It is in principle possible to compute the actual multi-dimensional densities. However, these would be time-and space-consuming to compute, store, and use. On the other hand, the densities obtained from a direct product need not be stored as they are implicitly defined from the one-mode densities. A further advantage of using this approximation to obtain multi-dimensional densities, is that multi-dimensional integrals of such densities with a sum-13 of-product potential representations can be easily factorized.