Basic rules

Given any hypothesis $ H$ (or $ H_i$ if we have many of them), also concerning the occurrence of an event, and a given state of information $ I$, probability assessments have to satisfy the following relations:
  1. \fbox{$0\le P(H\,\vert\,I) \le 1$}
  2. \fbox{$P(H\cup \overline H\,\vert\,I) = 1$}
  3. \fbox{$P(H_i\cup H_j\,\vert\,I) = P(H_i\,\vert\,I) + P(H_j\,\vert\,I)$\ \ \ if
$H_i$\ and $H_j$\ cannot be true together}
  4. \fbox{$P(H_i\cap H_j\,\vert\,I) = P(H_i\,\vert\,H_j,I)\cdot P(H_j\,\vert\,I)
= P(H_j\,\vert\,H_i,I)\cdot P(H_i\,\vert\,I)$}
The first basic rule represents basically a conventional scale of probability, also indicated between 0 and 100%.

Basic rule 2 states that probability 1 is assigned to a logical truth, because either is true $ H$ or its opposite (``tertium non datur''). Indeed $ H\cup \overline H$ represent a logical, tautological certainty (a tautology, usually indicated with $ \Omega$), while $ H\cap \overline H$ is a contradiction, that is something impossible, indicated by $ \emptyset$.

The first three basic rules are also known the `axioms' of probability,40while the inverses of the fourth one, e.g. $ P(H_i\,\vert\,H_j,I)=P(H_i\cap H_j\,\vert\,I)/P(H_j\,\vert\,I)$, are called in most literature ``definition of conditional probability''. In the approach followed here such a statement has no sense, because probability is always conditional probability (note the ubiquitous `$ I$' in all our formulae - for further comments see section 10.3 of Ref. [3]). Note that when the condition $ H_i$ does not change the probability of $ H_j$, i.e. $ P(H_i\,\vert\,H_j,I)= P(H_i\,\vert\,I)$, then $ H_i$ and $ H_j$ are said to be independent in probability. In this case the joint probability $ P(H_i\cap H_j\,\vert\,I)$ is given by the so-called product rule, i.e. $ P(H_i\cap H_j\,\vert\,I)=P(H_i\,\vert\,I)\cdot P(H_j\,\vert\,I)$.

These rules are automatically satisfied if probabilities are evaluated from favorable over possible, equally probably cases. Also relative frequencies of occurrences in the past respect these rules, with the little difference that the probabilistic interpretation of past relative frequencies is not really straightforward, as briefly discussed in the following appendix. That beliefs satisfy, in general, the same basic rules can be proved in several ways. If we calibrate our degrees of beliefs against `standards', as illustrated in section 3, this is quite easy to understand. Otherwise it can be proved by the normative principle of the coherent bet [10].

Giulio D'Agostini 2010-09-30