Re: Benford's Law

Re: Benford's Law

[Robert J. MacG. Dawson]

mjhealy@ece.unm.edu wrote:

> Does anybody on this list (including you, John) know of a connection
> between Benford's Law and any work in category theory?  I would really
> like to hear about it if so.

	I doubt if there is much of one.

	I suppose _some_ sort of connection might be made through the concept
of "invariance" -  Benford's law holds for distributions that are wide
enough not to have a "natural scale".  If you cannot give an approximate
answer to "how big is a (river/piece of string/data file/bank deposit)?"
  then the distribution of first digits in (say) centimeters should
[waving hands hard] be the same as that in furlongs or wavelengths of
green light; and from that property Benford's law follows.

	On the other hand, humans are approximately of a height, to the point
that the foot, hand, cubit, fathom, etc. can be used as rough units in
their natural form.  Thus there is a natural scale for human heights,
and we are not surprised that almost all human heights in meters have a
first digit 1 and very few do in inches.

	-Robert

Benford's law

[John Baez]

Mike writes:

>I haven't yet looked at Minhyong Kim's work, and I don't know how this
>fits in with number theory or categories, but a friend is encouraging me
>to go to the following conference on Benford's Law:
>http://www.ece.unm.edu/benford .
>
>Does anybody on this list (including you, John) know of a connection
>between Benford's Law and any work in category theory?  I would really
>like to hear about it if so.

I don't know any interesting connection between this law and category
theory or number theory.

I didn't know it was called "Benford's law", but I knew the idea:
if you take a table of widely spread numbers (say the gross national
products of nations, or the incomes of Americans), often about

log 2  ~  30%

will have 1 as their first digit, about

log 3 - log 2  ~  17%

will have 2 as their first digit, and so on.

It's easy to derive this law from the assumption that the data is
distributed in an approximately scale-invariant way within a certain
range.  (That is, the percentage of numbers in your table between
X and cX is about equal to the percentage between Y and cY, for c not
too big, and X and Y within some large but finite range.  Or: the
logarithms of the numbers are approximately uniformly distributed over
some interval.)

So, the mystery of Benford's law reduces to the mystery of this
fact: in practice, widely spread numbers are often distributed
in an approximately scale-invariant way, within some range.

(Perhaps some people find Benford's law mysterious because it's
impossible for a probability distribution to be *perfectly*
scale-invariant.  But that's a red herring.  It's enough to have
approximate scale-invariance within some range, for example a couple
powers of 10.)

Why is approximate scale-invariance so common?  People have
written books on this!  Here's a nice one:

Manfred Schroeder, Chaos, Fractals, Power Laws, W. H. Freeman, 1992.

Or, for starters:

http://en.wikipedia.org/wiki/Power_law

I would rather go to a conference on power laws than a conference
on Benford's law, which seems like just a spinoff.

Best,
jb

Benford's Law

[mjhealy]

This message was stimulated by John Baez's week257 which, though
interesting as usual, has one item of special interest to me at this time.
 I haven't yet looked at Minhyong Kim's work, and I don't know how this
fits in with number theory or categories, but a friend is encouraging me
to go to the following conference on Benford's Law: 
http://www.ece.unm.edu/benford .

Does anybody on this list (including you, John) know of a connection
between Benford's Law and any work in category theory?  I would really
like to hear about it if so.

Thanks,
Mike