Out-of-distribution data and anomalous inputs are vulnerabilities of machine
learning systems today, often causing systems to make incorrect predictions.
The diverse range of data on which these models are used makes detecting
atypical inputs a difficult and important task. We assess a tool, Benford's
law, as a method used to quantify the difference between real and corrupted
inputs. We believe that in many settings, it could function as a filter for
anomalous data points and for signalling out-of-distribution data. We hope to
open a discussion on these applications and further areas where this technique
is underexplored.

机器学习系统中的越界数据和异常输入是当前的漏洞，往往导致系统做出错误的预测。本文评估了 Benford's 法则作为一种用于量化真实输入和受损输入之间差异的方法。我们相信在许多情况下，它可以用作检测异常数据点和标识越界数据的过滤器。我们希望对这些应用进行讨论，并进一步探索该技术的其他领域。

利用统计技术在视觉模型中检测异常或超出分布的数据

On the Detection of Anomalous or Out-Of-Distribution Data in Vision  Models Using Statistical Techniques

The renowned proverb, Numbers do not lie, underscores the reliability and
insight that lie beneath numbers, a concept of undisputed importance,
especially in economics and finance etc. Despite the prosperity of Benford's
Law in the first digit analysis, its scope fails to remain comprehensiveness
when it comes to deciphering the laws of number. This paper delves into number
laws by taking the financial statements of China real estate as a
representative, quantitatively study not only the first digit, but also depict
the other two dimensions of numbers: frequency and length. The research
outcomes transcend mere reservations about data manipulation and open the door
to discussions surrounding number diversity and the delineation of the usage
insights. This study wields both economic significance and the capacity to
foster a deeper comprehension of numerical phenomena.

通过定量研究中国房地产的财务报表，本文深入探讨数字定律，不仅对第一位数字进行分析，还描绘了数字的另外两个维度：频率和长度。研究结果超越了对数据操纵的简单理解，为数字多样性和使用洞察力的界定打开了新的讨论空间。这项研究具有经济意义，能够促进对数字现象的更深刻理解。

探索数字规律：基于中国房地产的证据

Exploring the Law of Numbers: Evidence from China's Real Estate

Researchers have observed that the frequencies of leading digits in many
man-made and naturally occurring datasets follow a logarithmic curve, with
digits that start with the number 1 accounting for $\sim 30\%$ of all numbers
in the dataset and digits that start with the number 9 accounting for $\sim
5\%$ of all numbers in the dataset. This phenomenon, known as Benford's Law, is
highly repeatable and appears in lists of numbers from electricity bills, stock
prices, tax returns, house prices, death rates, lengths of rivers, and
naturally occurring images. In this paper we demonstrate that human speech
spectra also follow Benford's Law on average. That is, when averaged over many
speakers, the frequencies of leading digits in speech magnitude spectra follow
this distribution, although with some variability at the individual sample
level. We use this observation to motivate a new set of features that can be
efficiently extracted from speech and demonstrate that these features can be
used to classify between human speech and synthetic speech.

该研究观察到许多人造和自然数据集的前导数字频率遵循对数曲线，证明了人类语音频谱平均也遵循 Benford 定律，并使用这个观察结果来创建一组新的 speech 特征，以实现人类语音和合成语音的分类。