2.1.1 Axiomatic definition of probability

Probability is a measure of uncertainty. Once a random experiment is defined, we call probability of the event ${\mathcal E}$ the real number $\mbox{Prob}\left\{ \mathcal{E} \right\} \in [0,1]$ assigned to each event ${\mathcal E}$. The function $\mbox{Prob}\left\{ \cdot \right\} : \Omega \rightarrow [0,1]$ is called probability measure or probability distribution and must satisfy the following three axioms:

  1. $\mbox{Prob}\left\{ {\mathcal E} \right\} \ge 0$ for any ${\mathcal E}$.

  2. $\mbox{Prob}\left\{ \Omega \right\} =1$

  3. $\mbox{Prob}\left\{ {\mathcal E}_1+{\mathcal E}_2 \right\} =\mbox{Prob}\left\{ {\mathcal E}_1 \right\} +\mbox{Prob}\left\{ {\mathcal E}_2 \right\} $ if ${\mathcal E}_1$ and ${\mathcal E}_2$ are mutually exclusive.

These conditions are known as the axioms of the theory of probability [76]. The first axiom states that all the probabilities are nonnegative real numbers. The second axiom attributes a probability of unity to the universal event $\Omega $, thus providing a normalization of the probability measure. The third axiom states that the probability function must be additive, consistently with the intuitive idea of how probabilities behave.

All probabilistic results are based directly or indirectly on the axioms and only the axioms, for instance

\begin{align} & \mathcal{E}_1 \subset {\mathcal E}_2 \Rightarrow \mbox{Prob}\left\{ {\mathcal E}_1 \right\} \le \mbox{Prob}\left\{ {\mathcal E}_2 \right\} \\ & \mbox{Prob}\left\{ {\mathcal E}^ c \right\} =1-\mbox{Prob}\left\{ {\mathcal E} \right\} \end{align}

There are many interpretations and justifications of these axioms and we discuss briefly the frequentist and the Bayesian interpretation in Section 2.1.3. What is relevant here is that the probability function is a formalization of uncertainty and that most of its properties and results appear to be coherent with the human perception of uncertainty [69].

So from a mathematician point of view, probability is easy to define: it is a countably additive set function defined on a Borel field , with a total mass of one.

In practice, however, a major question remains still open: how to compute the probability value $\mbox{Prob}\left\{ {\mathcal E} \right\} $ for a generic event ${\mathcal E}$ ? The assignment of probabilities is perhaps the most difficult aspect of constructing probabilistic models. Although the theory of probability is neutral, that is it can make inferences regardless of the probability assignments, its results will be strongly affected by the choice of a particular assignment. This means that, if the assignments are inaccurate, the predictions of the model will be misleading and will not reflect the real behaviour of the modelled phenomenon. In the following sections we are going to present some procedures which are typically adopted in practice.