**B**ernard Delyon and François Portier just recently arXived a paper on population or evolutionary importance sampling, pointed out to me by Víctor Elvira. Changing the proposal or importance sampler at each iteration. And averaging the estimates across iterations, but also mentioning AMIS. While drawing a distinction that I do not understand, since the simulation cost remains the same, while improving the variance of the resulting estimator. (But the paper points out later that their martingale technique of proof does not apply in this AMIS case.) Some interesting features of the paper are that

- convergence occurs when the total number of simulations grows to infinity, which is the most reasonable scale for assessing the worth of the method;
- some optimality in the oracle sense is established for the method;
- an improvement is found by eliminating outliers and favouring update rate over simulation rate (at a constant cost). Unsurprisingly, the optimal weight of the t-th estimator is given by its inverse variance (with eqn (13) missing an inversion step). Although it relies on the normalised versions of the target and proposal densities, since it assumes the expectation of the ratio is equal to one.

When updating the proposal or importance distribution, the authors consider a parametric family with the update in the parameter being driven by moment or generalised moment matching, or Kullback reduction as in our population Monte Carlo paper. The interesting technical aspects of the paper include the use of martingale and empirical risk arguments. All in all, quite a pleasant surprise to see some follow-up to our work on that topic, more than 10 years later.