tag:blogger.com,1999:blog-6883415296937284014.post3978635556258655696..comments2024-01-08T02:16:57.647-08:00Comments on Religion, Sets, and Politics: Benford’s Law: Human Intuition, Randomness and FraudJoshuahttp://www.blogger.com/profile/00637936588223855248noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-6883415296937284014.post-64688570516825292532009-11-04T04:09:36.212-08:002009-11-04T04:09:36.212-08:00This comment has been removed by a blog administrator.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6883415296937284014.post-2713500230458673262009-10-31T17:39:04.003-07:002009-10-31T17:39:04.003-07:00Yes,
Although one can construct functions that de...Yes,<br /><br />Although one can construct functions that deliberately don't map well to S^1. But you have to work at it and they generally fail in an obvious fashion.Joshuahttps://www.blogger.com/profile/00637936588223855248noreply@blogger.comtag:blogger.com,1999:blog-6883415296937284014.post-89622287173931302052009-10-31T17:35:59.124-07:002009-10-31T17:35:59.124-07:00...so I guess then what's going on is that thi......so I guess then what's going on is that this works just as well with other functions, except that if we use any function other than a logarithm, we won't be able to draw any conclusions about the first digit?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6883415296937284014.post-24149135997357492792009-10-31T16:30:27.730-07:002009-10-31T16:30:27.730-07:00Shalmo, I'm sorry I just watched the first fiv...Shalmo, I'm sorry I just watched the first five minutes of that video. What does it have to do with the original post?<br /><br />Etienne,<br /><br />Yes, and in general you will expect that well behaved functions will have a uniform distribution. Thus, for example if I took sqrt(x) mod 1 as my function it turns out for natural data this will have a uniform distribution. What we are seeing in this particular case is a general example of that sort of behavior. <br /><br />The only specific reason we notice this in the case of Benford's law is because of the natural convenience of base 10 and the well behaved nature of the logarithm.Joshua Zelinskyhttps://www.blogger.com/profile/07716458882451866836noreply@blogger.comtag:blogger.com,1999:blog-6883415296937284014.post-59595748279499623322009-10-31T10:58:22.509-07:002009-10-31T10:58:22.509-07:00I'm still a bit confused... why do we expect f...I'm still a bit confused... why do we expect f(x) to be uniformly distributed between 0 and 1? Can't I pick any different f(x) : R -> S^1, repeat your argument, and derive a different Law?Etienne Vougahttps://www.blogger.com/profile/09782192943552381461noreply@blogger.comtag:blogger.com,1999:blog-6883415296937284014.post-63660526177245986742009-10-31T05:23:07.369-07:002009-10-31T05:23:07.369-07:00http://www.youtube.com/watch?v=69xYD8oWxYg&fea...http://www.youtube.com/watch?v=69xYD8oWxYg&feature=subShalmohttp://www.youtube.com/watch?v=69xYD8oWxYg&feature=subnoreply@blogger.comtag:blogger.com,1999:blog-6883415296937284014.post-37484359739581934692009-10-29T20:27:21.845-07:002009-10-29T20:27:21.845-07:00Here's another way to think about Benford'...Here's another way to think about Benford's Law. Suppose that, in some entry in a table of values, we find an entry that begins with the digit 1. By what fraction of the total value will we have to increase it in order to change that digit? (No peaking at the remaining digits of the number!) It could be anything up to 100%; we might have to double the original value to flip that leading digit to a 2. But if the leading digit is a 9, we will at most have to increase the total value by about 11% in order to change the leading digit to a 1.<br /><br />I suspect there is some fancy way to make the logarithmic values drop out of this consideration, but I'm too sleepy right now to work it out. Over to you!Unknownhttps://www.blogger.com/profile/04793417120098322488noreply@blogger.comtag:blogger.com,1999:blog-6883415296937284014.post-87655060102628408892009-10-29T19:57:55.870-07:002009-10-29T19:57:55.870-07:00This is probably the clearest way to explain Benfo...This is probably the clearest way to explain Benford's law. <br /><br />One of the nicest alternative ways has to do with scale invariance. If, in a given set of quantities, there is <i>any</i> distribution of first non-zero digits that is not merely an artifact of scale, it must be independent of the units in which the measurement is expressed. And if we take (say) the heights of buildings expressed in feet, in yards, etc., we find that they satisfy Benford's law regardless of the choice of units.<br /><br />There are also some fascinating applications of Benford's Law in <a href="http://primes.utm.edu/glossary/xpage/BenfordsLaw.html" rel="nofollow">probing the distribution of primes</a>.Unknownhttps://www.blogger.com/profile/04793417120098322488noreply@blogger.com