The Reverend Bayes and Solar Neutrinos

1. The Reverend Bayesand Solar Neutrinos Harrison B. Prosper Florida State University 27 March, 2000 CL Workshop, Fermilab CL Workshop, Fermilab, Harrison B. Prosper

2. Outline • The High Energy Physicist’s Problem • Bayesian Analysis: An Example • Final Comments CL Workshop, Fermilab, Harrison B. Prosper

3. The Problem • After $50 M, and half a decade, we find, alas, N = a few events, or maybe even zero. • But, we can still infer an upper limit on the cross section, and thereby perhaps exclude a theory or two. • How do we infer the upper limit? • How do we wish to interpret the probability? CL Workshop, Fermilab, Harrison B. Prosper

4. The “Standard Model” • Model • Likelihood • Prior information • What do the uncertainties mean? • Are they statistical, systematic, theoretical or some complicated combination of all three? CL Workshop, Fermilab, Harrison B. Prosper

5. Statistical Inference • Currently, statistical inference is based on probability • To be useful probability must be interpreted. • Relative Frequency (Venn, Fisher, Neyman, etc.) • Degree of Belief (Bayes, Laplace, Gauss, Jeffreys, etc.) • Propensity (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. CL Workshop, Fermilab, Harrison B. Prosper

6. Frequentist Inference • The Good • No “arbitrary” priors: Absence of prior anxiety! • Coverage property is powerful (some say beautiful) • There is a “badness of fit” test • One can play delightful MC games on a computer • The Bad • No systematic method to incorporate prior information • “Grosse Fuge” reasoning is difficult and unnatural • The Ugly • Difficult to teach • Doesn’t do what we want: Prob(Theory|Data) Grosse Fuge, Beethoven, 1825 CL Workshop, Fermilab, Harrison B. Prosper

7. Bayesian Inference • The Good • Natural model of inferential reasoning • General theory for handling uncertainty in all its forms • Results depend only on data observed • Does what we want: Prob(Theory|Data) • Easy to teach and understand • The Bad • Can be computationally demanding • Until recently, no “goodness of fit” test • The Ugly • Choosing prior probabilities can be, well, a “Grosse Fuge”! CL Workshop, Fermilab, Harrison B. Prosper

8. “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 CL Workshop, Fermilab, Harrison B. Prosper

9. Back to our Problem posterior likelihood prior Yes, but how do we encode this prior information? CL Workshop, Fermilab, Harrison B. Prosper

10. Bayesian Analysis: An ExampleSolar Neutrinos C. Bhat, P.C. Bhat, M. Paterno, H.B. Prosper, Phys. Rev. Lett. 81, 5056 (1998) CL Workshop, Fermilab, Harrison B. Prosper

11. 0.420 MeV Making Sunshine 0.862 MeV(90%), 0.383 MeV(10%) 14.06 MeV CL Workshop, Fermilab, Harrison B. Prosper

12. Solar Neutrino Spectrum Flux at Earth pp 6.0 7Be 0.49 8B 5.7x10-4 (1010 cm-2 s-1) J.N.Bahcalll John Bahcall CL Workshop, Fermilab, Harrison B. Prosper

13. Solar Neutrino Problem 1998 SNU SNU SNU http://www.sns.ias.edu/~jnb/Snviewgraphs/threesnproblems.html CL Workshop, Fermilab, Harrison B. Prosper

14. Super-K Electron Recoil Spectrum Super-Kamiokande Collaboration, Phys. Rev. Lett. 82, 2644 (1999) CL Workshop, Fermilab, Harrison B. Prosper

15. 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: CL Workshop, Fermilab, Harrison B. Prosper

16. Event rate in experiment i Total flux from neutrino source j Cross section for experiment i Normalized neutrino spectrum Neutrino survival probability The Model: Event Rates CL Workshop, Fermilab, Harrison B. Prosper

17. The Model: Electron Recoil Spectrum T measured electron kinetic energy t true electron kinetic energy R(T|t) Super-K resolution function CL Workshop, Fermilab, Harrison B. Prosper

18. Spectral Sensitivity CL Workshop, Fermilab, Harrison B. Prosper

19. Bayesian Analysis - I posterior likelihood prior CL Workshop, Fermilab, Harrison B. Prosper

20. Bayesian Analysis - II marginalization CL Workshop, Fermilab, Harrison B. Prosper

21. Pr(p|D): Active Neutrinos CL Workshop, Fermilab, Harrison B. Prosper

22. Pr(p|D): Sterile Neutrinos CL Workshop, Fermilab, Harrison B. Prosper

23. Final Comments • The criteria for choosing a particular theory of inference are ultimately subjective: • Does the theory do what we want? • Is the theory natural and easy to understand? • Is the theory powerful and general? • Is the theory well-founded? • Bayesian theory does what I want! • Prior probabilities can be arrived at in a principled manner. • However, not everyone will agree with your principles! • But even with conventional choices for prior probabilities it is possible to do real science. CL Workshop, Fermilab, Harrison B. Prosper