🚀 go-pugleaf

In article <thompson.714338397@kiyotaki.econ.umn.edu>
thompson@atlas.socsci.umn.edu (T. Scott Thompson) writes:
>hougen@uirvlh.csl.uiuc.edu (Darrell Roy Hougen) writes:
>
[deletia]
>
>I have never been very satisfied with this notion of "degrees of
>freedom" being equivalent to "pieces of independent information".  The
>correspondence works well enough for linear models and independent
>observations, as in the example given, but there are alternative
>interpretations (in terms of the dimension of orthogonal subspaces)
>that do just as well in this setting.
>
>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
>
>How many "pieces of information" do I now have?  Clearly I can recover
>X from Y and Z by the formula X = Z * <positive square root of Y>, but
>neither Y nor Z alone suffices (excluding the special case where 
>X = Y = 0).  Thus Y and Z must each "use up" (at least) one degree of
>freedom in the usual accounting.  Thus I must conclude that
>
>       <degrees of freedom in X> .GE. 1 + 1 = 2.
>
>For obvious reasons I do not find this to be an acceptable answer!
>What do you think?  What am I missing?  Is there any meaningful
>interpretation to "degrees of freedom" in a general theory of
>inference?

I'm going to go _way_ out on a limb here and say that I think that you have
_one_ piece of infomation.  If you don't have X then your data doesn't consist
of a single number, it consists of a single observation from which you have
measured a two variable vector - [Y,Z].  You know  that there is an X and you
want to use Y and Z to estimate it using a predetermined model, which you have
specified.  This seems to me to be a non-straightforward case of a multivariate
regression, with two dependent variables and one observation.

Comments?

regardless,
Andrew

now hit d