Combinatorial Characterization of Exponential Families of Lumpable Stochastic Matrices

2024-12-11

arXiv.org

AI Summary

This paper addresses the fundamental question of when the set of transition matrices of lumpable Markov chains over a finite state space—under a fixed aggregation function—forms an exponential family. Since this set generally fails to satisfy the structural requirements of an exponential family, we establish, for the first time, an efficient and verifiable combinatorial necessary and sufficient condition, thereby forging an exact correspondence between lumpability and exponential-family structure. Methodologically, we integrate tools from algebraic statistics, graph theory, and lumping theory for Markov chains; specifically, we characterize the problem via block-structured constraints induced by the aggregation and identify the associated sufficient statistics, yielding explicit combinatorial characterizations for the existence of lumpable exponential families. Our results provide a new theoretical foundation for interpretable stochastic modeling and model reduction, substantially enhancing both the statistical tractability and structural interpretability of aggregated models.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

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 combinatorial conditions for families of lumpable transition matrices to form exponential families.

Problem

• Characterize exponential families of lumpable stochastic matrices
• Find verifiable conditions for lumpable transition matrices
• Develop dimension-based method for exponential family determination

Innovation

• Dimension-based method for exponential families
• Efficiently verifiable lumpable conditions
• Characterization of lumpable stochastic matrices