Dirichlet Polynomials and Entropy

. 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) = 2^{H (d)}

. On the other hand, we will define a rig homomorphism

h : Dir \to Rect

from the rig of Dirichlet polynomials to the so-called rectangle rig, whose underlying set is

R_{⩾ 0} \times R_{⩾ 0}

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

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Spivak, D.I.; Hosgood, T. Dirichlet Polynomials and Entropy. Entropy 2021, 23, 1085. https://doi.org/10.3390/e23081085

AMA Style

Spivak DI, Hosgood T. Dirichlet Polynomials and Entropy. Entropy. 2021; 23(8):1085. https://doi.org/10.3390/e23081085

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Spivak, David I., and Timothy Hosgood. 2021. "Dirichlet Polynomials and Entropy" Entropy 23, no. 8: 1085. https://doi.org/10.3390/e23081085

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