Environmental Engineering Reference
In-Depth Information
The consequence of the main theorems - the formula of total probability
(FTP) and the Bayes formula - are widely used in solving a large number
of tasks.
The formula of total probability. If the results of experiments can be
used to propose n mutually exclusive hypotheses H 1 , H 2 ,...H n , representing
the complete group of incompatible events (for which Σ i= I n P ( i ) = 1), the
probability of event A , which can only come from one of these hypotheses,
is defined by:
PA PH PAH
()
=
( )( |
,
[1.13]
1
i
where P ( H i ) is the probability of hypothesis H i ; P ( A | H i ) is the conditional
probability of event A under hypothesis H i .
Since event A can occur with one of the hypotheses H 1 , H 2 , ... H n , then
A = AH 1 AH 2 ... AH n , but H 1 , H 2 ,... H n are incompatible, so
n
= ∧ ++ ∧
PA PA H
(
)
(
)
...
PA H
(
)
PAH
(
).
i
n
i
i
=
1
If event A depends on validity of hypothesis H i P ( AH i ) =P ( H i P ( A|H i ),
and this leads to expression [1.13].
The Bayes formula ( the formula of probability of hypotheses ). If the
probabilities of hypotheses H 1 , H 2 , ... H n prior to the experiment were
equal to P ( H 1 ), P ( H 2 ), ..., P ( H n ), and event A took place as a result of the
experiment, then the new (conditional) probabilities of the hypotheses are
evaluated:
PH PAH PH PAH
( )( |
)
( )( |
)
PAH
(|
)
=
i
i
=
i
i
.
i
n
[1.14]
PA
()
PH PAH
( )( |
)
i
i
The probabilities of the hypothesis before the start of the experiment
(initial) P ( H 1 ), P ( H 2 ), ..., P ( H n ) are called apriori , and tjose after the
experiment P ( H 1 | A ), ... P ( H n | A ) aposteriori .
The Bayes formula is used to reconsider the possibility of hypotheses
in the light of the experimental result.
The proof of the Bayes formula follows from the material discussed
previously. Since P ( H i A ) = P ( H i P ( A | H i ) = P ( H i P ( H i | A ):
i
=
I
￿ ￿ ￿ ￿ ￿
PH A PH PAH
(
)
( )( |
)
PH A
( |
)
=
i
i
=
i
.
[1.15]
i
PA
()
PA
()
If another experiment is caried out after the experiment which gave
the event A , and this second experiment can be carried whether or not the
event A 1 took place, then the conditional probability of the second event
is computed from [1.13], which does not include the former hypotheses
P ( H i ) and instead includes the new ones - P ( H i | A ):
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