I have now a deeper understanding of the method, and one fact that was not set straight (on the paper and for me at least), is that the set of particles only model an empirical distribution which we assume approximate a Gaussian distribution (true at t=0, not so true afterwards for non-Gaussian targets).
Therefore after training we can test the final distribution using Monte-Carlo estimation in two ways
Compute the approximated Gaussian distribution (taking mean and covariance of the empirical distribution) and then sample from it
Take the empirical distribution and use the particles as samples
Given that the empirical distribution allows to have multiple modes, the second option seems to be richer but this is something that we should probably explore
I have now a deeper understanding of the method, and one fact that was not set straight (on the paper and for me at least), is that the set of particles only model an empirical distribution which we assume approximate a Gaussian distribution (true at t=0, not so true afterwards for non-Gaussian targets). Therefore after training we can test the final distribution using Monte-Carlo estimation in two ways
Given that the empirical distribution allows to have multiple modes, the second option seems to be richer but this is something that we should probably explore