Probability, in its etymological sense,
is by nature doubly subjective. First, because
its essence is rooted in a “feeling” of the
“human understanding” (8).
Second, because its value depends on the information
available at a given moment on a given subject.
Many evaluations are based on the assumed properties
of `things' to behave in some ways rather than in others,
relying on symmetry judgments or on regularities
observed in the past and extended to the future
(at our own risk, hoping not to end up
like the inductivist turkey).
The question of whether there is
“such a thing as Chance in the world”(9)
(does God play dice?) is not easily settled,
but whatever the answer is,
“our ignorance of the real cause of any event has the same influence
on [our] understanding.” (9) So, at least for
pragmatic convenience, we can assign to `things' propensities,
seen as parameters of our models of
reality, just like physics quantities.
And they might change with time, as other parameters do.
Furthermore, it is a matter of fact that, besides text book stereotyped cases, propensities
are usually uncertain and we have to learn about them by doing
experiments and framing the observations in a (probabilistic)
causal model.
The key tool to perform the so-called probabilistic
inversion is Bayes rule and such models of reality go under the name of
Bayesian networks, in which probabilities
are attached to all uncertain quantities (possible observations, parameters
and hyper-parameters, which might have different meanings,
including that of propensity and of degree of belief, as when we
model the degree of reliability of a witness in Forensic Science
applications).
Predictions are then made by averaging values
of propensities
with weights equal to the probabilities we assign to each of them.
In this paper I have outlined this (in my opinion)
natural way of reasoning, which was that of the founding fathers
of probability theory, with a toy experiment.
Then, once we have mustered up the
courage to talk about probabilities of probabilities, as shyly done
nowadays by many, we extend them to related concepts, like odds
and Bayes factors.
I would like to end reminding de Finetti's
“Probability does not exist” (in the things),
adding that “propensity might, but it is in most cases
uncertain and it can change with time.”
“To make progress in understanding,
we must remain modest and allow that we do not know.
Nothing is certain or proved beyond all doubt.
...
The statements of science are not of what is true and what is not true,
but statements of what is known to different
degrees of certainty.”
(Richard Feynman)
