Biomedical Engineering Reference
In-Depth Information
fX; U; Ygis:
2
(n1)^
P
t
M
t=1
D
S
(t)g
PfjX; U ; Yg/Pfgf(
2
)
2
exp
d
S
f
2
2
P
t
M
t=1
[S(t)D
S
(t)]g
(1d
S
)
f
P
P
t
M
t=0
[R
S
(t)+
t
u=0
R
I
(u;t)]g
!
f
P
P
t
M
t=0
[S(t)+
t
u=0
I(u;t)]g
(1!)
f
t
M
1
Y
p
S
(t)
1d
S
]
I(0;t+1)
[1
p
S
(t)
1d
S
]
S(t)D
S
(t)I(0;t+1)
[
t=0
P
P
t
M
r=t+1
F
I
(t;r)
[d
I
(t)]
t
M
r=t+1
D
I
(t;r)
(t)
P
t
M
r=t+1
[I(t;r)F
I
(t;r)D
I
(t;r)]
g;
(1(t)d
I
(t))
(13)
P
n
j=1
A
j
[Y (j)A
j
]
2
.
For the prior distribution of the unknown parameters, we will assume
that a priori
2
;
i
; i = 1; 2; 3 are independently distributed of one an-
other. Furthermore, we will follow Box and Tiao
1
to assume P (
2
)/(
2
)
1
and assume natural conjugate priors for the other parameters. That is, we
assume:
2
=
1
n1
1
where ^
Pf
i
; i = 1; 2; 3g/d
S
a
S
1
(1d
S
)
b
S
1
!
a
0
1
(1!)
b
0
1
t
M
1
Y
p
S
(t)
1d
S
]
u
S
(t)1
[1
p
S
(t)
1d
S
]
v
S
(t)1
[
t=0
[(t)]
a
G
(t)1
[d
I
(t)]
b
G
(t)1
(1(t)d
I
(t))
v
G
(t)1
;
(14)
where the hyperparametersfa
S
; b
S
; a
0
; b
0
; u
S
(t); v
S
(t); a
G
(t); b
G
(t); v
G
(t)g
are positive real numbers. These hyperparameters can be estimated from
previous studies. In the event that prior studies and information are not
available, we will assume all these parameters to be 1 to reect the fact
that our prior information are vague and imprecise.
4.4. The Generalized Bayesian Method for Estimating
Unknown Parameters and State Variables
Using the above distribution results, the multi-level Gibbs sampling
procedures for estimating the unknown parameters (;
2
) and the state
variables X are given by the following loop:
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