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Generalised Benford Law

Benford's law [1] crops up in the most unexpected places. If you are not familiar with it, an informal place to get started may be Netflix's documentary "Digits" from the series "Connected: The Hidden Science of Everything". Why does Benford's law happen at all? Is there a more general law behind it? Using scale-invariance considerations, Pietronero et al. first discovered that a more general law is possible [2], and Barabesi and Pratelli then extended their work [3]. In our paper [4] we have in turn rediscovered and further extended these generalised results, and even uncovered a previously unknown result for the classic Benford distribution: a closed-form expression for the distribution of the j-th most significant digit (for j>=2).

The quintessential application of Benford's law is forensic analysis. In [4], we also question the optimality of using most significant digits in forensic detection tests: we show that every time that Benford's "law" holds for a dataset, an alternative "law" for the continued fraction coefficients of the same data also holds! This provides an alternative forensic test, completely analogous to using most significant digits. Furthermore, both approaches (i.e. using significant digits or using leading continued fraction coefficients) can be beaten in forensic detection tests...

See the video of the presentation that we gave in EUSIPCO 2021, and download a Matlab toolbox reproducing the results in [4] (and more). The README in the zip file should be enough to get you going. Enjoy!

Félix Balado and Guénolé Silvestre


[1] F. Benford, "The law of anomalous numbers", Proceedings of the American Philosophical Society, 78(4):551--572, 1938.

[2] L. Pietronero, E. Tosatti, V. Tosatti, and A. Vespignani, "Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf", Physica A: Statistical Mechanics and its Applications, 293(1):297--304, 2001.

[3] L. Barabesi and L. Pratelli, "On the generalized Benford law", Statistics & Probability Letters, 160:108702, 2020.

[4] F. Balado and G. Silvestre, "Benford's Law: Hammering a Square Peg Into a Round Hole?", 29th European Conference on Signal Processing (EUSIPCO), Dublin, Ireland, August, 2021, pp. 796--800. [full text]

Last updated: $Date: 2021/12/02 07:48:17 $