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

P(model|data)  Likelihood(data;model)  Prior(model) 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)

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).

Uniform Prior in Multidimensional Parameter Space Hypersphere: 6D space One knows nothing about the individual Cartesian coordinates x,y,z… 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. What do we known about the radius r =√(x^2+y^2+…) ?

Isospin Analysis : B→hh J. Charles et al. – hep-ph/0607246 Gronau/London (1990) MA: Modulus & Argument RI: Real & Imaginary Improper posterior

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?)

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.

BACKUP SLIDES

P(model|data) Likelihood(data,model)  Prior(model) 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) Hypothesis testing: given a model, assess the consistency of the data with a particular parameter value  1-CL curve (by varying the parameter value) [only repeatable events (Sampling Theory)] Bayesian: probability about the model (degree of belief), given the data P(model|data) Likelihood(data,model)  Prior(model)

D.R. Cox – PHYSTAT 05

D.R. Cox – PHYSTAT 05

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”