SOME STATE SPACE MODELS OF AIDS EPIDEMIOLOGY IN HOMOSEXUAL POPULATIONS - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 44

2. The State Space Models and the Generalized Bayesian
Approach
To illustrate, consider an infectious disease such as AIDS. Let X(t) be the
vector of stochastic state variables for key responses of the disease. Then,
X(t) is the stochastic model (stochastic process) for this disease. For this
process one can derive stochastic equations for the state variables of the
system by using basic biological mechanism of the disease; by using these
stochastic equations, one may also derive the probability distributions for
the state variables. If some observed data are available from this system,
one may also derive some statistical models to relate the data to the system.
Combining the stochastic model of the system with the statistical model,
one has a state space model for the system. That is, the state space
model of a system is a stochastic model consisting of two sub-models: The
stochastic system model which is the stochastic model of the system and
the observation model which is a statistical model relating some available
data to the system. It extracts biological information from the system via
its stochastic system model and integrates this information with those from
the data through its observation model.
2.1. Some Advantages of the State Space Models
The state space model of the system is advantageous over the stochastic
model of the system alone or the statistical model of the system alone in
several aspects. The following are some specic advantages:
(1) The statistical model alone or the stochastic model alone very often
are not identiable and can not provide information regarding some
of the parameters and variables. These problems usually do not
exist in state space models (see [2, 9, 15, 17]).
(2) State space model provides an optimal procedure to updating the
model by new data which may become available in the future. This
is the smoothing step of the state space models (see [3, 6]).
(3) The state space model provides an optimal procedure via Gibbs
sampling and the generalized Bayesian approach to estimate si-
multaneously the unknown parameters and the state variables of
interest; see Tan 10 , Tan and Ye 15 , Tan, Zhang and Xiong 17 . It is
optimal in the sense that the estimates are posterior mean values
which minimize the Bayesian risk under squared loss function.
(4) The state space model provides an avenue to combine information
from various sources (see [10]).