We examine data-processing of Markov chains through the lens of information geometry. We first establish a theory of congruent Markov morphisms within the framework of stochastic matrices. Specifically, we introduce and justify the concept of a linear right inverse (congruent embedding) for lumping, a well-known operation used in Markov chains to extract coarse information. Furthermore, we inspect information projections onto geodesically convex sets of stochastic matrices, and show that under some conditions, projecting (m-projection) onto doubly convex submanifolds can be regarded as a form of data-processing. Finally, we show that the family of lumpable stochastic matrices can be meaningfully endowed with the structure of a foliated manifold and motivate our construction in the context of embedded models and inference.

翻译：我们通过信息几何的视角审视马尔可夫链的数据处理。首先，在随机矩阵框架内建立了同余马尔可夫映射的理论。具体而言，针对马尔可夫链中用于提取粗粒度信息的常见操作——数据压缩（lumping），我们引入并论证了线性右逆（同余嵌入）这一概念。进一步地，我们研究了在随机矩阵测地凸集上的信息投影，并证明在某些条件下，向双凸子流形的投影（m-投影）可视为一种数据处理形式。最后，我们证明可数据压缩随机矩阵族能够有意义地赋予叶状流形结构，并在嵌入模型与推理背景下阐释了我们构造的合理性。