How should one simulate a continuously-marked point process by Birth and Death MCMC?

Question

As the question states, I am not sure how I should go about doing that. The main issue is how do I draw a sample from the marked space?

The way I am doing things now is as follow:

If we are in a birth step, I draw a point from the observation window according to some density, then I independently draw a realization from the existing marked space (kind of like bootstrapping) and attach it to the previously selected point and then calculate MH ratio to see whether it gets rejected or not.

But I am not sure whether this is the correct way of doing it. Specifically, can I independently draw a sample from the marked space and attach it to the selected point? (Assuming the marks are not a function of the position of the point, e.g. distance of the point to some existing structure)

The only literature I found that touched a little bit on Birth Death MH algorithm for marked point process is 'Perfect simulation for marked point process' by M. N. M. Van Lieshout and R Stoica.

In the paper, they said for a birth step, one would have to draw a new point $\xi$ following the distribution $\nu \times \nu_M/\nu(K) $ where $\nu$ is the Lebesgue measure on the compact subset of $K$ for position and $\nu_M$ is a probability measure on marked space $M$.

So it seems like what I did was treating the joint distribution for generating points and marks as the product of two marginals.

Now if I set certain interaction parameters in my model to be a function of the marks, will this method still give me the correct pattern?

Thanks a lot!

Answer 1

Since no satisfying answer was provided, I consulted one of the proposers of the Birth Death MH algorithm, Dr. Jesper Moller, to clarify this.

He confirmed that specifying the proposal distribution for a marked point is arbitrary since it is essentially in the same vein as choosing the proposal distribution in a standard MCMC. It's just that different proposal distribution lead to different speed in convergence.

This implies that if we assume a probability model that defines a dependence structure between the locations and the marks, the dependence structure in the simulated data will manifest itself according to the assumed model since it is a MCMC algorithm and it is guaranteed to converge asymptotically no matter the proposal distribution. Hence, the proposal distribution for point locations can be independent from that of the marks.

Hope this is help.