Statistical Criticality arises in Maximally Informative Representations.
We show that statistical criticality, i.e. the occurrence of power law frequency distributions, arises in samples that are most informative on the underlying generative 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. This differs from the entropy of the data, that we take as a measure of resolution. The entropy of the frequency, that we call relevance, provides an upper bound to the number of informative bits. Samples that maximise relevance at a given resolution - that we call most informative samples - exhibit statistical criticality. We show how this naturally arises from the concentration property of the Asymptotic Equipartition Property. Within a thermodynamic analogy, we find that most informative representations of high dimensional data arise from a principle of minimal entropy, at odds with equilibrium statistical mechanics where the entropy is maximised. This is why, contrary to statistical mechanics, statistical criticality requires no parameter fine tuning in most informative samples. In addition, Zipf's law arises at the optimal trade-off between resolution (i.e. compression) and relevance. As a byproduct, we derive an estimate of the maximal number of parameters that can be estimated from a dataset, in the absence of prior knowledge on the generative model. We finally show how our findings can be derived from an unsupervised version of the Information Bottleneck method.
Publisher URL: http://arxiv.org/abs/1808.00249
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