Tue Jul 10 12:34:06 EDT 2012

Understanding Density

Some things have popped up lately that are all related to the problem
of density: a bunch of discrete things interact in an intractable way,
but when this exact representation is integrated or averaged it can
represent a very nice model.  Usually when 1. the granularity is fine
enough and 2. the entropy is high enough, cross terms drop out and
there is a (information-reducing) "morphism" between operations in the
discrete domain and operations in the density domain.

I wonder, how to qualify "enough" for the properties of granularity
and entropy.

Some concrete examples:

- The use of money in the form of discrete transactions in time.  To
  map this to a density model (money stream) makes it easier to think

- Sigma/Delta modulators: all operations are discrete both in time and
  space: fixed sampling frequency and 0/1 bit levels, but the part of
  the signal we're interested in is a forgetful time integral = low
  pass filter.

- Sound is a phenomenon that can be modeled as fluctuations of
  densities of particle positions and velocities.

All density representations seem to be based on some form of smoothing
= convolution of a discrete distribution with a smooth, local kernel
like, e.g. a bell curve.