Transcript CKM Fits: What the Data Say (focused on B-Physics)

Bayesian Statistics at work: The
Troublesome Extraction of the
angle a
Stéphane T’JAMPENS
LAPP (CNRS/IN2P3 & Université de Savoie)
J. Charles, A. Hocker, H. Lacker, F.R. Le Diberder, S. T’Jampens, hep-ph-0607246
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Bayesian Statistics in 1 slide
The Bayesian approach is based on the use of inverse probability (“posterior”):
Bayesian: probability about the model (degree of belief), given the data
P(model|data)  Likelihood(data;model)  Prior(model)
Bayes’rule
Cox – Principles of Statistical Inference (2006)
 “it treats information derived from data (“likelihood”) as on exactly equal
footing with probabilities derived from vague and unspecified sources (“prior”).
The assumption that all aspects of uncertainties are directly comparable is often
unacceptable.”
 “nothing guarantees that my uncertainty assessment is any good for you - I'm
just expressing an opinion (degree of belief). To convince you that it's a good
uncertainty assessment, I need to show that the statistical model I created
makes good predictions in situations where we know what the truth is, and the
process of calibrating predictions against reality is inherently frequentist.”
(e.g., MC simulations)
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Uniform prior: model of ignorance?
Cox – Principles of
Statistical Inference (2006)
A central problem : specifying a prior distribution for a parameter about
which nothing is known  flat prior
Problems:
Not re-parametrization invariant (metric dependent):
uniform in q is not uniform in z=cosq
Favors large values too much [the prior probability for the range 0.1 to 1 is 10
times less than for 1 to 10]
Flat priors in several dimensions may produce clearly unacceptable answers.
In simple problems, appropriate* flat priors yield essentially same answer as
non-Bayesian sampling theory. However, in other situations, particularly those
involving more than two parameters, ignorance priors lead to different and
entirely unacceptable answers.
* (uniform prior for scalar location parameter, Jeffreys’ prior for scalar scale parameter).
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Uniform Prior in Multidimensional Parameter Space
Hypersphere:
6D space
One knows nothing about
the individual Cartesian
coordinates x,y,z…
What do we known
about the radius
r =√(x^2+y^2+…) ?
One has achieved the remarkable feat of learning
something about the radius of the hypersphere,
whereas one knew nothing about the Cartesian
coordinates and without making any experiment.
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Isospin Analysis : B→hh
J. Charles et al. – hep-ph/0607246
Gronau/London (1990)
MA: Modulus & Argument
RI: Real & Imaginary
Improper posterior
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Isospin Analysis: removing information from B0→p0p0
No model-independent constraint on a can be inferred in this case
 Information is extracted on a, which is introduced by the priors (where else?)
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Conclusion
PHYSTAT Conferences:
http://www.phystat.org
Statistics is not a science, it is mathematics (Nature will not decide for us)
[You will not learn it in Physics books  go to the professional literature!]
Many attempts to define “ignorance” prior to “let the data speak by themselves” but
none convincing. Priors are informative.
Quite generally a prior that gives results that are reasonable from various viewpoints
for a single parameter will have unappealing features if applied independently to
many parameters.
In a multiparameter space, credible Bayesian intervals generally under-cover.
If the problem has some invariance properties, then the prior should have the
corresponding structure.
specification of priors is fraught with pitfalls (especially in high dimensions).
Examine the consequences of your assumptions (metric, priors, etc.)
Check for robustness: vary your assumptions
Exploring the frequentist properties of the result should be strongly encouraged.
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BACKUP SLIDES
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Digression: Statistics
D.R. Cox, Principles of Statistical Inference, CUP (2006)
W.T. Eadie et al., Statistical Methods in Experimental Physics, NHP (1971)
www.phystat.org
Statistics tries answering a wide variety of questions  two main different! frameworks:
Frequentist: probability about the data (randomness of measurements),
given the model
P(data|model)
[only repeatable events
(Sampling Theory)]
Hypothesis testing: given a model, assess the consistency of the data with a
particular parameter value  1-CL curve (by varying the parameter value)
Bayesian: probability about the model (degree of belief), given the data
P(model|data) Likelihood(data,model)  Prior(model)
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D.R. Cox – PHYSTAT 05
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D.R. Cox – PHYSTAT 05
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Sujective/Objective
Cox/Hinkley – Theoretical
Statistics
“It is important not to be misled by somewhat emotive words like
subjective and objective. There are appreciable personal elements
entering into all phases of scientific investigations. So far as statistical
analysis is concerned, formulation of a model and of a question for
analysis are two crucial elements in which judgment, personal
experience, etc., play an important role. Yet they are open to rational
discussion. [...] Given the model and a question about it, it is,
however, reasonable to expect an objective assessment of the
contribution of the data, and to a large extent this is provided by the
frequentist approach”
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