In order to apply Bayes' theorem in one of its form
we need first to evaluate
.
Probability theory teaches us how to get it
[see Eq. (33) in Appendix A]:

(38) |

( could be due to a true evidence or to a fake one). Three new ingredients enter the game:

- , that is the probability of the evidence to be correctly reported as such.
- But the testimony could also be incorrect the other way around (it could be incorrectly reported, simply by mistake, but also it could be a `fabricated evidence'), and therefore also is needed. Note that the probabilities to lie could be in general asymmetric, i.e. , as we have seen in the AIDS problem of Appendix F, in which the response of the analysis models false witness well.
- Finally, since enters now directly, the `naïve' Bayes factor, only depending on , is not longer enough.

As expected, this formula is a bit more complicate that the Bayes factor calculated taking for granted, which is recovered if the lie probabilities vanish

(40) |

i.e. only when we are absolutely sure the witness does not err or lie reporting (but Peirce reminds us that ``absolute certainty, or an infinite chance, can never be attained by mortals'' [6]).

In order to single out the effects of the new ingredients, Eq. (39)
can be rewritten as^{62}

where

(42) |

under the

J | (43) |

Let us make some instructive limits of Eq. (41).

(44) | |||

(45) | |||

(46) | |||

(47) |

As we have seen, the ideal case is recovered if the lie factor vanishes. Instead, if it is equal to 1, i.e. J, the reported evidence becomes useless. The same happens if vanishes [this implies that vanishes too, being ].

However, the most remarkable limit
is the last one. It states
that, even if
is very high,
the effective Bayes factor cannot exceed the inverse of the lie factor:

(48) |

or, using logarithmic quantities

JL | J J | (49) |

At this point some numerical examples are in order (and those who claim they can form their mind on pure intuition get all my admiration...

how the

The table exhibits the limit behaviors we have seen analytically.
In particular,
if we fully trust the report, i.e.
J,
then
JL is exactly equal to
JL,
as we already know. But as soon as the absolute value of the lie factor
is close to
JL, there is a sizeable effect.
The upper bound can be the be rewritten as

min | (50) | ||

or | |||

JL | minJLJ | (51) |

a relation valid in the region of interest when thinking about an evidence in favor of , i.e. JL and J.

This upper bound is very interesting. Since minimum conceivable values
of
J for human beings can be of the order of
(to perhaps
or
,
but in many practical applications or can already be very generous!),
in practice the effective weights of evidence cannot exceed
values of about (I have no strong opinion on the exact
value of this limit, my main point is that *you consider
there might be
such a practical human limit*.)

This observation has an important consequence in the combination of evidences, as anticipated at the end of section 3.5. Should we give more consideration to a single strong piece of evidence, virtually weighing JL, or 10 independent weaker evidences, each having a JL of 1? As it was said, in the ideal case they yield the same global leaning factor. But as soon as human fallacy (or conspiracy) is taken into account, and we remember that our belief is based on and not on , then we realize that JL is well above the range of JL that we can reasonably conceive. Instead the weaker pieces of evidence are little affected by this doubt and when they sum up together, they really can provide a JL of about 10.

Giulio D'Agostini 2010-09-30