Given a target distribution $\pi$ and an arbitrary Markov infinitesimal generator $L$ on a finite state space $\mathcal{X}$, we develop three structured and inter-related approaches to generate new reversiblizations from $L$. The first approach hinges on a geometric perspective, in which we view reversiblizations as projections onto the space of $\pi$-reversible generators under suitable information divergences such as $f$-divergences. Different choices of $f$ allow us to recover almost all known reversiblizations while at the same time unravel and generate new reversiblizations. Along the way, we give interesting geometric results such as bisection properties, Pythagorean identities, parallelogram laws and a Markov chain counterpart of the arithmetic-geometric-harmonic mean inequality governing these reversiblizations. This also motivates us to introduce the notion of information centroids of a sequence of Markov chains and to give conditions for their existence and uniqueness. Building upon the first approach, we view reversiblizations as generalized means in the second approach, and construct new reversiblizations via different natural notions of generalized means such as the Cauchy mean or the dual mean. In the third approach, we combine the recently introduced locally-balanced Markov processes framework and the notion of convex $*$-conjugate in the study of $f$-divergence. The latter offers a rich source of balancing functions to generate new reversiblizations.

翻译：给定目标分布 $\pi$ 及有限状态空间 $\mathcal{X}$ 上的任意马尔可夫无穷小生成元 $L$，我们提出了三种结构相关的方法来基于 $L$ 构造新的可逆化。第一种方法基于几何视角，将可逆化视为在适当信息散度（如 $f$-散度）下向 $\pi$-可逆生成元空间的正交投影。通过选择不同的 $f$，我们不仅可以复现几乎全部已知的可逆化方法，还能同时揭示并生成新的可逆化。在此过程中，我们获得了有趣的几何结果，包括二分性质、毕达哥拉斯恒等式、平行四边形律以及这些可逆化所满足的马尔可夫链版本的算术-几何-调和平均不等式。这进一步促使我们引入马尔可夫链序列的信息质心概念，并给出其存在唯一性条件。基于第一种方法，我们在第二种方法中将可逆化视为广义均值，通过柯西均值或对偶均值等不同自然广义均值概念构造新的可逆化。在第三种方法中，我们将近期提出的局部平衡马尔可夫过程框架与 $f$-散度研究中凸共轭的概念相结合。后者为生成新的可逆化提供了丰富的平衡函数来源。