[R] MCMC validity question

[Thomas Mang]

Hello,

I have quite a tough problem, which might be able to be solved by MCMC. 
I am fairly new to MCMC (in the learning process) - so apologize if the 
answer is totally obvious, and any hints, links etc are greatly appreciated.

I'll illustrate the problem in a version cut-down to the essentials - 
the real problem is ways more complex.
Suppose I have a Markovian series of poisson processes, which generate 
(or kill) some objects. At t0, say I have initially x = 3 objects. In 
the next time step, t1, the number of total objects is 
Poisson-distributed, subject to a function of x at t0, covariates, and 
parameters. So x_t1 ~ Pois, with E(x_t1) = f(x_t0, covariates, 
parameters). Let's choose a very simple function for f, say just f = x * 
par1 * Covariate1. Now let this process be repeated for say 6 times, 
always with the number of objects obtained in a previous step as input 
(x) for the next step.
The problem is, at all time steps the total number of objects remains 
unobservable, because they are only detected with a certain, low 
probability (itself subject to covariates and parameters). So if you 
observe say 2 objects, the only thing you know is the figure must be >= 
2. Presume however that the detection prob is equal and independent for 
all objects at a time step, and so the observed number of objects is 
also Poisson-distributed. The likelihood-function is then built upon 
that figure.

The main problem is the input to function f: In the first step, I know 
what x is (or even don't know that, might again just be from a 
distribution). From that step on, I have only a Poisson-distribution, 
and lack the concrete realization; all I know is a pure minimum value. 
In general f is not a simply thing, and is quite impracticable to input 
a distribution itself; moreover, because of the inflation of variance, 
the output could not be treated as lambda of a Poisson-distribution any 
more. So the Poisson-distribution is lost, although physical knowledge 
tells you it really is (and would be, were it x was known precisely).

The question is therefore, how can I work around the fact that x is 
always known to be only from a distribution, not knowing the precise 
realization?

Ignoring above issue, and keeping in mind that I have shown only a 
simplified version of the model, MCMC methods seem a reasonable choice. 
I had an idea, which looks so strange, that I have strong doubts if it's 
valid to so: Say I have a present parameter set. Is it then possible to 
estimate the Poisson-lambda for t1 using f, then draw a random number 
from that distribution, and now treat that drawn figure as (fixed) input 
for calculation for t2, and so on, repeating it until t6 ? At the end, 
propose new parameters based on MCMC sampling, and repeat all over again.
Will a long sequence of MCMC iterations homogenize these fake 
(simulated) realizations of the poisson-distributions, and so the chain 
will converge to the posterior distribution of the parameters?


many many thanks for any inputs and thoughts,
cheers,
Thomas