# Presentation: 063399

## The Reverend Bayesand Solar Neutrinos

- Harrison B. Prosper
- Florida State University
- 27 March, 2000
- CL Workshop, Fermilab

## Outline

- The High Energy Physicist’s Problem
- Bayesian Analysis: An Example
- Final Comments

## 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?

## The “Standard Model”

- Model
- Likelihood
- Prior information
- What do the uncertainties mean?
- Are they statistical, systematic, theoretical or some complicated combination of all three?

## 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.

## 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

## 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”!

## “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

## Back to our Problem

- posterior

- prior

- likelihood

- Yes, but how do we encode this prior information?

## Bayesian Analysis: An ExampleSolar Neutrinos

- C. Bhat, P.C. Bhat, M. Paterno, H.B. Prosper, Phys. Rev. Lett. **81**, 5056 (1998)

## Making Sunshine

- 0.862 MeV(90%), 0.383 MeV(10%)

- 14.06 MeV

## Solar Neutrino Spectrum

- Flux at Earth
- pp   6.0
- 7Be 0.49
- 8B   5.7x10-4
- (1010 cm-2 s-1)

- John Bahcall

- J.N.Bahcalll

## Solar Neutrino Problem 1998

- http://www.sns.ias.edu/~jnb/Snviewgraphs/threesnproblems.html

- SNU

- SNU

- SNU

## Super-K Electron Recoil Spectrum

- Super-Kamiokande Collaboration, Phys. Rev. Lett. **82**, 2644 (1999)

## 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:

## The Model: Event Rates

## The Model: Electron Recoil Spectrum

**T  measured electron kinetic energy**
**t   true electron kinetic energy**
**R(T|t) Super-K resolution function**

## Spectral Sensitivity

## Bayesian Analysis - I

- posterior

- prior

- likelihood

## Bayesian Analysis - II

- marginalization

## Pr(p|D): Active Neutrinos

## Pr(p|D): Sterile Neutrinos

## 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.