Our importance Markov chain paper (with Charly Andral (PhD, Paris Dauphine), Randal Douc, and Hugo Marival (PhD, Telecom SudParis) has been published (on-line) by stochastic processes and their applications (SPA). Incidentally, my first publication in this journal. To paraphrase its abstract, it sort of bridges the (unsuspected?) gap between rejection sampling and importance sampling, moving from one to the other through a tuning parameter. Based on a modified sample of an instrumental Markov chain targeting an instrumental distribution (typically via a MCMC kernel), rather than the target of interest, the Importance Markov chain produces an extended Markov chain whose (first) marginal distribution converges to said target distribution. For instance, when targeting a multimodal distribution, the instrumental distribution can be chosen as a tempered version of the target and this frees the algorithm to explore its multiple modal regions more efficiently. We also derive a Law of Large Numbers and a Central Limit Theorem as well as prove geometric ergodicity for this extended kernel under mild assumptions on the instrumental kernel. Computationally, the algorithm is easy to implement and preexisting librairies can be used to sample from the instrumental distribution.
