Biomedical Engineering Reference
In-Depth Information
1; : : : ; k1g. From equations (7)-(9), the conditional posterior distribution
PfjX; U ; Ygof givenfX; U ; Ygis:
Y
m
k1
Y
P
P
t
M
1
t=0
t
M
1
t=0
B
(i
j
(t)g
i
f
M
1
(t)g
e
ft
M
i
g
[b
ji
]
f
PfjX; U; Yg/Pfg
i=1
j=1
P
P
t
M
1
t=0
D
(i)
j
t
M
1
t=0
R
(i)
j
[d
ji
]
f
(t)g
ji
f
(t)g
P
t
M
t=1
[I
j
(t)B
(i)
j
(t)D
(i)
j
(t)R
(i)
j
(1b
ji
d
ji
ji
)
f
(t)]g
;
(10)
where R
(i)
j
j
(t) + D
(i
j
(t).
For the prior distribution of the unknown parameters, we will assume
that a priori the parameters in are independently distributed of one
another. Furthermore, we will assume natural conjugate priors for all the
parameters. That is, we assume:
(t) = I
(i)
j
(t + 1)I
(i)
j
(t)B
(i)
m
k1
Y
Y
i
p
i
1
expf
i
q
i
g
[b
ji
]
u
ji
1
[d
ji
]
v
ji
1
ji
r
ji
1
Pfg/
i=1
j=1
(1b
ji
d
ji
ji
)
w
ji
1
; (11)
where the hyperparametersfp
i
; q
i
; u
ji
; v
ji
; w
ji
; r
ji
gare positive real num-
bers. These hyperparameters can be estimated from previous studies. In the
event that prior studies and information are not available, we will follow
Box and Tiao
6
to assume that Pf
i
; i = 1; : : : ; mg/
Q
m
i=1
(
i
)
1
and that
all other parameters are uniformly distributed to reect the fact that our
prior information are vague and imprecise.
5.4. The Generalized Bayesian Method for Estimating
Unknown Parameters and State Variables
Using the above distribution results, the multi-level Gibbs sampling proce-
dures for estimating the unknown parameters and the state variables X
are given by the following loop:
(i) Given the parameter values, we will use the stochastic equa-
tion (1) and the associated probability distributions to generate
a large sample offX; Ug. Then, by combining this large sample
with PfYjX; Ug, we selectfX; Ugfrom this sample through the
weighted Bootstrap method due to Smith and Gelfand
51
. This se-
lectedfX; Ugis then a sample generated from PfX; Uj; Ygal-
though the latter density is unknown (for proof, see Tan
56
, Chapter
3). Call the generated samplefX
()
; U
()
g.
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