On a Class of Tensor Markov Fields

Area of Science:

• * Statistical Physics
• * Probability Theory
• * Data Science

Background:

• * Markov Random Fields (MRFs) are foundational probabilistic graphical models.
• * Extending MRFs to tensor-valued data presents unique challenges and opportunities.
• * Existing models may not fully capture complex dependencies in multi-contextual data.

Purpose of the Study:

• * Introduce a new class of probabilistic graphical models: Tensor Markov Fields (TMFs).
• * Extend the theoretical framework of Markov Random Fields to tensor-valued random variables.
• * Establish a connection between TMFs and Gibbs fields, drawing from statistical mechanics.

Main Methods:

• * Generalization of Dobruschin, Hammersley, and Clifford theorems to tensor-valued fields.
• * Construction of TMFs based on information-theoretic principles for dependency structure inference.
• * Analysis of strictly positive probability measures for TMFs.

Main Results:

• * Demonstrated that Tensor Markov Fields are equivalent to Gibbs fields under strictly positive measures.
• * Established a direct link between TMFs and established results in theoretical statistical mechanics.
• * Provided a method for constructing TMFs from empirical data using information theory.

Conclusions:

• * Tensor Markov Fields offer a theoretically sound and generalizable framework for probabilistic modeling of tensor data.
• * The established connection to Gibbs fields facilitates the application of statistical mechanics principles.
• * TMFs provide a powerful and intrinsically simple tool for mathematical modeling and data analysis.