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
 (11) (12) (13) (14)

that, using the compact notation introduced above, we can rewrite in the following enlighting forms. The first is [Eq. (14)]
 (15)

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):

 (16) (17)

i.e.12
 (18)

Bayes factors due to independent13pieces 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

 (19)

i.e.
 (20)

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

Giulio D'Agostini 2010-09-30