标题： Pinsker 的适应总变差不等式 显示英文标题

标题： Pinsker's inequality for adapted total variation

摘要： Pinsker 的经典不等式断言，两个概率测度之间的总变异$TV(\mu, \nu)$被$\sqrt{ 2H(\mu|\nu)}$界定，其中$H$表示相对熵（或Kullback-Leibler散度）。 考虑到离散度量，$TV$可以被视为Wasserstein距离，并因此具有适应性变体$ATV$。 当$\mu, \nu$是随机过程$(X_k)_{k=1}^n, (Y_k)_{k=1}^n$的分布时，适应性Wasserstein距离相较于其经典对应项具有明显优势，并在随机控制到机器学习的众多应用中表现出色。 在本文中我们注意到适应性总变差距离$ATV$满足Pinsker型不等式$$ ATV(\mu, \nu)\leq \sqrt{n} \sqrt{2 H(\mu|\nu)}.$$ 显示英文摘要

摘要： Pinsker's classical inequality asserts that the total variation $TV(\mu, \nu)$ between two probability measures is bounded by $\sqrt{ 2H(\mu|\nu)}$ where $H$ denotes the relative entropy (or Kullback-Leibler divergence). Considering the discrete metric, $TV$ can be seen as a Wasserstein distance and as such possesses an adapted variant $ATV$. Adapted Wasserstein distances have distinct advantages over their classical counterparts when $\mu, \nu$ are the laws of stochastic processes $(X_k)_{k=1}^n, (Y_k)_{k=1}^n$ and exhibit numerous applications from stochastic control to machine learning. In this note we observe that the adapted total variation distance $ATV$ satisfies the Pinsker-type inequality $$ ATV(\mu, \nu)\leq \sqrt{n} \sqrt{2 H(\mu|\nu)}.$$