Geoscience Reference
In-Depth Information
the standard deviation of the posterior logit is 0.401. The posterior probability of the
unit cell containing a deposit becomes 0.069 with approximate standard deviation
equal to 0.069
0.028. In this way, a standard deviation can be estimated
for each posterior probability on the final integrated pattern. However, it will be
shown later that if one or more patterns are missing, the standard deviation of the
posterior probability should be increased due to lack of knowledge. Because no
information on geochemical signature is available for the unit cell in the preceding
example, the final standard deviation becomes 0.042 instead of 0.028. It is custom-
ary to produce
t
0.401
¼
p
/
˃
(
p
) maps that accompany the
p
maps with posterior proba-
bilities. Use of Student's
t
instead of standard normal
z
indicates that the 95 %
confidence limit of 1.96 for
z
is too small. However, the exact number of degrees of
freedom for Student's
t
is not known for this application. In practice, the 95 %
confidence limit for
t
is probably some value between 2.0 and 2.5 for a two-sided
test. For a one-sided test the 95 % confidence interval for
z
is 1.645 and the
corresponding value for Student's
t
would be somewhat greater.
The equation used for estimating the standard deviation of the contrast
C
is based
on the following asymptotic result for large
n
(see Bishop et al.
1975
, p. 377):
"
#
2
ðÞ
¼
α
1
pbd
1
pbd
þ
1
p bd
þ
1
p b d
2
˃
1
ðÞ
þ
n
Extrapolation of this variance to
C
¼
log
e
α
is only valid if it is small compared to
α
.
Box 5.3: Incorporation of Uncertainty Because of One or More
Missing Patterns
This refinement is based on a proposal by Spiegelhalter (
1986
, p. 37) to regard any
prior probability
p
(
d
) as the expectation of the possible final probabilities
p
(
d
|
x
)
that may be obtained on observing data
x
. In general,
p
(
d
)
¼
R
p
(
d
|
x
)
¼
E
x
[
p
(
d
|
X
)]
p
(
x
)
dx
. In the situation that
there are th
re
e
m
ap layers as in Fig.
5.1
:
X
ij
pd
b
i
c
j
pb
i
c
j
¼
pd
bc
pbðÞþd
bc
p bc
þ d
bc
pbðÞþd
bc
p bc
pðÞ
¼
p
(
d
))]
2
p
(
b
i
c
j
). If only
B
is unknown, the information on
C
can be added to the prior probability in
order
˃
2
[
p
(
d
)]
¼
∑
ij
[
p
(
d
|
b
i
c
j
with corresponding variance
to
obtain
updated
prior
probabilities
p
b
(
d
) with
variance:
pd
b
pd
b
2
pd
2
p b
as follows from
2
˃
1
pd
½
ðÞ
¼
pd
ðÞ
ðÞþ
pd
ðÞ
∑
j
p
(
d
|
b
i
c
j
)
p
(
b
i
c
j
)
¼
p
(
d
|
b
i
)
p
(
b
i
).
˃
2
2
(two patterns
missing) derived in Box
5.3
are independent of any other patterns for which data
were available and used to change the prior probability. In the example that resulted
˃
1
2
(one pattern missing) and
The expressions for the variances
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