The Importance Markov chain is a novel algorithm bridging the 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), the Importance Markov chain produces an extended Markov chain where the marginal distribution of the first component converges to the target distribution. For example, when targeting a multimodal distribution, the instrumental distribution can be chosen as a tempered version of the target which allows the algorithm to explore its modes more efficiently. We obtain a Law of Large Numbers and a Central Limit Theorem as well as 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.

翻译：重要性马尔可夫链是一种新颖的算法，它架起了拒绝采样与重要性采样之间的桥梁，通过一个调谐参数实现两者的过渡。该算法基于对面向工具性分布（通常通过MCMC核）的工具性马尔可夫链的修正样本，生成一个扩展马尔可夫链，其中第一个分量的边际分布收敛至目标分布。例如，当目标分布呈现多峰形态时，可将工具性分布选为目标分布的调整温度版本，从而使算法更高效地探索其模态。在工具性核的温和假设条件下，我们获得了该扩展核的大数定律、中心极限定理以及几何遍历性。从计算角度看，该算法易于实现，且可借助现有库对工具性分布进行采样。