It is known that the set of lumpable Markov chains over a finite state space, with respect to a fixed lumping function, generally does not form an exponential family of stochastic matrices. In this work, we explore efficiently verifiable necessary and sufficient combinatorial conditions for families of lumpable transition matrices to form exponential families.

翻译：已知在有限状态空间上，对于固定的聚合函数，可聚合马尔可夫链的集合通常不构成随机矩阵的指数族。本文中，我们探讨了可聚合转移矩阵族构成指数族时，高效可验证的必要与充分组合条件。