REPLY Getting a Good Problem Statement (SD6554)

[SDMAIL Ken Lloyd]

Posted by  "Ken Lloyd" <kalloyd at wattsys.com>

Hello everyone.  

Wonderful discussion. I believe this is my first posting to your group.

David Rees makes some good points.  My contribution is a small attempt at
refinement on these points. In my work, there is a critical distinction
between data and models.  Data represents "known" values from measurement.
While data is known, it may contain measurement errors and noise.  Compare
this with models.  Models represents a temporal procession of a-priori
information believed true, but unknown or uncertain.  This at the start,
this is based on two (assumed faulty) premises 1) that the data = model +
error (d = m + e) and that the data is only a function of the model d =
f(m). Since we are certain that models contain modeling errors (referring
minimally to Godelian incompleteness or inconsistency), we need to separate
measurement errors from modeling errors.  After this, the model may be
refined and validated.

Using a hybridization of network graphs to represent models, it is possible
to create knowledge models.  The subtle distinction (critical IMO) is that
in dealing with unknowns, even those we believe true, is that we move from
the world of trajectories and determinism, to the world of ensembles,
probabilities and distributions.   

In reference to Rees statement that everything is connected to everything
else, there is a distinction between geodesic connection length (an
enumeration) and the L2 Norm length (a Euclidean distance).  The shorter L2
length implying a tighter degree of (possibly non-linear) coupling distance.
Furthermore there exist "small-world" network paths that can be leveraged
(see Newman, Watts, Barabasi).

Therefore I suggest that the boundary conditions are not monolithic "in or
out", but result from network graph clustering phenomena related to the
previous paragraph.  This yields a probabilistic distribution of model
ensembles.  The only challenge left is to decide which models under
investigation are "most likely".  This is derived from a measurable
probability on a distribution of probabilistic models - again, ensembles,
not trajectories.

This can be accomplished by relating the distributions of known data, to the
distributions of unknown models, through cyclic refinement of Bayesian
forward and inverse processes.  For those unfamiliar, see J. Scales,
"Geophysical Inverse Theory" or works by A. Tarantola or P. Mosterman. This
yields the likeyhood measurement useful for model "tolerancing".

I hope that this makes some sense to those who predominately think in
deterministic patterns.  I propose this aligns better with statistical
mechanics, (graph) thermodynamics, and physics (both classical and quantum).

Ken

=============================
Kenneth A. Lloyd
CEO and Director of Systems Science
Watt Systems Technologies Inc.
Albuquerque, NM USA
Posted by  "Ken Lloyd" <kalloyd at wattsys.com>
posting date  Sun, 2 Sep 2007 08:37:41 -0600