Obviously, none can state initial odds with very
high precision.^{19}
But this does not matter (table 1
can help to get the point).
We want to understand
if they are of the order of 1 (equally likely),
of the order of a few units (one is a bit more likely than the other one),
or of suitable powers of 10 (much more or much
less likely than the other one). If one has doubts about the final
result, one can make a `sensitivity analysis', i.e. vary the
value inside a wide but still believable range and check
how the result changes. The sensitivity
(or insensitivity) will depend also on the
other pieces of evidence to draw the final conclusion.
Take for example two different evidences, characterized by
Bayes factors of versus
very high (e.g. )
or very small (e.g. ),
corresponding to JL's of or , respectively
(for the moment we assume all subjects agree
on the evaluation of these factors).
Given these values, it is easy to check that,
for many practical purposes,
the conclusions will be the same
even if the initial odds are in the range to ,
i.e. a JL between and , that can be stated
as
JL. Adding
`weights of evidence' of or , we get
final JL's of or ,
respectively.^{20}

The limit case in which the Bayes factor is zero or infinity (i.e. JL's or ) makes the conclusion absolutely independent from priors, as it seems obvious.

Giulio D'Agostini 2010-09-30