科学与数学en.wikipedia.org
本福德定律
Benford's Law
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导读
Benford's Law（本福特定律）描述许多真实世界数据集中首位数字的非均匀分布：数字 1 出现得最频繁，约为 30%，随后依次递减。维基百科页面介绍其定义、数学解释、常见应用与局限，并提到它被用于审计和舞弊检测，也被卷入选举数据争议。
为什么经久不衰：它用极简的数学规律解释了大量日常数据中的反直觉现象，例子直观、应用广泛，而且围绕它的误用与争议至今仍有讨论价值。
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用最大熵和尺度参数解释 Benford 律为何自然，而不是神秘现象。
Benford 定律、Zipf 定律和其他幂律常让人着迷，但它们体现的是很简单的东西：当事情在乘法操作上均匀分布，而不是在加法操作上均匀分布时的行为。也就是说，x 大 x% 与 x 小 x% 的概率相同，而不是更常见的均匀分布定义（大 x 个单位与小 x 个单位概率相同）。对许多测量来说，在乘法上均匀分布很自然；对于不能为负的量，这被视为最均匀的“最大熵”分布。一般来说，位置参数倾向于在加法/减法上均匀，而尺度参数倾向于在乘法上均匀（对 log(x) 均匀），这也是有些图用对数轴的原因。
BenoitEssiambre查看英文原文
Benford's law, Zipf's law and other power laws tend to fascinate but they are a manifestation of something really simple. It's how things behave when they are evenly distributed over multiplicative operations instead of additive operations. By this I mean things are as likely to be x% bigger than x% smaller instead of the more common definition of evenly distributed (things x unit larger are as likely as x unit smaller). Being evenly distributed over multiplication is natural for many measures. I think it is considered the most even, "maximum entropy" distribution for things that can't be negative. The most even, maximum entropy distribution for an unbounded (-infinity to +infinity) prior is an unbounded uniform distribution but you can't apply it, for example, in the case of the size of objects since it would imply that negative sizes are as likely as positive sizes (and objects don't have negative sizes). In general position parameters tend to be uniformly distributed over addition/subtraction whereas scale parameters tend to be evenly distributed over multiplication (uniform over log(x)). It's the reason some graphs use log axis. For Bayesians, it's a fairly straightforwards concept. See also improper priors (https://en.wikipedia.org/wiki/Prior_probability#Improper_priors)
道琼斯指数例子直观展示：对数增长下首位 1 出现时间最长。
《纽约时报》引用 Nigrini 的道琼斯指数例子：若道指为 1000，首位是 1；要变成首位 2 需涨到 2000，即 100% 增长。按每年约 20% 增长，要五年才从 1 到 2。若从首位 5 开始，从 5000 到 6000 只需 20% 增长，一年就达到；到 9000 后，只需 11% 增长、约七个月就到 10000，首位又回到 1。之后又要翻倍到 20000 才到首位 2。所以在对数式增长序列中，数字 1 在每一步都占主导。
ronenlh查看英文原文
Quoted from The New York Times, Tuesday, August 4, 1998 http://www.rexswain.com/benford.html (linked in Wikipedia): To illustrate Benford's Law, Dr. Mark J. Nigrini offered this example: "If we think of the Dow Jones stock average as 1,000, our first digit would be 1. "To get to a Dow Jones average with a first digit of 2, the average must increase to 2,000, and getting from 1,000 to 2,000 is a 100 percent increase. "Let's say that the Dow goes up at a rate of about 20 percent a year. That means that it would take five years to get from 1 to 2 as a first digit. "But suppose we start with a first digit 5. It only requires a 20 percent increase to get from 5,000 to 6,000, and that is achieved in one year. "When the Dow reaches 9,000, it takes only an 11 percent increase and just seven months to reach the 10,000 mark, which starts with the number 1. At that point you start over with the first digit a 1, once again. Once again, you must double the number -- 10,000 -- to 20,000 before reaching 2 as the first digit. "As you can see, the number 1 predominates at every step of the progression, as it does in logarithmic sequences."
尝试把 Benford 洞察用于低功耗加法器设计，但内存访问熵太高，收益有限。
我在硕士论文里用了这个想法——数字常在指数空间均匀分布。由于计算机加法等运算常涉及至少一个小数，加法器的开关活动应集中在低位。于是计划让低位更慢但更省电，高位更快但功耗更高，在满足最坏情况时钟目标的同时降低平均功耗。可惜内存操作看起来接近纯熵，且足够常见，收益不值得设计投入。
Symmetry查看英文原文
I used this, well the idea that numbers often tend to be distributed evenly in exponential space, in my MEng thesis. The idea was that because so often in computers our additions or other arithmetic operations involve at least one small number then most of the switching activity in an adder should be concentrated in the lower order bits. So the plan was to make the low order bits of the adder slower and more power efficient. Make the high order bits faster and less power efficient. Meet the same overall clock rate target given by the worst case but use less power in the average case. Sadly memory operations, which look a lot like pure entropy, were sufficiently common that the gains just weren't worth the design effort.
提醒 Benford 定律用于选举舞弊检测有争议，不能当作强证据。
Benford 定律作为选举舞弊检测技术受到广泛批评。一位政治学家写过一篇描述变体的论文，用 2006 年墨西哥选举数据仅作说明。批评者广泛用该论文作为舞弊证据，但他们无法用其他方式证明舞弊。论文从未打算作为强证明，主要是探索性的。
corpMaverick查看英文原文
Benford's Law has been widely criticized as a technique to detect election fraud. A political scientist wrote a paper describing a variation. http://www-personal.umich.edu/~wmebane/pm06.pdf He used some data from the 2006 election in Mexico just to illustrate the proposed technique. The paper has been used extensively by critics of the election as a proof of fraud. But they haven't been able to prove fraud any other way. The paper was never meant to be used as a strong proof of anything, it was mostly exploratory.
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