Subject: Re: Real Subsets with the 0-1 Intersection Pro
From: wollan@mayo.edu (Peter Wollan)
Date: 15 Nov 1996 21:49:01 GMT
In article frt@senator-bedfellow.MIT.EDU, lones@lones.mit.edu (Lones A Smith) writes:
>This really concerns probability theory.
>
>Let S(x,c) be a closed Borel subset of [0,1], for any x in [0,1] & real c.
>
>Suppose S has the "0-1 intersection property": For any c1 and c2, and for
>all x1 <> x2, S(x1,c1) and S(x2,c2) have either zero or one point in common.
>
>CLAIM: {x in [0,1]|union of S(x,c) over all real c has measure >0} is countable
>
What about S(x,c) = {c, if c is in [0,1], x else}?
Then for any x, union over c of S(x,c) = [0,1], but each S(x,c) is a single point so
must satisfy the 0-1 intersection prop.
Am I missing something?
Peter Wollan
wollan@mayo.edu
Subject: Re: Attitude responses
From: Warren
Date: 15 Nov 1996 21:56:02 GMT
Nick,
Unless you have reason to believe the categories are equally spaced, I
would worry a little about numbering just as you said. And the general
test of association may not tell you what you need to know...how do the
proportions relate to each other on an ordinal scale.
You have a couple of options, I would think.
In PROC FREQ in SAS, you can select scores different from 1,2,3,4 (CMH),
but you don't seem to have any strong belief in how much distance there
is between categories.
You could use ridits which assign scores that take into account the
number in each category...but I think we are back to numbering the
categories if I remember the ridit procedure...it does some kind of
midranking as I recall.
You could use a straight ranking
procedure...like Kruskal-Wallis-Mann-Whitney.
Another technique for analyzing these data (your example fits the classic
mold very well) is something called a "proportional odds model." It
would be the best choice if the assumptions are met. SAS will do this
type of model using PROC LOGISTIC and test the "PO" assumption. If the
proportional odds assumption isn't reasonable, you can do a "generalized
logits" model using PROC CATMOD...you could compare the "not at all"
group to all the rest. Agresti would be a good place to look for
references, but Agresti isn't a good "elementary" text...I haven't seen
his new book but it might be a little more "elementary". All of these
would require a little reading on your part concerning the assumptions.
n.w.nelson@education.leeds.ac.uk (nick nelson) wrote:
>Say I have responses on an attitude scale
>
>eg Do you like this? lots / some / a little / not at all
>
>and two groups eg men and women. What is the best way to
>establish whethere the two groups differ significantly?
>
>On approach I have seen involves numbering the responses 4,3,2,1
>and working with the means, but this seems dubious due to the
>non-interval nature of the scale.
>
>Alternatively you could cast the reponses in a 4x2 table and do
>a chi2 on it, but this ignores the order information altogether.
>
>Is there a middle path?
>
>Nick.
Subject: Re: Need help on a upper bound question
From: "Raymond V. Liedka"
Date: Fri, 15 Nov 1996 16:00:06 -0700
On 14 Nov 1996, Bob Lee wrote:
> Hi I have a question that I am having trouble with. I'd appreciate any help.
>
> Suppose the average family income of an area is $10,000.
>
> a) Find and upper bound for the percentage of families with incomes
> over $50,000.
> b) Find a better upper bound if it is known that the standard
> deviation of incomes is $8,000.
>
> I assume that some kind of distribution must be assumed.
>
I assume that your instructor expects you to do your own homework, or is
this a take-home test? You will learn more if you really try to figure
things out rather than try and get others to do your work!
Raymond V. Liedka
Department of Sociology
University of New Mexico
Subject: Re: Occam's razor & WDB2T [was Decidability question]
From: radford@cs.toronto.edu (Radford Neal)
Date: 16 Nov 96 17:31:49 GMT
>< Radford Neal:
><
>< One often sees people using priors that are such that the
>< effective complexity of the model increases as the amount of
>< data increases. This makes no sense - it amounts to using a
>< prior that one knows is going to be contradicted by future
>< data.
Neil Nelson wrote:
>... Of course the difficulty here is the
>determination of the prior probabilities and algorithmic
>relation, for which our only effective recourse is an analysis
>of the previously and currently available data. This implies
>that our prior probabilities and algorithm may change depending
>on any increase in the available data; or more simply, we would
>not want to hold to our previous judgment if new information
>indicated we were previously in error.
This is not the case for a full Bayesian analysis, since the prior
decided on before any data is collected will implicitly contain all
the revisions of judgement that would be prompted by any possible data
set.
In practice, a Bayesian is likely to use a model and prior that do not
contain certain possibilities that seem very unlikely at first, simply
because formalising all these possibilities is too much work. If the
actual data indicate that these possibilities need to be considered,
then the Bayesian might revise the prior and model, perhaps adopting a
more complex one.
However, I think that this scenario has little to do with the usual
reasons why people think that you can't use complex models with small
datasets. The usual reasons are not compatible with a Bayesian viewpoint.
Radford Neal
----------------------------------------------------------------------------
Radford M. Neal radford@cs.utoronto.ca
Dept. of Statistics and Dept. of Computer Science radford@utstat.utoronto.ca
University of Toronto http://www.cs.utoronto.ca/~radford
----------------------------------------------------------------------------
Subject: Variance of convex function convex ??
From: ahmed shabbir
Date: Sat, 16 Nov 1996 14:58:45 -0600
A function f(x,w), where w is a random variable and x is deterministic,
is convex in x for fixed w, and is also convex in w for fixed x. We know
that the expectation: E[f(x,w)] is then convex in x. Is the variance:
Var[f(x,w)] convex in x ?
Any ideas, suggestions, references will be greatly appreciated.
SHABBIR Ahmed
