196
Appendix A
Then 1
1
,
,
,
(
)
(
)
n
n
n
S
S
=
x
x
␥
␥
␥
and from Equation A.64,
(
) (
) ( ) ( ) (
)
r l
l
∞
−
= +
=
∫
1
0
P
, ,
!
G
,
l
l
f
n
n
N n l
e l
d
S
d
x
␥
␥
␥
␥
(A.73)
for l 5 0, 1, 2, . . . . Hence, E(N | d, , ␥) 5 n 1 ␥ S r (, ␥ | x n ) / S 1 (, ␥ | x n )
and the mixture density in Equation A.68 is given as
(
) ( )
( )
{
} ( ) (
)
«
=
−
−
, ,
exp
1
G
,
f
f
n
n
d
S
d
x
r
l
r l
l r l
l
␥
␥
(A.74)
The EM iteration
+
+
→
( )
( )
( 1)
( 1)
,
,
v
v
v
v
␥
␥
is given as
(
) (
)
+
= +
( 1)
( )
( )
( )
( )
( )
1
,
,
v
v
v
v
v
v
n
n
n
S
S
x
x
r
␥
␥
␥
␥
(A.75)
and
(
)
(
)
( 1)
( 1)
( )
( )
( 1)
( 1)
1
E
l o g
,
,
0
v
v
v
v
r
n
n
n
r
n
n
f A
+
+
+
+
∂
+ −
=
∂
U
x
d
u
␥
␥
␥
(A.76)
where r 5 1, 2, ... , m. In the case of the lognormal superpopulation,
Equation A.76 is reduced to Equation A.31 and Equation A.32 with N
replaced by ␥
(v+1)
. The conditional expectations of log A and (log A – m)
2
given data d and (', ␥') are given by Equations A.33 and A.34 with
j-density replaced by that in Equation A.74. The EM iterations will
always produce a pair of estimates (u ˆ
n
, g ˆ n ).
From a computational point of view, the EM algorithm for the
superpopulation approach to N is not any more diffi cult than for fi xed
N. The basic computation still lies in the accurate evaluation of the general gamma density d G n (l).
This approach for predicting N is suffi ciently general to include the
usual Bayesian/subjective approach. In this case, the prior distribution
can be any arbitrary but completely specifi ed distribution, and it need
not be a member of a parametric family. The posterior distribution of
N given the data d is that given in Equation A.64 with P(•|g) replaced
by the prior probability distribution. The EM algorithm for solves
Equation A.71 via this posterior distribution and f ( y|). This approach
to the estimation of N is attractive in petroleum resource evaluation
Appendix A
Then 1
1
,
,
,
(
)
(
)
n
n
n
S
S
=
x
x
␥
␥
␥
and from Equation A.64,
(
) (
) ( ) ( ) (
)
r l
l
∞
−
= +
=
∫
1
0
P
, ,
!
G
,
l
l
f
n
n
N n l
e l
d
S
d
x
␥
␥
␥
␥
(A.73)
for l 5 0, 1, 2, . . . . Hence, E(N | d, , ␥) 5 n 1 ␥ S r (, ␥ | x n ) / S 1 (, ␥ | x n )
and the mixture density in Equation A.68 is given as
(
) ( )
( )
{
} ( ) (
)
«
=
−
−
, ,
exp
1
G
,
f
f
n
n
d
S
d
x
r
l
r l
l r l
l
␥
␥
(A.74)
The EM iteration
+
+
→
( )
( )
( 1)
( 1)
,
,
v
v
v
v
␥
␥
is given as
(
) (
)
+
= +
( 1)
( )
( )
( )
( )
( )
1
,
,
v
v
v
v
v
v
n
n
n
S
S
x
x
r
␥
␥
␥
␥
(A.75)
and
(
)
(
)
( 1)
( 1)
( )
( )
( 1)
( 1)
1
E
l o g
,
,
0
v
v
v
v
r
n
n
n
r
n
n
f A
+
+
+
+
∂
+ −
=
∂
U
x
d
u
␥
␥
␥
(A.76)
where r 5 1, 2, ... , m. In the case of the lognormal superpopulation,
Equation A.76 is reduced to Equation A.31 and Equation A.32 with N
replaced by ␥
(v+1)
. The conditional expectations of log A and (log A – m)
2
given data d and (', ␥') are given by Equations A.33 and A.34 with
j-density replaced by that in Equation A.74. The EM iterations will
always produce a pair of estimates (u ˆ
n
, g ˆ n ).
From a computational point of view, the EM algorithm for the
superpopulation approach to N is not any more diffi cult than for fi xed
N. The basic computation still lies in the accurate evaluation of the general gamma density d G n (l).
This approach for predicting N is suffi ciently general to include the
usual Bayesian/subjective approach. In this case, the prior distribution
can be any arbitrary but completely specifi ed distribution, and it need
not be a member of a parametric family. The posterior distribution of
N given the data d is that given in Equation A.64 with P(•|g) replaced
by the prior probability distribution. The EM algorithm for solves
Equation A.71 via this posterior distribution and f ( y|). This approach
to the estimation of N is attractive in petroleum resource evaluation