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Dirichlet Polynomials and Entropy

Topos Institute, Berkeley, CA 94704, USA
Author to whom correspondence should be addressed.
Academic Editor: Karl Svozil
Entropy 2021, 23(8), 1085;
Received: 14 July 2021 / Revised: 18 August 2021 / Accepted: 19 August 2021 / Published: 21 August 2021
(This article belongs to the Section Information Theory, Probability and Statistics)
A Dirichlet polynomial d in one variable y is a function of the form d(y)=anny++a22y+a11y+a00y for some n,a0,,anN. We will show how to think of a Dirichlet polynomial as a set-theoretic bundle, and thus as an empirical distribution. We can then consider the Shannon entropy H(d) of the corresponding probability distribution, and we define its length (or, classically, its perplexity) by L(d)=2H(d). On the other hand, we will define a rig homomorphism h:DirRect from the rig of Dirichlet polynomials to the so-called rectangle rig, whose underlying set is R0×R0 and whose additive structure involves the weighted geometric mean; we write h(d)=(A(d),W(d)), and call the two components area and width (respectively). The main result of this paper is the following: the rectangle-area formula A(d)=L(d)W(d) holds for any Dirichlet polynomial d. In other words, the entropy of an empirical distribution can be calculated entirely in terms of the homomorphism h applied to its corresponding Dirichlet polynomial. We also show that similar results hold for the cross entropy. View Full-Text
Keywords: bundle; weighted geometric mean; category theory; Dirichlet polynomial bundle; weighted geometric mean; category theory; Dirichlet polynomial
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MDPI and ACS Style

Spivak, D.I.; Hosgood, T. Dirichlet Polynomials and Entropy. Entropy 2021, 23, 1085.

AMA Style

Spivak DI, Hosgood T. Dirichlet Polynomials and Entropy. Entropy. 2021; 23(8):1085.

Chicago/Turabian Style

Spivak, David I., and Timothy Hosgood. 2021. "Dirichlet Polynomials and Entropy" Entropy 23, no. 8: 1085.

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