Biomedical Engineering Reference
In-Depth Information
As an example, in this section we will illustrate how to develop a state
space model for the extended multi-event model given in Section (3.1) with
the observation model being based on the observed number of cancer inci-
dence over time. For this state space model, the stochastic system model
is specied by the stochastic equations given by (1) with the probability
distribution of state variables being given in Section (4.3). The observa-
tion model is a statistical model based on the number of cancer cases
(y i (j); i = 1; : : : ; m; j = 1; : : : ; n) over n dierent age groups and m dif-
ferent exposure levels.
5.1. The Stochastic System Model, the Augmented State
Variables and Probability Distribution
The probability distribution for the state variables in equation (4) is ex-
tremely complicated involving many summations. For implementing the
Gibbs sampling procedures to estimate the unknown parameters and the
state variables, we thus expand the model by augmenting the dummy
un-observable variables U (t) =fB r (t); D r (t); r = 1; : : : ; k1gand put
U =f U (t); t = 0; : : : ; t M 1g. Then, from the distribution results in Sec-
tion (4.3), we have:
k1
Y
Pf U (t)j X (t)g=
f i fB i (t); D i (t)jI i (t)g;
i=1
k1
Y
Pf X (t + 1)j U (t); X (t)g= g 0 fb 0 (t); 0 (t)g
h i1 fb i (t)jB i (t); D i (t)g;
i=2
where for i = 1; : : : ; k1; b i1 (t) = I i (t + 1)I i (t)B i (t) + D i (t).
The joint density offX; Ugis
t M
Y
PfX; Ug=
Pf X (t)j X (t1); U (t1)gPf U (t1)j X (t1)g: (5)
t=1
5.2. The Observation Model and the Probability
Distribution of Cancer Incidence
The observation model is based on y ij , where y ij is the observed number
of new cancer cases in the jth age group [t j1 ; t j ) under exposure to the
carcinogen with dose level s i . Let n i (j) be the number of normal people
from whom the y ij are generated. To derive the probability distribution of
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