Stats on volcanic eruptions show pattern called Benford's Law

Benford's law is not odd if you are believer in the Bayesian interpretation of probability theory.

The shape of Benford's distribution actually follows the shape of a maximum entropy, most uninformative distributions (the “flatest distribution” in an information theoretic way) for a _magnitude_ value (a value that can only be positive).

For example: sizes, lengths and volumes. These units can't go in the negative. Alternatively, positions can go in the negative, the ignorance distribution for position is flat horizontal line and numbers representing a position generally do not follow benford's law.

It is not that unintuitive. Take street lengths. Assuming that they follow Benford's law (a log prior) simply means that, for any length of street L, if you pick another random street, you are as likely to pick a street within the length range (L/2 to L) than (L to 2*L).

From the original street length, to get a street twice as short, you need to subtract much less than you would have to add to get a street twice as long. That is why this distribution is not flat. Or rather it is flat on the multiplication or division operation, not on addition or subtraction.

If instead you'd assume that a street X meter longer is as likely as a street X meter shorter you would end up with impossible probabilities. For example, when picking a random street relative to a street of 1 km, a (1 to 3) kilometer street would be as likely as a (-1 to 1) km street??? If you assumed probabilities were equal for all lengths between 0 and infinity that would mean you think there are likely as many streets measuring a tredecillion billion km long as there are street 5 km long. This is simply not how things are sized in the universe. Smaller things are in greater numbers. Log priors are one of these areas where the math predicts the universe logically and the universe is mirrored by the math beautifully iff you do your calculations properly (Using Bayesianity).

Of course they do. Benford's Law is just an artifact of how counting works.

Let's say we are measuring widget production. Right now we're making 1 widget per period. Each period we increase production by 10%. After a random number of periods, what do we expect the MSD (most significant or left-most digit) to be? After a random (large) number of periods, if we add up how frequently different numbers show up as the MSD, what will the distribution be?

Well, let's think about it. It takes 8 periods before our production number gets over two (and thus stops having an MSD of 1), and another 4 periods before our production number gets over three (and thus stops having an MSD of 2). If we keep going, the results look like this:

  1 -> 8 periods
  2 -> 4 periods
  3 -> 3 periods
  4 -> 2 periods
  5 -> 2 periods
  6 -> 2 periods
  7 -> 1 periods
  8 -> 1 periods
  9 -> 2 periods
  1 -> 7 periods
  2 -> 4 periods
  3 -> 3 periods
  4 -> 3 periods

And so on. If we were to stop there, we have counted 41 periods. The most common MSD is 1 (taking place 36% of the time), followed by 2 (19%), 3 (14%), 4, and so on. Benford's Law! Of course, you might say, that just shows up because I chose to stop there.

But actually, it works to a greater or lesser extent, pretty much regardless of when we stop (or where we start). It works regardless of the units being used. It also works if we're adding a flat amount instead of a percentage increase, or if we're adding a random amount, or if the number is not monotonically increasing. It also works in other bases. (In fact, thinking about the example of binary may be illuminating. The MSD is always 1 for any number > than 0.)

It also works if we're generating random samples between 1 and some BIG_NUM. If you generate 1000 numbers between 1 and, say, 2^24, you might find that that the MSD of 48% of those samples is 1, while no other digit is found as the MSD in more than 7% of samples. Pick a different BIG_NUM, and you'll see a different distibution. 2^20 gives you a much flatter distribution; in the test I just ran I saw 16% with a MSD of 1, versus 9-12% for all other digits. But the point is, lots of potential BIG_NUMs produce samples which skew heavily towards having an MSD of 1, and thus a random BIG_NUM has a highly skewed expected distribution.

And so on. :) It's actually kind of fun to play with. But at the end of the day, this is just a product of using a positional number system.

An excellent visual tour of Benford's Law: http://www.datagenetics.com/blog/march52012/index.html

My understanding of this law is that any collection of real-world data/metrics should follow this distribution pattern. There may be exceptions in which there are other factors influencing the data, but most should follow this "law". In that case, I don't see what is surprising about volcanic eruptions following this law. I would have assumed it already did (well, I suppose you have to already know about this law first to assume that).

Simple explanation for Benford's Law: When numbers expand out into an additional digit through doublings, a "1" is always present, "2" is usually present, and so on down the line. The law is present in proportion to the degree that the sample set possesses logarithmic distribution.

The distribution doesn't have to be limited to doublings. The law also applies to triplings, quadruplings, halvings, etc. The key is logarithmic/exponential/geometric distribution. Try it for yourself on the calculator and see how rarely 9s come up as the first digit for every x^n.

A good real world example would be the distribution of frequencies of musical pitch in equal temperament. In this case, the exponential multiplier (x) for x^n is 21/12, or about 1.059. Here's (http://en.wikipedia.org/wiki/Piano_key_frequencies) a list of the frequencies; notice how the first digits have tons of ones and almost no nines.

Benford's Law seems pretty obvious once you think about it. Though I love its application to "forensic accounting".

Do you think it could be used to judge the "truthiness" of media websites online? I don't think it works with rounded numbers that are more likely to be found in the news.

As a side note, the main photo in the article is stunning! I need to find a desktop-size version.