Biomedical Engineering Reference
In-Depth Information
4.1. The Stochastic System Model, the Augmented State
Variables and Probability Distributions
The probability distribution of the state variables in equation (6) is quite
complicated and not manageable. To implement the multi-level Gibbs sam-
pling procedure to estimate unknown parameters and state variables, we
thus expand the model by augmenting some un-observable dummy state
variables U(t) =fF
S
(t); D
S
(t); F
I
(u; t); D
I
(u; t); u = 0; 1; : : : ; tg
0
. By the
distribution results in Section (3.1), it is easily seen that the conditional
density of U(t) given X(t) is
S(t)
D
S
(t); F
S
(t)
d
S
D
S
(t)
p
S
(t)
F
S
(t)
(1d
S
PfU(t)jX(t)g=
Y
t
I(u; t)
F
I
(u; t); D
I
(u; t)
p
S
(t))
S(t)D
S
(t)F
S
(t)
(u)
F
I
(u;t)
f
u=0
[d
I
(u)]
D
I
(u;t)
(1(u)d
I
(u))
I(u;t)F
I
(u;t)D
I
(u;t)
g: (8)
From the model and distribution results in Section (3.1), the conditional
density of X(t + 1) givenfX(t); U(t)gis
t
Y
PfX(t + 1)jX(t); U(t)g= f
S
(c
S
(t); t)g
f
I
(c
I
(u; t); u; t);
(9)
u=0
where c
S
(t) = Max(0; S(t + 1)S(t) + I(0; t + 1) + F
S
(t) + D
S
(t)) and
c
I
(u; t) = Max(0; I(u + 1; t + 1)I(u; t) + F
I
(u; t) + D
I
(u; t)).
Put U =fU(t); t = 0; 1; : : : ; t
M
1g. Then, the joint density offX; Ug
is
t
M
1
Y
PfX; Ujg= PfX(0)g
PfX(i + 1)jX(i); U(i)gPfU(i)jX(i)g: (10)
i=0
Notice that the equation in (10) is a product of densities of negative bi-
nomials, multinomials and binomial variables so that the above distribution
is referred to as a chain negative binomial-multinomial distribution.
4.2. The Observation Model
Let Y (j) be the observed number of new AIDS cases during the time period
[t
j1
; t
j
) j = 1; : : : ; n. Then, the equation for the observation model of the
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