Geometric aspects of data-processing of Markov chains

Author:

Wolfer Geoffrey1,Watanabe Shun2

Affiliation:

1. Center for AI Project , RIKEN, Tokyo, Japan

2. Department of Computer and Information Sciences , Tokyo University of Agriculture and Technology, Tokyo, Japan

Abstract

Abstract 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.

Publisher

Oxford University Press (OUP)

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