Biomedical Engineering Reference
In-Depth Information
(1) Given the parameter values, we will use the stochastic equations
(1)-(3) 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 Gelfant
8
. This se-
lectedfX; Ugis then a sample generated from PfX; Uj; Ygal-
though the latter density is unknown (for proof, see Tan
10
, Chapter
3). Call the generated samplefX
()
; U
()
g.
(2) On substitutingfU
()
; X
()
gwhich are generated numbers from
the above step, generatef;
2
gfrom the conditional density
Pf;
2
jX
()
:U
()
; Yggiven by equation (13).
(3) Withf;
2
gbeing generated from Step 2 above, go back to Step
1 and repeat the above (1)-(2) loop until convergence.
The convergence of the above algorithm has been proved in Tan
10
, Chap-
ter 3. At convergence, one then generates a random sample offX; Ugfrom
the conditional distribution PfX; UjYgoffX; Uggiven Y , independent
of and a random sample of from the posterior distribution PfjYg
of given Y , independent offX; Ug. Repeat these procedures one then
generates a random sample of size N offX; Ugand a random sample of
size M of . One may then use the sample means to derive the estimates
offX; Ugand and use the sample variances as the variances of these
estimates.
5. Some Illustrative Examples
To illustrate the usefulness of state space models of AIDS epidemic given
in Section 4, in this section we apply the model and method to the Swiss
AIDS data sets of homosexual men. To assess the AIDS epidemic in this
population we will estimate simultaneously the death rates, the immigration
rate, the HIV infection distribution, the HIV incubation distribution and
the numbers of S people, I people and AIDS cases in these populations.
Given in Table 1 are the AIDS incidence data from 1981 until 1995 for the
homosexual population in Switzerland. (To avoid the problem of reporting
delay, we have used the data only up to December 1995.)
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