The Importance Markov Chain

The Importance Markov chain is a new algorithm bridging the gap between rejection sampling and importance sampling, moving from one to the other using a tuning parameter. Based on a modified sample of an auxiliary Markov chain targeting an auxiliary target (typically with a MCMC kernel), the Importance Markov chain amounts to construct an extended Markov chain where the marginal distribution of the first component converges to the target distribution. We obtain the geometric ergodicity of this extended kernel, under mild assumptions on the auxiliary kernel. As a typical example, the auxiliary target can be chosen as a tempered version of the target, and the algorithm then allows to explore more easily multimodal distributions. A Law of Large Numbers and a Central limit theorem are also obtained. Computationally, the algorithm is easy to implement and can use preexisting librairies to simulate the auxiliary chain.