Subjective definition of probability

This is the kind of definition that one finds in Bayesian books[11,19,29,30,31] and the formulation cited here is that given in the ISOa measure of thedegree of beliefthat an event will^{3.2}occur.

At first sight this definition does not seem to be superior to
the combinatorial or the frequentistic ones.
At least they give some practical
rules to calculate ``something''. Defining
probability as *``degree of belief''* seems too vague to
be of any use. We need, then, some explanation of its
meaning; a tool to evaluate it - and
we will look at this tool (Bayes' theorem)
later. We will end this section with some
explanatory remarks on the definition, but
first let us discuss the
__advantages__ of this definition.

- It is natural, very general and can be applied to any thinkable event, independently of the feasibility of making an inventory of all (equally) possible and favourable cases, or of repeating the experiment under conditions of equal probability.
- It avoids the linguistic schizophrenia of having to distinguish ``scientific'' probability from ``non scientific'' probability used in everyday reasoning (though a meteorologist might feel offended to hear that evaluating the probability of rain tomorrow is ``not scientific'').
- As far as measurements are concerned, it allows
us to talk about the probability of the
*true value*of a physical quantity, or of any scientific hypothesis. In the frequentistic frame it is only possible to talk about the probability of the*outcome*of an experiment, as the true value is considered to be a constant. This approach is so unnatural that most physicists speak of `` probability that the mass of the top quark is between '',__although__they believe that the correct definition of probability is the limit of the frequency. - It is possible to make a very general theory of uncertainty which can take into account any source of statistical or systematic error, independently of their distribution.

To get a better understanding of the subjective definition of
probability let us take a look at __odds in betting__.
The higher the
degree of belief
that an event will occur, the higher
the amount of money that someone (``a rational better'')
is ready to pay in order to receive a sum of money if the event
occurs. Clearly the bet must be acceptable
in both directions (*``coherent''*
is the correct adjective), i.e. the amount of money
must be smaller or equal to
and not negative (who would accept such a bet?).
The cases of and mean that the events are considered
to be false or true, respectively,
and obviously it is not worth betting on certainty.
They are just limit cases, and in fact they can be
treated with standard logic.
It seems reasonable^{3.3}
that the amount of money that one is willing to pay
grows linearly
with the degree of belief.
It follows that if someone thinks that
the probability of the event is , then he
will bet to get
if the event occurs, and to lose
if it does not. It is easy to
demonstrate that the condition of ``coherence''
implies that
.

What has gambling to do with physics? The
definition of probability through
betting odds has to be considered *operational*, although there is no
need to make a bet (with whom?) each time one
presents a result. It has the important role of forcing
one to make an
*honest* assessment of the value of probability that
one believes. One could replace money with other forms
of gratification or penalization, like the increase or
the loss of scientific reputation. Moreover, the
fact that this operational procedure is not to
be taken literally should not be surprising. Many
physical quantities are defined in a similar way.
Think, for example, of the textbook definition of
the electric field, and try to use it
to measure
in the proximity of an electron.
A nice example comes from the definition of a poisonous chemical
compound: it *would be lethal if ingested*.
Clearly it is preferable to keep this operational definition
at a hypothetical level, even though it is the
best definition of the concept.