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
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