🚀 go-pugleaf

Andrew.Robinson@anu.edu.au writes:

>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]
>>
>>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

>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?

Well first, I don't have to estimate X since I can calculate it
exactly from Y and Z.  Secondly, I agree that the single data point X
contains the exact same information as the vector (Y,Z).  I will even
agree (although I am not sure that this is so obvious) that the number
of "pieces of information" in X (hence in (Y,Z) is one.  After all,
the latter could simply be a normalization of units.

The problem is to relate this to the usual degrees of freedom
accounting which seems to require that independent "pieces of
information" add up in some way.  For example, we seem to think of Y
and Z as independent pieces of information because knowledge of one is
totally uninformative about the other.  According to the adding up
rule, then, (Y,Z) should be 2 pieces of information, not one.  Clearly
we agree that this is not so.  The question is "why not?"

I think that the definition of "independent pieces of information"
that is implicitly used in the informal interpretation of "degrees of
freedom" does not satisfy an adding up rule, and thus fails to provide
a good interpretation for what "degrees of freedom" really represent.
Assuming they represent anything.  I am skeptical about this.
-- 
T. Scott Thompson              email:  thompson@atlas.socsci.umn.edu
Department of Economics        phone:  (612) 625-0119
University of Minnesota        fax:    (612) 624-0209
--
T. Scott Thompson              email:  thompson@atlas.socsci.umn.edu
Department of Economics        phone:  (612) 625-0119
University of Minnesota        fax:    (612) 624-0209