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I need a reference for the Metropolis algorithm for finding
Markov transition matrices fro a given limiting vector.
The algorithm works as follows.
Let $p = (\p_{1}, ...)$  be the given limiting vector.
Let $Q=[q_{ij}]$ be any symmetric transition matrix.
Define $a_{ij}$ to be 1 if $p_{j}/p_{i} >1$ and
to be $p_{j}/p_{i}$ if $p_{j}/p_{i} <1$.
Let $P=[p_{ij}]$ have entries $p_{ij}=a_{ij}q_{ij}$ if
$i \neq j$ and $p_{ii}=1-\sum{j: j \neq i}p_{ij}$.
Then P satisfies the desired condition.
Reference please??
( I also have an algorithm due to Barker. Reference? )
-- 
                   Myron Hlynka
                   Dept. of Math. & Stat.
                   University of Windsor
                   Windsor, Ontario, Canada

Dr Michael Mattes wrote:
> 
> Christopher McKinstry (chris@clickable.com) wrote:
> > I believe probability is relativistic.
> 
> >  Here’s a simple experiment you can try at home to demonstrate
> > probability changing with speed:
> 
> > 1)  Let "V" the maximum number of digits you can write per second.
> > 2)  In "T" seconds, write down a random number in decimal form.
> 
> > Now, the maximum number you could have written is "9" repeated VT times.
> > This is your Reality Radius "RR". The fact you even have one proves
> > probability behaves relativistically.
> 
> > Consider, as V decreases, there comes a point at which the only number
> > you have time to write is "1".
Why do you think it takes more time to write "9" than "1"? Also it does
not take more time to write "pi", i.e. to indicate a number with infinitely many
digits than 1. So your idea is not very well-founded.

Dear colleagues,
Suppose we have two sets
X: { x1, x2, x3, x4, x5 }
and
Y: { y1, y2, y3, y4, y5 }.
I would like to test whether elements of set X have smaller values
than those of set Y. One of the most popular methods for this is
Wilcoxson's U-test, which requires no assumption on the distribution
of samples unlike t-test.
The question is the following: if the values of elements include
measurement error and the probability distiribution of this error is
known, how can we execute nonparametric tests to compare the above
sets? Can we say "the probability that Nthe null hypothesis is rejected
at significant level 5% is 70%?" Is there a standard method to handle 
with this problem? Any comments and recommendation of textbooks
are welcome.
**
*
*  Yukio Sadahiro
*  Research Center for Advanced Science and Technology
*  University of Tokyo
*  Phone: 81-3-3481-4541
*  FAX: 81-3-3481-4582
*  e-mail: sada@uesgc5.rcast.u-tokyo.ac.jp
*  URL: http://www.urban.rcast.u-tokyo.ac.jp/ues/sada/sada-e.html
*
**

This question may highlight greater ignorance on my part than I realise.
My apologies if this is so...
My reading of Bayesian theory and issues concerning Beta distributions
all give formulae similar to this for its density function:
p(x) = x^{a-1}(1-x)^{b-1}/B(a,b)           (0 < x < 1)
(with slight modification sometimes, which I *think* doesn't affect
the sense of my question.) The beta function, B(a,b) is defined in
terms of gamma functions, thus:
B(a,b) = Gamma(a)Gamma(b)/Gamma(a+b)
Normally, the values given to a and b in the examples I have seen are
whole number, integer values. However, within certain *other* restrictions,
the gamma and beta functions do not have this constraint. The case I am
interested in (when a and b might take on any positive real value) seems to
fall within the permitted region of these other restrictions. As far as I
can see, there is also no intrinsic restriction or constraint for a and b
to take on whole number integer values when they are used, as above, in
the beta probability distribution. It seems to then be possible that a
justification might exist for allowing a and b to take on fractional,
positive real values in the beta probability distributions, above. I feel
fairly sure that this follows (which may show my ignorance), and that the
only obstacle to using beta distributions where one has fractional, positive
real values for the parameters, a and b, would be in justifying its use
in a particular application ("proving" or substantiating the isomorphism
between the pure statistical model and the practical application, I
suppose.) If necessary, for the sake of argument, I would prefer us to
take as axiomatic the assumption that Bayesian techniques are *not* ruled
out on other grounds?
My question is: does the above reasoning seem valid? i.e., the weak part
to any use of beta distributions with fractional, positive real values
for the parameters a and b, as defined above, would be in justifying the
application of such a distribution to a practical real-life problem? I
do have one such application in mind, and feel that the link between
theory and application is the one I need to pay *hard* attention to.
Many thanks
-- 
David Stretch: Greenwood Institute of Child Health, Univ. of Leicester, UK.
dds@leicester.ac.uk     Phone:+44 (0)116-254-6100   Fax:+44 (0)116-254-4127

Hi,
	Could someone suggest on how to tackle the following problem.
Q.	The receiver of a digital communication system is designed to 
operate with a BER (bit error rate) of 10^(-10).  The receiver must 
decide whether the observed signal is greater than the decision level and 
hence a digital one (or mark) is present or whether the observed signal 
is less than the decision level and hence a digital zero (or space) is 
present.  To operate at a BER of x, the receiver can only make a mistake 
a fraction x of the time.  If the reciever has an rms noise level of 
12mV, and the noise is caused by a multitude of factors, find the 
decision level that just permits operation at the specified BER.
Thanks in advance.

In article <32710408.2CF2@er4.eng.ohio-state.edu>, Ashutosh Sabharwal
 wrote:
> Hi,
> 
> I am looking for any general purpose 
> resampling code in matlab (public domain
> code would be great !!).
> 
> Thanks for help in advance.
> Cheers
> -- 
> Saby
The MATLAB Statistics Toolbox version 2.0 has a function called 
bootstrp.m.
-- 
==== Brad Jones ========================= brad@mathworks.com ======
|    The MathWorks, Inc.                    info@mathworks.com    |
|    24 Prime Park Way                http://www.mathworks.com    |
|    Natick, MA 01760-1500                   ftp.mathworks.com    |
==== Tel: 508-653-1415 ==== Fax: 508-653-6971 =====================

This is a classical problem in signal detection theory.  McMaster
has a fine electrical engineering department.  A good book in this
area has been written by Simon Haykin of your university.
jim bucklew

David Stretch  in <54ss3t$g2d@hawk.le.ac.uk> asks
about using non-integer parameter values for the Beta distribution.
As you have noticed any shape parameters greater than zero are
mathematically permitted by the Beta distribution function. In particular,
half-integer values are very often convenient.
There is no particular Bayesian reason to prefer integer values except in
the very special balls-in-urn problem that is often used to introduce the
Beta distribution in textbooks.
Aaron Brown
Aaron C. Brown
New York, NY

sada@okabe.t.u-tokyo.ac.jp (Yukio Sadahiro) in
<54pi3o$e13@t-server.t.u-tokyo.ac.jp> writes:
> Suppose we have two sets: X: { x1, x2, x3, x4, x5 } and
> Y: { y1, y2, y3, y4, y5 }. I would like to test whether elements
> of set X have smaller values than those of set Y. . . .[I]f the
> values of elements include measurement error and the
> probability distiribution of this error is known, how can we
> execute nonparametric tests to compare the above sets?
> Can we say "the probability that Nthe null hypothesis is
> rejected at significant level 5% is 70%?"
You could say that, although I do not think it is a useful formulation for
communicating results. Generally we prefer to put all uncertainty into one
number, either significance level or confidence. Except in very
specialized situations, there is not much use distinguishing between
measurement error and sample uncertainty.
This means you must choose between either (1) throwing away the
information about the distribution of measurement error or (2) using a
parametric test for differences. The reason is that the distribution of
measurement error is irrelevant unless you know (or assume) something
about the distribution of X-Y differences.
Aaron C. Brown
New York, NY

In article <54ss3t$g2d@hawk.le.ac.uk>,
David Stretch   wrote:
>My reading of Bayesian theory and issues concerning Beta distributions
>all give formulae similar to this for its density function:
>
>p(x) = x^{a-1}(1-x)^{b-1}/B(a,b)           (0 < x < 1)
>
>(with slight modification sometimes, which I *think* doesn't affect
>the sense of my question.) The beta function, B(a,b) is defined in
>terms of gamma functions, thus:
>
>B(a,b) = Gamma(a)Gamma(b)/Gamma(a+b)
>
>Normally, the values given to a and b in the examples I have seen are
>whole number, integer values. However, within certain *other* restrictions,
>the gamma and beta functions do not have this constraint. The case I am
>interested in (when a and b might take on any positive real value) seems to
>fall within the permitted region of these other restrictions. As far as I
>can see, there is also no intrinsic restriction or constraint for a and b
>to take on whole number integer values when they are used, as above, in
>the beta probability distribution.
That's right.  The beta distribution is defined as above for any real
values for a and b that are greater than zero.  In a Bayesian
analysis, there is no reason not to use such values if they seem
appropriate for the problem.  If a and/or b are less than one, the
density function has singularities at 0 and/or 1, but the integrals
are finite, so no problem arises in a fully Bayesian analysis.  The
non-Bayesian procedure of using the maximum a posteriori probability
estimate may not make much sense in the presence of such singularies,
however.
----------------------------------------------------------------------------
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
----------------------------------------------------------------------------

I want to see whether the correlation structure of certain financial markets changed after a 
certain event.  I can use Box's M, but that test is so powerful for anything other than matrices 
very small in dimension, that it rejects equality almost all of the time.  I could put a 
confidence interval around each correlation in the first set and see whether the correlations 
from the second set fall within, but that assumes the first set is "true" and ignores the 
simultaneous inference problem.  Is there some F test I can perform on the correlations or 
covariances on a univariate basis (what is the distribution of a sample covariance, anyway?)  Is 
there any way around the power of Box's M?

Myron Hlynka (hlynka@uwindsor.ca) wrote:
: I need a reference for the Metropolis algorithm for finding
: Markov transition matrices fro a given limiting vector.
: 
: Reference please??
The original historical reference is
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., and
Teller, E. (1953) Equations of state calculations by fast computing
machine. J. Chem. Phys., 21, 1087-1091.
I suggest getting the recent volume of collected papers Markov Chain Monte
Carlo in Practice (1996) edited by W.R. Gilks, S. Richardson, and D.J.
Spiegelhalter, London:  Chapman & Hall.
-- 
Michael P. Cohen                       home phone   202-232-4651
1615 Q Street NW #T-1                  office phone 202-219-1917
Washington, DC 20009-6331              office fax   202-219-2061
mcohen@cpcug.org

In article <54ltnb$ei3@cnn.princeton.edu>,
>Okay.  Now a number of guests equal to the number of real numbers between
>0 and 1 arrive at the hotel.  All of a sudden, the inn is full!!
>
Among those numbers, if all irrational numbers are asked to leave, then 
the inn can accomdate all the rational numbers left.
-- 
Satoru Hayasaka
WWW   : http://alcor.concordia.ca/~s_hayas

DA0503  wrote:
> Two people are playing a game in which a coin
> is tossed 11 times.  The first player gets a
> point for a toss of heads.  The other player
> gets a point for a toss of tails.  Whoever
> gets 6 points first wins.  Suppose that so far
> the first player has 2 points and the second
> player has 4 points.  What is the probability 
> that the first player wins the game?
6/32 (assuming a fair coin)
Dick