🚀 go-pugleaf

In article <thompson.714338397@kiyotaki.econ.umn.edu>
thompson@atlas.socsci.umn.edu (T. Scott Thompson) writes:
>To put out a concrete example that illustrates the nature of my
>qualms, consider the following.  Suppose my data consist of a single
>number, say X.  Suppose that I am given two pieces of information
>about X (but not X itself):
>
>           (1)  I can observe Y := X**2
>           (2)  I can observe Z := {  1 if X > 0
>                                   { -1    otherwise

It seems to me that the concept of degrees of freedom is closely related to
that of dimension.  Given that, i would say that X and Y have one degree of
freedom, and Z has *no* degrees of freedom, because it's discrete.

On the other hand, it's often more useful to talk about bits of information.
If X is symmetrically distributed, Z gives you one bit of information.  If we
observe Y to some accuracy, we can divide that accuracy by dY/dX to get the
accuracy of X.  Then we compare that with our previous accuracy of X to see
how much information we have gained.  Believe it or not, this actually works
out in the end.

--
Joe Keane, amateur mathematician
jgk@osc.com (uunet!amdcad!osc!jgk)