We consider the problem of sampling from a probability distribution $π$. It is well known that this can be written as an optimisation problem over the space of probability distribution in which we aim to minimise the Kullback--Leibler divergence from $π$. We consider several partial differential equations (PDEs) whose solution is a minimiser of the Kullback--Leibler divergence from $π$ and connect them to well-known Monte Carlo algorithms. We focus in particular on PDEs obtained by considering the Wasserstein--Fisher--Rao geometry over the space of probabilities and show that these lead to a natural implementation using importance sampling and sequential Monte Carlo. We propose a novel algorithm to approximate the Wasserstein--Fisher--Rao flow of the Kullback--Leibler divergence and conduct an extensive empirical study to identify when these algorithms outperforms other popular Monte Carlo algorithms.

翻译：我们考虑从概率分布 $π$ 中采样的问题。众所周知，这可以表述为在概率分布空间上的一个优化问题，其目标是最小化与 $π$ 的Kullback--Leibler散度。我们研究了几种偏微分方程，其解是Kullback--Leibler散度相对于 $π$ 的最小化器，并将它们与著名的蒙特卡洛算法联系起来。我们特别关注通过在概率空间上考虑Wasserstein--Fisher--Rao几何所导出的偏微分方程，并表明这些方程自然地引出了使用重要性采样和序贯蒙特卡洛的实现。我们提出了一种新颖的算法来近似Kullback--Leibler散度的Wasserstein--Fisher--Rao流，并进行了广泛的实证研究，以确定这些算法在何时优于其他流行的蒙特卡洛算法。