The Reverend Bayes and Solar Neutrinos Harrison B
The Reverend Bayes and Solar Neutrinos Harrison B. Prosper Florida State University 27 March, 2000 CL Workshop, Fermilab 1 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
Outline l The High Energy Physicist’s Problem l Bayesian Analysis: An Example l Final Comments 2 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
The Problem l After $50 M, and half a decade, we find, alas, N = a few events, or maybe even zero. l But, we can still infer an upper limit on the cross section, and thereby perhaps exclude a theory or two. l How do we infer the upper limit? l How do we wish to interpret the probability? 3 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
The “Standard Model” l Model l Likelihood l Prior information l What do the uncertainties mean? l Are they statistical, systematic, theoretical or some complicated combination of all three? 4 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
Statistical Inference l Currently, statistical inference is based on probability l To be useful probability must be interpreted. • Relative Frequency • Degree of Belief • Propensity l l l (Venn, Fisher, Neyman, etc. ) (Bayes, Laplace, Gauss, Jeffreys, etc. ) (Popper, etc. ) The validity of these interpretations cannot be decided by an appeal to Nature. Statistical inference is based on principles that can always be challenged by anyone who doesn’t find all of them compelling. Again, Nature cannot help. Statistical inference cannot be fully objective. 5 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
Frequentist Inference l l l The Good l No “arbitrary” priors: Absence of prior anxiety! l Coverage property is powerful (some say beautiful) l There is a “badness of fit” test l One can play delightful MC games on a computer The Bad l No systematic method to incorporate prior information l “Grosse Fuge” reasoning is difficult and unnatural The Ugly l Difficult to teach l Doesn’t do what we want: Prob(Theory|Data) Grosse Fuge, Beethoven, 1825 6 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
Bayesian Inference l l l The Good l Natural model of inferential reasoning l General theory for handling uncertainty in all its forms l Results depend only on data observed l Does what we want: Prob(Theory|Data) l Easy to teach and understand The Bad l Can be computationally demanding l Until recently, no “goodness of fit” test The Ugly l Choosing prior probabilities can be, well, a “Grosse Fuge”! 7 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
“A Frequentist uses impeccable logic to answer the wrong question, while a Bayesian answers the right question by making assumptions that nobody can fully believe in. ” P. G. Hamer Bayesian Frequentist 8 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
Back to our Problem posterior likelihood prior Yes, but how do we encode this prior information? 9 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
Bayesian Analysis: An Example Solar Neutrinos C. Bhat, P. C. Bhat, M. Paterno, H. B. Prosper, Phys. Rev. Lett. 81, 5056 (1998) 10 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
Making Sunshine 0. 420 Me. V 0. 862 Me. V(90%), 0. 383 Me. V(10%) 14. 06 Me. V 11
Solar Neutrino Spectrum Flux at Earth pp 6. 0 7 Be 0. 49 8 B 5. 7 x 10 -4 (1010 cm-2 s-1) J. N. Bahcalll John Bahcall 12 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
Solar Neutrino Problem 1998 SNU SNU http: //www. sns. ias. edu/~jnb/Snviewgraphs/threesnproblems. html 13 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
Super-K Electron Recoil Spectrum Super-Kamiokande Collaboration, Phys. Rev. Lett. 82, 2644 (1999) 14 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
The Model: Survival Probability The neutrino survival probability is: The probability that a solar neutrino of a given energy En arrives at the Earth. We shall model the probability as follows: 15 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
The Model: Event Rates Event rate in experiment i Total flux from neutrino source j Cross section for experiment i Normalized neutrino spectrum Neutrino survival probability 16 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
The Model: Electron Recoil Spectrum T measured electron kinetic energy t true electron kinetic energy R(T|t) Super-K resolution function 17 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
Spectral Sensitivity 18 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
Bayesian Analysis - I posterior likelihood prior 19
Bayesian Analysis - II marginalization 20
Pr(p|D): Active Neutrinos 21 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
Pr(p|D): Sterile Neutrinos 22 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
Final Comments l The criteria for choosing a particular theory of inference are ultimately subjective: l Does theory do what we want? l Is theory natural and easy to understand? l Is theory powerful and general? l Is theory well-founded? l Bayesian theory does what I want! l Prior probabilities can be arrived at in a principled manner. l However, not everyone will agree with your principles! l But even with conventional choices for prior probabilities it is possible to do real science. 23 27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper
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