Ziv-Merhav Theorem Extension in Thermodynamic Formalism

Core Concepts

Asymptotic results for a modified cross-entropy estimator beyond Markovian settings.

Abstract

The content discusses the extension of the Ziv-Merhav theorem to decoupled measures over finite alphabets, proving strong asymptotic consistency for modified estimators. It introduces the concept of cross-entropic pressure and its relation to large deviations in thermodynamic formalism. The paper outlines a modification to the original estimator, providing more general results without auxiliary parsing. The structure, key insights, and implications are detailed in a comprehensive manner.
Structure:

Introduction to Entropy and Cross Entropy Estimation
Ziv-Merhav Parsing Procedure Overview
Modification of Ziv-Merhav Estimator (QN)
Central Theorem on Almost Sure Convergence
Open Problem Discussion on Rigidity in Cross-Entropy Analogue of Shannon–McMillan–Breiman Theorem
Examples and Applications Discussion

Key Insights:

Introduction to entropy, cross entropy, and relative entropy in information theory.
Ziv-Merhav procedure for estimating cross entropy between ergodic measures.
Extension of estimator QN for decoupled measures with strong asymptotic consistency.
Utilization of cross-entropic pressure inspired by thermodynamic formalism.
Implications for hidden Markov models and ψ-mixing measures.

Stats

"cN = o(N) under our assumptions."

Quotes

Key Insights Distilled From

On the Ziv-Merhav theorem beyond Markovianity II

by Nich... at arxiv.org 03-26-2024

https://arxiv.org/pdf/2312.02098.pdf

Deeper Inquiries

How does the modification proposed in this work improve upon the original Ziv-Merhav estimator

The modification proposed in this work improves upon the original Ziv-Merhav estimator by providing more general results and stronger consistency guarantees. While the original estimator was limited to specific cases like ergodic Markov measures, the modified estimator extends its applicability to a broader class of decoupled measures on shift spaces over a finite alphabet. This enhancement allows for a wider range of applications and scenarios where the estimation of cross entropy is required.

What are the practical implications of utilizing the cross-entropic pressure in information theory

Utilizing the cross-entropic pressure in information theory has practical implications in various areas. The cross-entropic pressure serves as a rescaled cumulant-generating function that offers insights into large deviations within the thermodynamic formalism. In practical terms, it provides a way to analyze and estimate relative entropies between different sources or systems, which is crucial in fields like statistical physics, dynamical systems, pattern recognition, and information theory. By leveraging techniques from thermodynamic formalism, researchers can gain deeper understanding and make more accurate estimations related to entropy, cross entropy, and relative entropy.

How can the findings of this study be applied to real-world scenarios beyond theoretical frameworks

The findings of this study have significant implications for real-world scenarios beyond theoretical frameworks. For example:

Data Compression: Improved estimators for cross entropy can enhance data compression algorithms by providing better estimates of information content between sources.

Pattern Recognition: In fields like machine learning and artificial intelligence, accurate estimation of entropic quantities is essential for tasks such as pattern recognition and anomaly detection.

Network Security: Understanding cross entropy relationships can help improve network security protocols by identifying unusual patterns or behaviors that deviate significantly from expected distributions.

Financial Modeling: Applying these findings to financial data analysis can lead to better risk assessment models based on information divergence between different market conditions or investment strategies.

Biomedical Research: Estimating relative entropies between biological sequences or datasets could aid in identifying similarities or differences critical for genetic research or disease diagnosis.

By applying the concepts developed in this study to real-world scenarios across diverse domains, researchers and practitioners can enhance their analytical capabilities and decision-making processes based on robust information-theoretic principles.