From: Don Groom (deg@panisse.lbl.gov)
Date: Thu Jun 05 2003 - 18:34:23 PDT
I was discussing stuff with Alex about the meaning of the likelihood
function, and have gotten a little disturbed by the \P at the bottom of p
14 of the HST paper. [That is *not* to say I have a problem with it at
this point.]
From the RPP02 discussion of statistics, last edited by Glen Cowan and
previously by Bob Cousins, p. 230 (from the input \TeX file for RPP02):
"The likelihood function is the joint p.d.f.\ for the data, evaluated with
the data obtained in the experiment and regarded as a function of the
parameters. Note that the likelihood function is {\it not} a p.d.f.\ for
the parameters $\vec{\theta}$; in frequentist statistics this is not
defined. In Bayesian statistics one can obtain from the likelihood the
posterior p.d.f.\ for $\vec{\theta}$, but this requires multiplying by a
prior p.d.f. (see Sec.~\use{Sec.bayes})."
For the parameters \theta read [\Omega_M,\Omega_\Lambda, {\cal M},
\alpha]. The reason for emphasis on the likelihood being the pdf for the
data but NOT for the parameters is that this is a common error that
Glen Cowan, Fred James, and others find dismaying, just like biasing
distributions by the use of bad priors.
I don't know how to "integrate over the nuisance parameters," but would
look at it very very carefully, and maybe discuss it with Cowan or
Cousins. Besides being a statistics nut, Bob is part of NOMAD, and has
made a living doing 2D CL plots for the collaboration.
Also sorry not to have picked this up before.
D
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Don Groom (Particle Data Group, Supernova Cosmology Project)
DEGroom(at)lbl.gov www-ccd.lbl.gov Voice: 510/486-6788 FAX: 510/486-4799
Analog: 50-308//Berkeley Lab//Berkeley, CA 94720
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