From Boltzmann to Zipf through Shannon and Jaynes
Corral, Álvaro (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
García del Muro, Montserrat (Universitat de Barcelona. Institut de Nanociència i Nanotecnologia)

Data:	2020
Resum:	The word-frequency distribution provides the fundamental building blocks that generate discourse in natural language. It is well known, from empirical evidence, that the word-frequency distribution of almost any text is described by Zipf's law, at least approximately. Following Stephens and Bialek (2010), we interpret the frequency of any word as arising from the interaction potentials between its constituent letters. Indeed, Jaynes' maximum-entropy principle, with the constrains given by every empirical two-letter marginal distribution, leads to a Boltzmann distribution for word probabilities, with an energy-like function given by the sum of the all-to-all pairwise (two-letter) potentials. The so-called improved iterative-scaling algorithm allows us finding the potentials from the empirical two-letter marginals. We considerably extend Stephens and Bialek's results, applying this formalism to words with length of up to six letters from the English subset of the recently created Standardized Project Gutenberg Corpus. We find that the model is able to reproduce Zipf's law, but with some limitations: the general Zipf's power-law regime is obtained, but the probability of individual words shows considerable scattering. In this way, a pure statistical-physics framework is used to describe the probabilities of words. As a by-product, we find that both the empirical two-letter marginal distributions and the interaction-potential distributions follow well-defined statistical laws.
Drets:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.
Llengua:	Anglès
Document:	Article ; recerca ; Versió publicada
Matèria:	Maximum entropy principle ; Two-letter interactions ; Boltzmann factor ; Word-frequency distribution ; Zipf's law ; Quantitative linguistics ; Power laws
Publicat a:	Entropy, Vol. 22, Issue 2 (February 2020) , art. 179, ISSN 1099-4300

DOI: 10.3390/e22020179
PMID: 33285954

19 p, 1.1 MB

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