##

Adding pieces of evidence

Imagine now the following variant of the previous toy
experiment. After the white ball is observed, you put it again
in the box, shake well and make a second extraction.
You get white the second time too. Calling and the
two observations,
we have now:^{11}

that,
using the compact notation introduced above,
we can rewrite in the following enlighting forms.
The first is
[Eq. (14)]

that is,
*the final odds after the first inference become the
initial odds of the second inference*
(and so on, if there are several pieces of evidence).
Therefore, beginning
from a situation in which was thirteen times more credible
than is exactly equivalent to having
started from unitary odds updated
by a factor 13 due to a piece of evidence.
The second form comes from Eq. (13):

i.e.^{12}

*Bayes factors* due to
*independent*^{13}pieces of evidence *multiply*. That is,
two independent pieces of evidence ( and ) are equivalent
to a single piece of evidence (`
'), whose Bayes factor
is the product of the individual ones.
In our case
.
In general, if we have several hypotheses and
several *independent*
pieces of evidence, , , ..., ,
indicated all together as
,
then
Eq. (4) becomes

i.e.

where stand for `product'
(analogous to for sums).

Giulio D'Agostini
2010-09-30