Re: [UAI] Is it luck or is it skill - my resolution
Hi Rich,
In your analysis you present a frequentist and a Bayesian approach,
arguing that the paradox exists only for the frequentist case. Fair
enough. I would just like to point out that the frequentist approach
(orthodox hypothesis testing) is even more problematic than that, in that
it effectively makes assumptions it claims not to:
In the frequentist exposition, you state: I have no idea whether my
population includes clairvoyants (or at least I do not want to impose my
prior beliefs). You then give us an example of a circumstance under which
you would reject the null hypothesis. However, from your example we can
calculate bounds on your prior belief that a randomly chosen individual is
clairvoyant:
P(clairvoyant) 0.5. (To explain your default belief in the null hypothesis).
P(clairvoyant) 1/10001 (approximately .0001). (Otherwise it would be
irrational to reject the null hypothesis on observing success - the
alternative would still be less likely.)
If you are willing to use the commonly used p value threshold of 0.01, we
get a stronger bound: P(clairvoyant) 1/101.
Here I am assuming that you are willing to believe a hypothesis whenever
it has probability 0.5; if instead you prefer to build in a grey area
where you do not accept any beliefs, the bounds on your prior again become
more stringent.
So despite the explicit denial, this method does impose your prior
beliefs.
regards,
Konrad
On Tue, 28 Jun 2005, Rich Neapolitan wrote:
I thank all those who responded to my query and discussed the matter with
me. Here is my resolution.
First, I'll re-describe the problem using some numbers and terminology
provided by Francisco Javier Diez. Suppose there is some task such that
P(success) = .0001 if someone is not clairvoyant and P(success) = 1 if
someone is clairvoyant. I have no idea whether my population includes
clairvoyants (or at least I do not want to impose my prior beliefs). Mike
claims he is one. My null hypothesis is that he is not. When he succeeds a
very unlikely event has occurred (.0001) if the null hypothesis is true. So
I reject that hypothesis and believe Mike probably is one. Next I have
10,000 people making claims they are clairvoyants. My null hypothesis is
that none are. If the null hypothesis is true, the probability of at least
one succeeding is
1-(.)^10,000 = .63. So if Mike alone succeeds I have no reason to
reject the null hypothesis. I need quite few people succeeding to reject
it. So I have little reason for believing Mike or anyone else in the group
is clairvoyant.
There is no way out of this if we insist on obtaining our beliefs from
hypothesis testing. However, if as I.J. Good said, we don't sweep our prior
beliefs under the carpet, we can solve the problem using Bayes' Theorem.
Suppose we believe that there is a .01 probability some individual (say
Mike) is clairvoyant. Then
P(clairvoyant|success)
=
P{success|clairvoyant)P(clairvoyant)/[P{success|clairvoyant)P(clairvoyant)
+ P{success|not clairvoyant)P(not clairvoyant)
=1 x .01 / [1x .01 + .0001 x .99] = .99.
So when Mike succeeds we believe he is probably a clairvoyant regardless of
how many other people attempt the task or succeed.
In applications to situations like Buffet predicting stock performance I
think with a little analysis we can formulate reasonable priors, etc. and
analyze the problem this second way. In applications like coin tossing we
can also assign extremely small priors to someone having special ability.
Actually out of a large group I could see where someone could have some
talent for forcing heads. So I really mean a random experiment in which we
control for all known tricks. There still could be some very small
probability that someone has psychic ability.
Rich
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Post Doctoral Research Assistant in Artificial Intelligence Operations Research TITLE OF THE PROJECT The role of management practices in closing the productivity gap START DATE: October 1, 2005 Applications are invited for one Post Doctoral Research Assistant for 2½ years to work with the Automated Scheduling, Optimisation and Planning Research Group (ASAP) in the School of Computer Science and Information Technology at the University of Nottingham, on an EPSRC- funded project entitled The role of management practices in closing the productivity gap. PROJECT DESCRIPTION The project focuses in particular on the role of management practices in the productivity gap. For example, is it the case that US-owned retail outlets operating in the UK are more productive than UK-owned outlets here because the US companies implement and use appropriate management practices more effectively? The project adopts an inter- disciplinary approach and involves a mix of case study and survey methods. As such this project will attempt to: * focus on selected significant parts of the service sector in the UK; * focus on the micro level by examining the role of management practices within companies and sites, but including more macro variables as part of the models; * explore techniques and theories from different academic disciplines; * compare UK and USA owned companies working in the UK. The research will be carried out under the supervision of Dr Uwe Aickelin, http://www.cs.nott.ac.uk/~uxa COLLABORATORS This is a joint research proposal arising from a recent EPSRC IDEAS factory event. The partners are Sheffield University (Psychology), Cambridge University (Judge Institute) and Aston University (Business School). RESEARCH GROUP The Automated Scheduling, Optimisation and Planning Research group (ASAP) is one of four major groupings within the School. The School obtained a grade 5 in the 2001 Research Assessment exercise. ASAP is concerned with investigating and developing Artificial Intelligence and Operational Research approaches to a vide variety of scheduling and optimisation problems. It has been at the forefront of research in this area over the last few years and is internationally recognised for its research work. The group comprises 9 members of academic staff, 11 Post Doctoral Research Associates, 35 PhD students and 1 secretary. Further details are available on the WWW at: http://www.asap.ac.uk/ POST DOCTORAL RESEARCH ASSISTANT The Post Doctoral Research Assistant will investigate the potential of a variety of novel artificial intelligence and operational research methods, with an emphasis on mathematical modelling and heuristic optimisation, to generate the most appropriate model and simulation of the productivity problem. The ideal candidate for this project will hold a PhD in a subject related to the project, i.e. mathematical modelling, (meta-) heuristic optimisation or simulation. The starting salary is in the range £19,460 - £21,640 per annum. This post will be offered on a fixed-term contract for a period of 2½ years. Please quote ref. UweAickelin/01 and the title of the project. APPLICATIONS Applicants should send a detailed curriculum vitae together with the names and addresses of two referees who can support their application. Applications should be sent to: Ms Emma-Jayne Dann School of Computer Science IT, University of Nottingham Jubilee Campus Wollaton Road Nottingham NG8 1BB, UK e-mail: [EMAIL PROTECTED] CLOSING DATE for applications is 26 August, 2005. = Dr Uwe Aickelin School of Computer Science (ASAP) University of Nottingham Nottingham NG8 1BB UK Phone +44(0) 11595 14215 Fax +44(0) 11584 67591 Email [EMAIL PROTECTED] Web http://www.aickelin.com This message has been checked for viruses but the contents of an attachment may still contain software viruses, which could damage your computer system: you are advised to perform your own checks. Email communications with the University of Nottingham may be monitored as permitted by UK legislation. ___ uai mailing list uai@ENGR.ORST.EDU https://secure.engr.oregonstate.edu/mailman/listinfo/uai
