194
Appendix A
0
1
0
( ,
', ')
(
)
+ E
E
log (
) , , ', '
, ', '
0
1, 2, ...,
1, 2, ...,
r
n
i
n
n
r
n
Q
N
N n
f Y
N
N
i
k r
m
g
+
∂
=
∂



 

−
∂

 =



 

∂








=
=
U
x
x
d
u
␪ ␥ ␪ ␥
␪
␪
␪ ␥
␪ ␥
(A.59)
where N 0 5 E(N|d, ␪', ␥'). Note that when P(N|␥) is a point mass at N 0 ≥ n,
Equation A.59 reduces to Equation A.28.
We now derive the conditional expectations. Given for (␪, ␥), x n and
N–n 5 l for l 5 0, 1, 2, … , defi ne
(
) ( ) ( ) (
)
l
r l
l
= +
+
≥
,
!
P
! ,
0
l
l
f
q
n l
n l
l
␪ ␥
␪
␥
(A.60)
(
)
(
) ( )
l
l
∞
= ∫
0
,
,
G
l
n
l
n
S
q
d
x
␪ ␥
␪ ␥
(A.61)
(
)
(
)
∞
=
= ∑
0
,
,
n
l
n
l
s
S
x
x
␪ ␥
␪ ␥
(A.62)
Then the likelihood of (␪, ␥) given the data d is
( ) ( )
(
)
=


=


∏
1
w w
b
,
n
j
j
j
n
j
L
f x
s
x
␪
␪␥
(A.63)
and the conditional probability function of N, given d, is
(
) (
) (
)
P
, ,
,
,
l
n
n
N n l
S
s
␪ ␥
␪ ␥
␪ ␥
= +
=
d
x
x
(A.64)
Now, for l ≥ 0 and l ≥ 0, defi ne a density similar to Equation A.19 as
(
) (
) ( ) (
)
j l
=
, , ,
,
G
,
n
l
n
l
n
l
q
d
S
x
x
l
l
␪ ␥
␪ ␥
␪ ␥
(A.65)
Let h (a|l, ␪) be the density given by Equation A.22. Then the conditional density of Y n11 , given N 5 n 1 1 and x n at Y n11 5 a, is
(
) (
) (
)
h l
j l
l
∞
0
= +
= ∫
, , ,
,
, , ,
n
n
f a N n l
a
l
d
x
x
␪ ␥
␪
␪ ␥
(A.66)
Therefore the second term in the solution of Equation A.59 is equal to