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

2 27 March 2000CL Workshop, Fermilab, Harrison B. Prosper Outline l The High Energy Physicist’s Problem l Bayesian Analysis: An Example l Final Comments

3 27 March 2000CL 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?

4 27 March 2000CL 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?

5 27 March 2000CL 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 (Venn, Fisher, Neyman, etc.) Degree of Belief (Bayes, Laplace, Gauss, Jeffreys, etc.) Propensity (Popper, etc.) l The validity of these interpretations cannot be decided by an appeal to Nature. l 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. l Statistical inference cannot be fully objective.

6 27 March 2000CL Workshop, Fermilab, Harrison B. Prosper Frequentist Inference 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 l The Bad l No systematic method to incorporate prior information l “Grosse Fuge” reasoning is difficult and unnatural l The Ugly l Difficult to teach l Doesn’t do what we want: Prob(Theory|Data) Grosse Fuge, Beethoven, 1825

7 27 March 2000CL Workshop, Fermilab, Harrison B. Prosper Bayesian Inference 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 l The Bad l Can be computationally demanding l Until recently, no “goodness of fit” test l The Ugly l Choosing prior probabilities can be, well, a “Grosse Fuge”!

8 27 March 2000CL 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 Frequentist Bayesian

9 27 March 2000CL Workshop, Fermilab, Harrison B. Prosper Back to our Problem posteriorposteriorpriorpriorlikelihoodlikelihood Yes, but how do we encode this prior information?

10 27 March 2000CL 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)

11 27 March 2000CL Workshop, Fermilab, Harrison B. Prosper MeV MeV(90%), MeV(10%) MeV Making Sunshine

12 27 March 2000CL Workshop, Fermilab, Harrison B. Prosper Solar Neutrino Spectrum Flux at Earth pp Be B 5.7x10 -4 (10 10 cm -2 s -1 ) Flux at Earth pp Be B 5.7x10 -4 (10 10 cm -2 s -1 ) John Bahcall J.N.Bahcalll

13 27 March 2000CL Workshop, Fermilab, Harrison B. Prosper Solar Neutrino Problem SNU

14 27 March 2000CL Workshop, Fermilab, Harrison B. Prosper Super-K Electron Recoil Spectrum Super-Kamiokande Collaboration, Phys. Rev. Lett. 82, 2644 (1999)

15 27 March 2000CL Workshop, Fermilab, Harrison B. Prosper The Model: Survival Probability The neutrino survival probability is: The probability that a solar neutrino of a given energy E arrives at the Earth. We shall model the probability as follows:

16 27 March 2000CL 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 Event rate in experiment i Total flux from neutrino source j Cross section for experiment i Normalized neutrino spectrum Neutrino survival probability

17 27 March 2000CL 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 T measured electron kinetic energy t true electron kinetic energy R(T|t) Super-K resolution function

18 27 March 2000CL Workshop, Fermilab, Harrison B. Prosper Spectral Sensitivity

19 27 March 2000CL Workshop, Fermilab, Harrison B. Prosper Bayesian Analysis - I posteriorposteriorpriorpriorlikelihoodlikelihood

20 27 March 2000CL Workshop, Fermilab, Harrison B. Prosper Bayesian Analysis - II marginalizationmarginalization

21 27 March 2000CL Workshop, Fermilab, Harrison B. Prosper Pr(p|D): Active Neutrinos

22 27 March 2000CL Workshop, Fermilab, Harrison B. Prosper Pr(p|D): Sterile Neutrinos

23 27 March 2000CL Workshop, Fermilab, Harrison B. Prosper Final Comments l The criteria for choosing a particular theory of inference are ultimately subjective: l Does the theory do what we want? l Is the theory natural and easy to understand? l Is the theory powerful and general? l Is the 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.