We analyze the information geometric structure of time reversibility for parametric families of irreducible transition kernels of Markov chains. We define and characterize reversible exponential families of Markov kernels, and show that irreducible and reversible Markov kernels form both a mixture family and, perhaps surprisingly, an exponential family in the set of all stochastic kernels. We propose a parametrization of the entire manifold of reversible kernels, and inspect reversible geodesics. We define information projections onto the reversible manifold, and derive closed-form expressions for the e-projection and m-projection, along with Pythagorean identities with respect to information divergence, leading to some new notion of reversiblization of Markov kernels. We show the family of edge measures pertaining to irreducible and reversible kernels also forms an exponential family among distributions over pairs. We further explore geometric properties of the reversible family, by comparing them with other remarkable families of stochastic matrices. Finally, we show that reversible kernels are, in a sense we define, the minimal exponential family generated by the m-family of symmetric kernels, and the smallest mixture family that comprises the e-family of memoryless kernels.

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可逆马尔可夫链的信息几何

我们分析了马尔可夫链不可约转移核参数族的时间可逆性信息几何结构。我们定义和表征了马尔可夫核的可逆指数族，并证明了不可约和可逆马尔可夫核既形成了一个混合族，又可能令人惊讶地形成了所有随机核集合中的指数族。我们提出了可逆内核的整个流形的参数化，并检查可逆测地线。我们定义了可逆流形上的信息投影，并推导出 e 投影和 m 投影的封闭形式表达式，以及关于信息发散的毕达哥拉斯恒等式，从而导致马尔可夫核可逆化的一些新概念。我们展示了与不可约和可逆核有关的边缘度量系列也在对分布之间形成指数系列。我们通过将可逆矩阵与其他非凡的随机矩阵族进行比较，进一步探索可逆族的几何特性。最后，我们证明可逆内核在我们定义的某种意义上是由对称内核 m 系列生成的最小指数族，以及包含无记忆内核 e 系列的最小混合族。