SOME STATE SPACE MODELS OF AIDS EPIDEMIOLOGY IN HOMOSEXUAL POPULATIONS - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 51

state space model is given by the equation:

t j 1 t j

X X X

t

Y (j) = A j + e(j) = A(t) + e(j) = F I (u; t) + e(j)

t=t j1 t=t j1 u=0

where e(j) is the measurement error (reporting error for reporting AIDS
incidence) for observing Y (j) and A j =

P

t j 1
t=t j1 A(t).
After correcting for reporting delay and under reporting for the AIDS
incidence data, one may assume that the e(j)'s are independently dis-
tributed as normal random variables with mean zero and with variance
j depending on A j . Since AIDS may be assumed as an inated Poisson
process (variance is much greater than the mean), one may assume that
W j =fY (j)A j g=

p
A j as normal with mean 0 and variance 2 so that
j = A j 2 . It follows that the conditional density of Y = (Y (j); j =
1; : : : ; n) givenfX; Ugis PfYjX; Ug=

Q

n
j=1 gfY (j)jX; Ug, where

gfY (j)jX; Ug= gfY (j)jA j g= (2A j 2 ) 2
expf 1

2A j 2 [Y (j)A j ] 2 g:

(11)

The the joint density offX; U ; Ygis

n

Y

P rfX; U ; Yg= = PfX(0)g gfY (j)jA j g

j=1

t j 1

Y

PfX(t + 1)jX(t); U(t)gPfU(t)jX(t)g: (12)

t=t j1

The above distribution will be used to derive the conditional posterior
distribution of the unknown parameters =f; 2 ggivenfX; U ; Yg.
Notice that because the number of parameters is very large, the classical
sampling theory approach by using the likelihood function PfYjX; Ugis
not possible without making assumptions about the parameters; however,
this problem can easily be avoided by new information from the stochastic
system model and the prior distribution of the parameters.

4.3. The Posterior Distribution of the Unknown
Parameters and State Variables
Let Pfgbe the prior distribution of =f; 2 g. From equations
(11)-(12), the conditional posterior distribution PfjX; U; Ygof given