We consider the problem of sampling from a probability distribution $π$ which admits a density w.r.t. a dominating measure. It is well known that this can be written as an optimisation problem over the space of probability distributions in which we aim to minimise a divergence from $π$. The optimisation problem is normally solved through gradient flows in the space of probability distributions with an appropriate metric. We show that the Kullback--Leibler divergence is the only divergence in the family of Bregman divergences whose gradient flow w.r.t. many popular metrics does not require knowledge of the normalising constant of $π$.

翻译：我们考虑从概率分布$π$中采样的问题，该分布关于某个控制测度具有密度函数。众所周知，这可以表述为概率分布空间上的一个优化问题，其目标是最小化与$π$的散度。该优化问题通常通过具有适当度量的概率分布空间中的梯度流来求解。我们证明，在Bregman散度族中，Kullback--Leibler散度是唯一一种其关于多种常用度量的梯度流无需知道$π$的归一化常数的散度。