Bending Benford, Part 1

Appreciating Mathematical Beauty Volume 3

"Democracy is based upon the conviction that there are extraordinary possibilities in ordinary people." -Harry Emerson Fosdick

And I believe that [that there are extraordinary possibilities in ordinary people]. I believe that confederations and democracies of power are important for other reasons, some of which are simply game theoretic in nature. I also believe that equilibria break and that challenges the human spirit and species to rise to a higher form. We can't just wish for goodness the same way we cannot wish for bandits not to ride down from the hills and take the crops. We must repeatedly address the incentive structures, and our investment in the framework is a misunderstood technology.

The Right and Wrong Definitions of Technology

The other day, the 2000 Mules documentary came out (no, I haven't watched it, yet) and there is renewed interest in examining election fraud. I honestly do not know whether or not I'm the first to make all the observations I present here, but after a few dozen conversations with mathematicians, I think that some of my techniques are original. I was working on a paper prior to being sucked into vaccine statistics work. Really, I don't care whether this gets "published" beyond this substack. I just want to get the idea out there so that somebody (anybody) can make use of it.

Apologies in advance for the quick write-up. I have far too much to do to make this thorough and complete.

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A Statistical Warning

"Absolute certainty is a privilege of uneducated minds-and fanatics. It is, for scientific folk, an unattainable ideal." -Cassius J. Keyser

Context is everything.

Before I go any further, I want to make this statement about statistics: there is no such thing as a correct or incorrect "method" in statistics—there is only the judgment of an observer as to how meaningfully the results match reality. I say this in order to pre-empt all conversation about whether or not Benfor's law (BL) "can" be applied to election results. There are (at least) two crappy arguments.

1. BL always provides a test for the sanctity of election results. (Technically false, though it often provides us with interesting observations and starting points.)

2. BL cannot be used as a test for the sanctity of election results. (Sort of technically true, but only in a vacuous sense that is horrifically misleading.)

Always look at statistical results with respect to the big picture. Always. Numbers don't mean anything in a vacuum. Of course statistical tests can be applied. And they certainly narrow down interesting localities for investigation. But any good investigation involves further examination of the facts. Context is everything.

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Benford's Law

"Most people use statistics like a drunk man uses a lamppost; more for support than illumination." -Andrew Lang

Benford's law became famous (or "famous" relative to what can possibly happen with clever mathematical tricks) in the aftermath of the 2020 U.S. election. Numerous investigators noted observations of voting patterns that seemed to defy expectations.

First, let's talk about what BL says, and why. According to Wikipedia,

Benford's law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small.[1] In sets that obey the law, the number 1 appears as the leading significant digit about 30 % of the time, while 9 appears as the leading significant digit less than 5 % of the time. If the digits were distributed uniformly, they would each occur about 11.1 % of the time.[2] Benford's law also makes predictions about the distribution of second digits, third digits, digit combinations, and so on.

The unsophisticated version of BL is that in distributions that have a particular property called scale invariance (okay, that's not entirely unsophisticated), we see a non-uniform pattern among the leftmost digits of the numerals that represent the data. This isn't really surprising to a statistician, but it can be befuddling to nearly everyone upon a first encounter. Here is an example: exactly 15 out of 50 consecutive powers of 2 have a leftmost digit of 1:

1, 16, 128, 1024, 16384, 131072, 1048576,...

In fact, this is true for almost any 50 consecutive powers of 2, though once in a long while, there will be 16 with the leftmost digit of 1 instead of 15. In this exponential distribution (exponential distributions are nearly always the kind to which we apply BL), we can begin to form an intuition as to why BL works: A power of 2 is [approximately] 0.301 powers of 10.

Or, written differently, the base 10 logarithm (the "anti-exponential" given the inverse function relationship) is

There are tons of distributions to which BL applies precisely because there are tons of exponential distributions. The two most common examples are populations and money. These do not even have to be growing exponentially at any one particular moment to conform relatively well to BL! The 26% of American states with populations that have a leftmost digit of 1 is fairly typical of such data sets, though 30.1% would be the expectation.

The nice thing about Benford's law is that it doesn't take somebody with a PhD to apply. Here are some BL results run by one of my college debate partners (who is more quantitative than 95% or more of college graduates, and uses data in his work, but has not done the work of a theoretician).

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Benford's Law and Election Results

Here is where things get controversial. There were some anomalous BL signals during the 2020 election. These signals were met with overly simplistic arguments from both sides (and a tiny few really good arguments). I have pre-empted these, so we can proceed in a sane manner.

Before we go very far, I want to point out this 2018 blog post about 2016 election results in which counties in Wisconsin that conformed poorly to BL just happened to be those that used Dominion voting machines.

This is already enough to establish context (meaning for further investigation), though this context may or may not extend to all circumstances. Here is a github repository of BL test results that I saved during the election controversy. There is discussion here about the results. Some of these show Biden voting not conforming to BL very well. For example:

But the arguments begin here with observations that the sizes of districts throw off these distributions. And they do! In particular, they should throw off the majority vote the most. For instance, if each voting district has between 400 and 1,200 residents, then there is not a full range (order of magnitude) of population. If the result were the most serious kind of landslide, we would not even see 2 or 3 as leftmost digits, and we would not see 1 often enough.

So, do we just give up applying BL in these situations?

Not if we are clever, and that gets us to the point of this story…

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Widening the Application of Benford's Law: Other Number Bases

There are actually a whole bunch of ways to make Benford's law a more incisive tool for forensics work. One simple one is to test the leftmost digit rule of a set of numbers after converting them to another number base, like 8, 9, 11, or 12. There may be cases in which the criminals know just enough statistics to fake the base 10 distribution, but not enough to generate their dataset as an exponential distribution plus randomness [of some kind], which is what would be required for Benford's law to be thwarted when numbers get numerically expressed in any ordinary base number system.

The "leftmost digit(s)" stats are a little different, however. For instance, while 30.1% of population data should begin with (leftmost digit) a 1 when expressed in base 10, we should expect for that proportion to be 31.5% when the same numbers are expressed in base 9.

That's cool and all, but we can go further…

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Widening the Application of Benford's Law: Funny Number Bases

Here is where we get freaky with Benford's law. Ready?

There is nothing about BL that requires it to be applied to numbers expressed in base 10. In fact, we don't even need to use an integer base! We can use fractional bases, and even bases less than 2, which allows for us to examine the results in districts of constrained size. I know that this is where I lose the general reader, but the mathematicians/statisticians reading here will get it. Here is a paper that I was writing up over a year ago, but quit because duty called with respect to vaccine research:

(Oops, much of the paper disappeared in the newsletter. Fixing now…)

I think I have to type words before the sections to get them all to show up…

Typing more words…

Some day in the future I hope to come back to this work and finish the paper I started.

Sorry for the typos and unfinished nature of the paper. I believe that this is enough for the smart people working on election results to understand the BL filter. Some people apply BL to one or two digits only, but this filter may require examination of a dozen or more digits. But that's not really a big deal to a good programmer.

Have fun with it. And if you know somebody doing statistical work on election results, please forward this.