---
_id: '7130'
abstract:
- lang: eng
text: "We show that statistical criticality, i.e. the occurrence of power law frequency
distributions, arises in samples that are maximally informative about the underlying
generating process. In order to reach this conclusion, we first identify the frequency
with which different outcomes occur in a sample, as the variable carrying useful
information on the generative process. The entropy of the frequency, that we call
relevance, provides an upper bound to the number of informative bits. This differs
from the entropy of the data, that we take as a measure of resolution. Samples
that maximise relevance at a given resolution—that we call maximally informative
samples—exhibit statistical criticality. In particular, Zipf's law arises at the
optimal trade-off between resolution (i.e. compression) and relevance. As a byproduct,
we derive a bound of the maximal number of parameters that can be estimated from
a dataset, in the absence of prior knowledge on the generative model.\r\n\r\nFurthermore,
we relate criticality to the statistical properties of the representation of the
data generating process. We show that, as a consequence of the concentration property
of the asymptotic equipartition property, representations that are maximally informative
about the data generating process are characterised by an exponential distribution
of energy levels. This arises from a principle of minimal entropy, that is conjugate
of the maximum entropy principle in statistical mechanics. This explains why statistical
criticality requires no parameter fine tuning in maximally informative samples."
acknowledgement: We acknowledge interesting discussions with M Abbott, E Aurell, J
Barbier, R Monasson, T Mora, I Nemenman, N Tishby and R Zecchina. This research
was supported by the Kavli Foundation and the Centre of Excellence scheme of the
Research Council of Norway (Centre for Neural Computation) (RJC and YR), by the
Basic Science Research Program through the National Research Foundation of Korea
(NRF), funded by the Ministry of Education (2016R1D1A1B03932264) (JJ), and, in part,
by the ICTP through the OEA-AC-98 (JS).
article_number: '063402'
article_processing_charge: No
article_type: original
author:
- first_name: Ryan J
full_name: Cubero, Ryan J
id: 850B2E12-9CD4-11E9-837F-E719E6697425
last_name: Cubero
orcid: 0000-0003-0002-1867
- first_name: Junghyo
full_name: Jo, Junghyo
last_name: Jo
- first_name: Matteo
full_name: Marsili, Matteo
last_name: Marsili
- first_name: Yasser
full_name: Roudi, Yasser
last_name: Roudi
- first_name: Juyong
full_name: Song, Juyong
last_name: Song
citation:
ama: 'Cubero RJ, Jo J, Marsili M, Roudi Y, Song J. Statistical criticality arises
in most informative representations. Journal of Statistical Mechanics: Theory
and Experiment. 2019;2019(6). doi:10.1088/1742-5468/ab16c8'
apa: 'Cubero, R. J., Jo, J., Marsili, M., Roudi, Y., & Song, J. (2019). Statistical
criticality arises in most informative representations. Journal of Statistical
Mechanics: Theory and Experiment. IOP Publishing. https://doi.org/10.1088/1742-5468/ab16c8'
chicago: 'Cubero, Ryan J, Junghyo Jo, Matteo Marsili, Yasser Roudi, and Juyong Song.
“Statistical Criticality Arises in Most Informative Representations.” Journal
of Statistical Mechanics: Theory and Experiment. IOP Publishing, 2019. https://doi.org/10.1088/1742-5468/ab16c8.'
ieee: 'R. J. Cubero, J. Jo, M. Marsili, Y. Roudi, and J. Song, “Statistical criticality
arises in most informative representations,” Journal of Statistical Mechanics:
Theory and Experiment, vol. 2019, no. 6. IOP Publishing, 2019.'
ista: 'Cubero RJ, Jo J, Marsili M, Roudi Y, Song J. 2019. Statistical criticality
arises in most informative representations. Journal of Statistical Mechanics:
Theory and Experiment. 2019(6), 063402.'
mla: 'Cubero, Ryan J., et al. “Statistical Criticality Arises in Most Informative
Representations.” Journal of Statistical Mechanics: Theory and Experiment,
vol. 2019, no. 6, 063402, IOP Publishing, 2019, doi:10.1088/1742-5468/ab16c8.'
short: 'R.J. Cubero, J. Jo, M. Marsili, Y. Roudi, J. Song, Journal of Statistical
Mechanics: Theory and Experiment 2019 (2019).'
date_created: 2019-11-26T22:36:09Z
date_published: 2019-06-17T00:00:00Z
date_updated: 2021-01-12T08:11:57Z
day: '17'
doi: 10.1088/1742-5468/ab16c8
extern: '1'
external_id:
arxiv:
- '1808.00249'
intvolume: ' 2019'
issue: '6'
keyword:
- optimization under uncertainty
- source coding
- large deviation
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1808.00249
month: '06'
oa: 1
oa_version: Preprint
publication: 'Journal of Statistical Mechanics: Theory and Experiment'
publication_identifier:
issn:
- 1742-5468
publication_status: published
publisher: IOP Publishing
quality_controlled: '1'
status: public
title: Statistical criticality arises in most informative representations
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2019
year: '2019'
...