Geoscience Reference
In-Depth Information
In fact, the posterior probability, although much higher than the prior, is still
small:
pB
ð
j
A
Þ ¼
pA
ð
j
B
Þ
pB
ðÞ=
pA
ðÞ ¼
0
:
99
0
:
0001
=
0
:
0010989
¼
0
:
09009
:
Facing these results you may decide to change your prior to p(B) = 0.09 and take
the blood test a second time. If the result is the same, then you should become
worried. The posterior probability p(B|A) would now increase to around 0.99.
This underlines the importance of repetition to contest the prior, what was what
Rev. Thomas Bayes emphasized when using his theorem, in the
first half of the 18th
century.
By combining Total Probability Theorem with Bayes Theorem posterior
probabilities of all the elements of a partition can be obtained:
X
p
ð
A 1 B
j Þ ¼½
p
ð
BA 1
j
Þ
p(A 1 Þ=
½
p
ð
BA j
Þ
p(A j Þ:
i 2 N
Cumulative Distribution Functions
For the evaluation of preferences the most important attributes are evaluated by real
numbers, and the most important events are numerical intervals. In that case the
probabilities of preference will be de
ned in the sigma algebra of Borel, which is
the set B of all countable unions of intervals and complements of intervals in the
real line (that is, the sigma generated by the intervals
a sigma algebra generated
by a given basis is the smallest sigma algebra containing it). To identify a
probability p in this sigma algebra, it is enough to inform the probability of the
intervals of the form (
−∝
, x] for every x
R.
, x) = F(x), any rightcontinuous nondecreasing function F
from R into [0, 1] satisfying lim x −∝ F(x) = 0 and lim x + F(x) = 1 determines a
unique probability p on (R, B). Functions of this kind are called cumulative
probability functions or cumulative distribution functions (cdf).
Specially easy to use are the cdf which are not only rightcontinuous but
absolutely continuous. Absolutely continuous are those functions F in the real
domain for which there is another function in the real domain f such that, for all x,
x
Conversely, by p(
−∝
ðÞ ¼ R x
a
;
ðÞ
:
Fx
f. f is called the density of F.
If F, absolutely continuous, is the cdf of p, the density f o F is determined by
fu
du
fx
ðÞ ¼lim
e! 0 ð
p
ð
x
e;
x
þ eÞ=ð
2
eÞ:
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