article
Statistical criticality arises in most informative representations
published
yes
Ryan J
Cubero
author 850B2E12-9CD4-11E9-837F-E719E66974250000-0003-0002-1867
Junghyo
Jo
author
Matteo
Marsili
author
Yasser
Roudi
author
Juyong
Song
author
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.
IOP Publishing2019
eng
optimization under uncertaintysource codinglarge deviation
Journal of Statistical Mechanics: Theory and Experiment
1742-5468
1808.0024910.1088/1742-5468/ab16c8
20196
yes
Cubero, R. J., Jo, J., Marsili, M., Roudi, Y., & Song, J. (2019). Statistical criticality arises in most informative representations. <i>Journal of Statistical Mechanics: Theory and Experiment</i>. IOP Publishing. <a href="https://doi.org/10.1088/1742-5468/ab16c8">https://doi.org/10.1088/1742-5468/ab16c8</a>
Cubero, Ryan J, Junghyo Jo, Matteo Marsili, Yasser Roudi, and Juyong Song. “Statistical Criticality Arises in Most Informative Representations.” <i>Journal of Statistical Mechanics: Theory and Experiment</i>. IOP Publishing, 2019. <a href="https://doi.org/10.1088/1742-5468/ab16c8">https://doi.org/10.1088/1742-5468/ab16c8</a>.
R.J. Cubero, J. Jo, M. Marsili, Y. Roudi, J. Song, Journal of Statistical Mechanics: Theory and Experiment 2019 (2019).
Cubero RJ, Jo J, Marsili M, Roudi Y, Song J. Statistical criticality arises in most informative representations. <i>Journal of Statistical Mechanics: Theory and Experiment</i>. 2019;2019(6). doi:<a href="https://doi.org/10.1088/1742-5468/ab16c8">10.1088/1742-5468/ab16c8</a>
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.” <i>Journal of Statistical Mechanics: Theory and Experiment</i>, vol. 2019, no. 6, 063402, IOP Publishing, 2019, doi:<a href="https://doi.org/10.1088/1742-5468/ab16c8">10.1088/1742-5468/ab16c8</a>.
R. J. Cubero, J. Jo, M. Marsili, Y. Roudi, and J. Song, “Statistical criticality arises in most informative representations,” <i>Journal of Statistical Mechanics: Theory and Experiment</i>, vol. 2019, no. 6. IOP Publishing, 2019.
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