Entropy as a Universal Monoidal Natural Transformation: A Categorical Characterization
Keskeiset käsitteet
The essential properties of entropy (monotonicity, additivity, and subadditivity) are consequences of entropy being a monoidal natural transformation from the under category functor to an integrally closed partially ordered abelian group.
Tiivistelmä
The paper provides a categorical characterization of entropy, showing that the key properties of entropy - monotonicity, additivity, and subadditivity - arise naturally from entropy being a monoidal natural transformation.
The main insights are:
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For the category of finite probability spaces FinProb, the Shannon entropy H1 can be characterized as the universal monoidal natural transformation from the under category functor −/FinProb to the category of integrally closed partially ordered abelian groups. This provides a succinct characterization of Shannon entropy as a reflection arrow.
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The characterization can be generalized to other monoidal categories with a monoidal structure on their under categories, such as the category of finite abelian groups, the category of finite dimensional vector spaces, and the augmented simplex category. This allows defining entropies over these categories as well, and connecting them via the naturality of the universal entropy.
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For the category of ρ-th-power-summable discrete probability distributions LProbρ (0 < ρ < 1), the Shannon entropy alone is the universal monoidal natural transformation, without the need for the Hartley entropy.
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The conditional Shannon entropy can also be characterized as a universal functor to an integrally closed partially ordered abelian group, based on the chain rule, without requiring any continuity assumptions.
The paper provides a clean, category-theoretic characterization of entropy that unifies and generalizes previous results.
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arxiv.org
A Characterization of Entropy as a Universal Monoidal Natural Transformation
Tilastot
H1(X) ≥ H1(Y) for discrete random variables X, Y where Y is a function of X.
H1(X) + H1(Y) = H1(X, Y) for independent discrete random variables X, Y.
H1(X) + H1(Y) ≥ H1(X, Y) for any jointly-distributed discrete random variables X, Y.
Lainaukset
"The Shannon entropy can be characterized as the universal monoidal natural transformation from −/LProbρ to the category of integrally closed partially ordered abelian groups (a reflective subcategory of the lax-slice 2-category over MonCatℓin the 2-category of monoidal categories), providing a succinct characterization of Shannon entropy as a reflection arrow."
"We can likewise define entropy for every monoidal category with a monoidal structure on its under categories (e.g. the category of finite abelian groups, the category of finite inhabited sets, the category of finite dimensional vector spaces, and the augmented simplex category) via the reflection arrow."
Syvällisempiä Kysymyksiä
How can the categorical characterization of entropy be extended to continuous probability distributions or more general measure spaces
The categorical characterization of entropy can be extended to continuous probability distributions or more general measure spaces by considering the category of measurable spaces and measurable functions. In this context, the objects would be measurable spaces (sets equipped with a sigma-algebra) and the morphisms would be measurable functions between these spaces. The tensor product in this setting could be defined as the product sigma-algebra, and the entropy functor would map these measurable spaces to a suitable codomain category, such as ordered commutative monoids or ordered abelian groups. The coherence maps and natural transformations would need to be adapted to the continuous setting, ensuring that the properties of entropy hold for continuous distributions as well.
What are the implications of treating information as a "commodity" that can be exchanged, as suggested by the entropy of the opposite category of finite sets
Treating information as a "commodity" that can be exchanged has interesting implications for the study of entropy. By considering the entropy of the opposite category of finite sets, we are essentially quantifying the amount of information contained in different sets. Viewing information as a commodity suggests that it can be traded or transferred between different entities, similar to how commodities are exchanged in economic systems. This perspective opens up possibilities for analyzing information flow, transfer, and transformation in various contexts, shedding light on the dynamics of information exchange and processing.
Can the universal entropy monad provide insights into the relationship between different notions of entropy, such as the differential entropy and the information dimension for Gaussian random variables
The universal entropy monad can provide valuable insights into the relationship between different notions of entropy, such as the differential entropy and the information dimension for Gaussian random variables. By considering the unit of the idempotent monad as the universal entropy, we can connect these different entropies as components of a single natural transformation. This allows us to establish a unified framework for understanding and comparing various entropy measures across different categories, including Gaussian distributions. The universal entropy monad serves as a unifying concept that links diverse entropy measures and facilitates a deeper understanding of their interrelationships.
