Some Order Preserving Inequalities for Cross Entropy and Kullback-Leibler Divergence.

Area of Science:

• Information Theory
• Machine Learning
• Statistical Analysis

Background:

• Cross entropy and Kullback-Leibler (K-L) divergence are core concepts in information theory.
• Cross entropy is utilized in machine learning for optimization, being equivalent to maximizing likelihood.
• K-L divergence serves as a metric to quantify differences between probability distributions.

Purpose of the Study:

• To derive new inequalities involving cross entropy and K-L divergence.
• To leverage the relationship between cross entropy and weighted geometric means for novel theoretical insights.
• To expand the mathematical toolkit for analyzing probability distributions.

Main Methods:

• Utilized the rearrangement inequality.
• Applied a recent theorem on weighted Kolmogorov means.
• Developed a new theorem specifically for inequalities between K-L divergences.

Main Results:

• Established new mathematical inequalities for cross entropy and K-L divergence.
• Demonstrated the application of these inequalities through numerical examples.
• Provided a deeper theoretical understanding of the relationships between these information-theoretic measures.

Conclusions:

• The paper presents novel inequalities that advance the theoretical understanding of cross entropy and K-L divergence.
• These findings have potential implications for optimization and distribution comparison in various scientific fields.
• The introduced methods offer new analytical tools for researchers in information theory and machine learning.