Applying the updating reasoning to our box game,
the Bayes factor of interest is
Role of priors
As it was remarked, this number would give the required odds
if the hypotheses were initially equally likely.
But how strong are the initial relative beliefs
on the two hypotheses? `Unfortunately',
we cannot perform a probabilistic inversion
if we are unable to assign somehow prior probabilities to the
hypotheses we are interested in.9
Indeed, in the formulation of the problem I on purpose
passed over the relevant pieces of information
to evaluate the prior probabilities (it was said that
``there are two types of boxes'',
not ``there are two boxes''!). If we specify that
we had boxes of type
and of the other kind,
then the initial odds
are and the final ones will be
Graphical representation of the causal connections
, where are the effects
(White/Black at each extraction). These effects are then causes
of other effects (), the reported colors, where `'
stands for `testimony'.
The arrows connecting the various `nodes' represent
conditional probabilities. The model will be fully exploited
in Appendix J.
from which we get (just requiring that the probability of the two
hypotheses have to sum up to one10)
If the two hypotheses were initially considered equally likely,
then the evidence makes 13 times more believable
than , i.e.
, or approximately 93%.
On the other hand, if was a priori much less
credible than ,
for example by a factor 13,
just to play with round numbers, the same evidence
made and equally likely. Instead, if we were
initially in strong favor of , considering it for instance
13 times more plausible than , that evidence
turned this factor into 169, making us 99.4% confident
- highly confident, some would even
say `practically sure'! -
that the box is of type .