to individual, but it can change especially if the state of information he possesses changes.
First Bruno de Finetti (1906-1985) and then Leonard Savage (1917-1971) developed a subjective conception of probability.
Particularly strong is Bruno de Finetti's (1930) critical position towards the concepts seen previously: “The classical view is based on physical consideration 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 became meaningful is 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 since 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”.
For the avoidance of doubt, we make it clear that subjective does not mean arbitrary. This setting of probability can be applied, for example, if repeated tests can be performed, when tables with the results of statistical surveys on a certain phenomenon are available, as in the case of mortality tables (tables that show, for each age, what is the probability that a person of that age dies before his next birthday) or in the presence of odds defined by bookmakers based on certain algorithms; in other words, the subjective probability is not in contrast with the approaches examined above, but contains them as special cases.
The subjectivist conception of probability leads to a definition that could be formulated as follows: “the probability of an A event is the measure of the degree of belief that coherent individual attributes to the occurrence of A”. The individual must be coherent, that is, not prey to preconceptions and able to attribute the same value of probability to similar phenomena, but above all, in a bet, must be willing to invert roles: whoever accepts to pay the sum s to receive in exchange the sum S must be willing to invert roles. In summary: the probability of an event is the price that an individual deems right to pay to receive a unit of account if the event occurs.
In general, following Bruno de Finetti's (1930) operational definition, if a certain event within a consistent and fair bet is given “x versus y”, the subjective probability will be equal to:
This type of approach is, on the one hand, consistent, but above all applicable to most problems.