Importance sampling (IS) is a powerful Monte Carlo methodology for the approximation of intractable integrals, very often involving a target probability density function. The performance of IS heavily depends on the appropriate selection of the proposal distributions where the samples are simulated from. In this paper, we propose an adaptive importance sampler, called GRAMIS, that iteratively improves the set of proposals. The algorithm exploits geometric information of the target to adapt the location and scale parameters of those proposals. Moreover, in order to allow for a cooperative adaptation, a repulsion term is introduced that favors a coordinated exploration of the state space. This translates into a more diverse exploration and a better approximation of the target via the mixture of proposals. Moreover, we provide a theoretical justification of the repulsion term. We show the good performance of GRAMIS in two problems where the target has a challenging shape and cannot be easily approximated by a standard uni-modal proposal.

翻译：重要性采样（IS）是一种强大的蒙特卡洛方法，用于近似难以直接计算的积分问题，其通常涉及目标概率密度函数。IS的性能在很大程度上取决于样本生成所依据的提议分布的合理选择。本文提出了一种名为GRAMIS的自适应重要性采样器，能够迭代优化提议集合。该算法利用目标的几何信息来调整这些提议的位置和尺度参数。此外，为实现协同自适应，我们引入了一个排斥项，以促进对状态空间的协调探索。这转化为更多样化的探索以及通过提议混合对目标的更好近似。同时，我们为排斥项提供了理论依据。在两个目标形状复杂且难以通过标准单模态提议近似的问题中，我们展示了GRAMIS的优异性能。