Quasi-Monte-Carlo vs. variable dimensionality?

Question

I've been looking through the Matlab documention on using quasi-random sampling of N-dimensional unit cubes. This represents a problem with N stochastic parameters. Based on the fact that it is a unit cube, I presume that I need to use the inverse CDF of each parameter to map from the [0,1] domain to the value range of each parameter.

I would like to try this on a problem for which I now use Monte Carlo. Unfortunately, the problem I'm analyzing does not have a fixed number of dimensions. For each instantiation of the problem, I generate a variable number of widgets (say) using a Poisson distribution. Only after that do I randomly generate the parameters for each widget. That whole process yields one instance of the problem to be analyzed, so the number of parameters varies from one instance to the next.

Is this kind of problem still amenable to Quasi-Monte-Carlo?

Answer 1

What I used once was to get highest possible dimension of the problem d, generate Sobol sequence in d and use whatever number of points necessary for a particular sampling. I would say it helped somewhat...

Answer 2

From talking to a much smarter colleague, we need to consider the various combinations of widget counts for each widget type. For example, if we have 2 of widget type#1, 4 of widget type #2, 1 of widget type #3, etc., that constitutes one combination. QMC can be applied to that one combination. We are assuming that number of widget#i is independent of the number of widget#j for i<>j, so the probability of each combination is just the product of p(2 widgets of type#1), p(4 widgets of type#2), p(1 widget of type#3), etc. The individual probabilities are easy to get from their Poisson distributions (or their flat distributions, or whatever distribution is being used). If there are N widget types, this is just a joint PMF in N-space. This probability is then used to weight the QMC result for that particular combination. Note that even when the exactly combination is nailed down, QMC is still needed because there each widget is associated with 3 stochastic parameters.