Important relations that follow from the basic rules are
( is also a generic hypothesis):
|
|
|
(29) |
|
|
0 |
(30) |
|
|
|
(31) |
|
|
|
(32) |
|
|
|
(33) |
|
|
|
(34) |
|
|
idem |
(35) |
The first two rules are quite obvious. Eq. (31)
is an extension of the third basic
rule in the case two hypotheses are not mutually exclusive.
In fact, if this is not case, the probability of
is double counted and needs to be subtracted.
Eq. (32) is also
very intuitive, because either is true together with or with
its opposite.
Formally, Eq. (33) follows from Eq. (32)
and basic rule 4. Its interpretation is that the probability of
any hypothesis can be seen as `weighted average' of conditional
probabilities, with weights given by the probabilities of the
conditionands [remember that
and therefore Eq. (33)
can be rewritten as
that makes self evident its
weighted average interpretation].
Eq. (34) and (35) are simple extensions
of Eq. (32) and (33) to a generic `complete class',
defined as a set of mutually exclusive hypotheses
[
, i.e.
],
of which at least one must be true [
,
i.e.
]. It follows then that Eq. (35)
can be rewritten as the (`more explicit') weighted average
[Note that any hypothesis
and its opposite
form a complete class,
because
and
.]
Giulio D'Agostini
2010-09-30