* Benford's Law
@ 2007-10-16 17:07 mjhealy
0 siblings, 0 replies; 3+ messages in thread
From: mjhealy @ 2007-10-16 17:07 UTC (permalink / raw)
To: categories
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
^ permalink raw reply [flat|nested] 3+ messages in thread
* Re: Benford's Law
@ 2007-10-17 12:08 Robert J. MacG. Dawson
0 siblings, 0 replies; 3+ messages in thread
From: Robert J. MacG. Dawson @ 2007-10-17 12:08 UTC (permalink / raw)
To: categories
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
^ permalink raw reply [flat|nested] 3+ messages in thread
* Benford's law
@ 2007-10-17 2:51 John Baez
0 siblings, 0 replies; 3+ messages in thread
From: John Baez @ 2007-10-17 2:51 UTC (permalink / raw)
To: categories
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
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