Characterization of Exponential Families of Lumpable Stochastic Matrices

TL;DR

This paper investigates conditions under which families of lumpable Markov chain transition matrices form exponential families, providing a dimension-based method to verify this property efficiently.

Contribution

It introduces necessary and sufficient conditions for lumpable matrices to be exponential families and develops a general dimension-based verification method.

Findings

Identifies conditions for lumpable matrices to be exponential families

Provides a dimension-based method for verification

Enhances understanding of the structure of lumpable Markov chains

Abstract

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 conditions for families of lumpable transition matrices to form exponential families. To this end, we develop a broadly applicable dimension-based method for determining whether a given family of stochastic matrices forms an exponential family.