##

Bayes theorem and Bayes factor

The `probabilistic inversion'
can *only*^{3} be performed
using
the so-called *Bayes' theorem*, a simple consequence
of the fact that, given the *effect* and some
*hypotheses* concerning its possible cause,
the joint probability of and ,
conditioned by the *background
information*^{4}
, can be written as

where `' stands for a logical `AND'.
From the second equality of the last
equation we get

that is one of the ways to express Bayes'
theorem.^{5}
Since a similar expression holds for
any other hypothesis ,
dividing member by member the two expressions
we can restate the theorem in terms of the
relative beliefs, that is

the initial ratio of beliefs (`odds') is updated by the
so-called *Bayes factor*, that depends on how likely
*each* hypothesis can produce that effect.^{6}Introducing and
BF,
with obvious meanings, we can rewrite
Eq. (4) as

Note that, if the initial odds are unitary, than
the final odds are equal to the updating factor. Then, *Bayes factors
can be interpreted as odds due only to an individual
piece of evidence, if the two hypotheses were
considered initially equally
likely*.^{7}
This allows us to rewrite
BF as
, where the tilde is to remind that *they
are not properly odds*, but rather
`*pseudo-odds*'. We get then an expression in which all terms
have *virtually* uniform meaning:

If we have only two hypotheses, we get simply
.
If the updating factor is unitary, then the piece of
evidence does not modify our opinion on the two hypotheses
(no matter how small can numerator and denominator be,
as long as their ratio remains finite and unitary! - see Appendix G
for an example worked out in details);
when
vanishes, then
hypothesis becomes impossible
(``*it is falsified*''); if instead it is infinite
(i.e. the denominator vanishes), then it is the other
hypothesis to be impossible. (The undefined case
means that we have to look for other hypotheses
to explain the effect.^{8})

Giulio D'Agostini
2010-09-30