Variability as a better characterization of Shannon entropy

Abstract: The Shannon entropy, one of the cornerstones of information theory, is widely used in physics, particularly in statistical mechanics. Yet its characterization and connection to physics remain vague, leaving ample room for misconceptions and misunderstanding. We will show that the Shannon entropy can be fully understood as measuring the variability of the elements within a given distribution: it characterizes how much variation can be found within a collection of objects. We will see that it is the only indicator that is continuous and linear, that it quantifies the number of yes/no questions (i.e. bits) that are needed to identify an element within the distribution, and we will see how applying this concept to statistical mechanics in different ways leads to the Boltzmann, Gibbs and von Neumann entropies.

• The paper redefines Shannon entropy as a measure of variability by quantifying the number of yes/no questions required to pinpoint an element.
• It verifies that the entropy formula uniquely satisfies properties such as invariance, continuity, and additivity, underscoring its fundamental role.
• The analysis bridges statistical mechanics and information theory by clarifying entropy applications in both discrete and continuous domains.

Essay: Understanding Shannon Entropy Through Variability

The paper by Carcassi et al. provides an in-depth analysis of Shannon entropy, aiming to refine its characterization within the context of both information theory and physics. By interpreting Shannon entropy as a measure of variability, the authors propose a unified perspective that seeks to resolve the conceptual ambiguities often associated with this cornerstone of information theory. In statistical mechanics, Shannon entropy serves as a precursor to more specialized entropy concepts such as those pioneered by Boltzmann, Gibbs, and von Neumann.

Variability as a Measure of Entropy

The authors advocate for viewing Shannon entropy not simply as an abstract formula but as a precise indicator of variability within a distribution. In this context, "variability" refers to the diversity of elements in a distribution, assessed through the number of yes/no questions required to pinpoint an element's identity—a concept intrinsically linked to bits of information. This approach to entropy circumvents the imprecise terms such as "uncertainty" or "disorder" that traditionally accompany discussions of entropy but often create confusion.

Theoretical Implications

The paper asserts that the Shannon entropy formula, $H = -\sum p_i \log p_i$, is not arbitrary. Instead, it emerges as the sole continuous and linear indicator fulfilling specific properties: it is insensitive to relabeling elements, continuous with respect to weight variations, and additive across independent distributions. This positions Shannon entropy as a fundamental measure of variability across diverse scientific domains.

Applications and Interconnections in Physics

When translating the concept of Shannon entropy to a physical context, particularly statistical mechanics, the paper distinguishes between two types of distributions. The Boltzmann entropy pertains to variations at the level of single particles within a macrostate. In contrast, Gibbs and von Neumann entropies pertain to microstate distributions within a larger ensemble, whether classical or quantum. These applications illuminate how the treatment of microscopic configurations contributes to our understanding of macroscopic thermodynamic phenomena.

Addressing Continuous Variables

The paper carefully addresses the extension of entropy to continuous domains, noting that for continuous variables, entropy can take negative values depending on the choice of units—a fact often overlooked. The authors emphasize that phase space properties, such as volume conservation, ensure coordinate invariance of the Shannon entropy, reinforcing its suitability in describing physical systems.

Challenges and Clarifications

In addressing misconceptions, the authors reframe the role of information in both physics and computer science. They argue that "information" in an information-theoretic sense does not inherently contain knowledge but pertains to encoded data requiring context for meaningful interpretation. This distinction is critical when discussing entropy across disciplines, ensuring clarity when entropy-related terms cross theoretical boundaries.

Conclusion and Implications

Carcassi et al.’s analysis posits that reinterpreting Shannon entropy as variability creates a more cohesive narrative across disciplines. This perspective not only aligns more congruently with the mathematical structure of the entropy formula but also augments its intuitive appeal. In the realms of physics and information theory, this reformulation promises to facilitate a deeper understanding of entropy, potentially informing future advancements in theoretical and practical applications of entropy within AI and beyond. The work accentuates the significance of standardized interpretations to bridge gaps between differing domains of scientific inquiry.