Motivated by information geometry, a distance function on the space of stochastic matrices is advocated. Starting with sequences of Markov chains the Bhattacharyya angle is advocated as the natural tool for comparing both short and long term Markov chain runs. Bounds on the convergence of the distance and mixing times are derived. Guided by the desire to compare different Markov chain models, especially in the setting of healthcare processes, a new distance function on the space of stochastic matrices is presented. It is a true distance measure which has a closed form and is efficient to implement for numerical evaluation. In the case of ergodic Markov chains, it is shown that considering either the Bhattacharyya angle on Markov sequences or the new stochastic matrix distance leads to the same distance between models.

翻译：受信息几何启发，本文提出了一种定义在随机矩阵空间上的距离函数。从马尔可夫链序列出发，我们主张将Bhattacharyya角作为比较马尔可夫链短期与长期运行的自然工具。我们推导了该距离收敛性和混合时间的界。基于比较不同马尔可夫链模型（尤其是在医疗过程建模场景中）的需求，本文提出了一种新的随机矩阵空间距离函数。该函数是一个具有闭式解、便于数值计算高效实现的真实距离度量。对于遍历马尔可夫链，研究表明：无论是基于马尔可夫序列的Bhattacharyya角，还是采用新的随机矩阵距离，最终得到的模型间距离是一致的。