- how the new thing differs from from our initial beliefs;
- how strong our initial beliefs are.

Probability theory teaches us how to update the degrees of
belief on the different causes that might
be responsible of an `event' (read `experimental data'),
as simply explained by Laplace in his
*Philosophical essay*[17]
(`VI principle'^{14}
at pag. 17 of the original book,
available at book.google.com - boldface is mine):

This is the famous``The greater the probability of an observed event given any one of a number of causes to which that event may be attributed, the greater the likelihood^{15}of that cause {given that event}. The probability of the existence of any one of these causes {given the event} is thus a fraction whose numerator is the probability of the event given the cause, and whose denominator is the sum of similar probabilities, summed over all causes. If the various causes are not equally probablea priory, it is necessary, instead of the probability of the event given each cause, to use the product of this probability and the possibility of the cause itself.This is the fundamental principle of that branch of the analysis of chance that consists of reasoning.''a posteriorifrom events to causes

This formula teaches us that what matters is

- how much compares with , where and are two distinguished causes that could be responsible of the same effect;
- how much compares to .

the odds are updated by the observed effect by a factor (`

In particular, we learn that:

- It makes no sense to speak about how the probability
of changes if:
- there is no alternative cause ;
- the way how might produce has not been modelled, i.e. if has not been somehow assessed.

- The updating depends only on the Bayes factor,
a function of the probability of given either
hypotheses, and
*not on the probability of other events that have not been observed and that are even less probable than*(upon which p-values are instead calculated). - One should be careful not to confuse
with , and in general,
with . Or, moving to continuous variables,
with , where `' stands,
depending on the contest,
for a
*probability function*or for a*probability density function*, while and stand for an observed quantity and a true value, respectively.

Giulio D'Agostini 2012-01-02