Statistical criticality arises in most informative representations

R.J. Cubero, J. Jo, M. Marsili, Y. Roudi, J. Song, Journal of Statistical Mechanics: Theory and Experiment 2019 (2019).


Journal Article | Published | English
Author
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Abstract
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. Furthermore, 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.
Publishing Year
Date Published
2019-06-17
Journal Title
Journal of Statistical Mechanics: Theory and Experiment
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).
Volume
2019
Issue
6
Article Number
063402
ISSN
IST-REx-ID

Cite this

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
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, 2019(6). https://doi.org/10.1088/1742-5468/ab16c8
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 2019, no. 6 (2019). https://doi.org/10.1088/1742-5468/ab16c8.
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, 2019.
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.
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.

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