**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]

[back]