The Reverend Bayes and Solar Neutrinos

1
The Reverend Bayesand Solar Neutrinos

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

2
Outline

• The High Energy Physicists Problem
• Bayesian Analysis An Example
• Final Comments

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?

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?

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 doesnt
  find all of them compelling. Again, Nature cannot
  help.
• Statistical inference cannot be fully objective.

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
• Doesnt do what we want Prob(TheoryData)

Grosse Fuge, Beethoven, 1825
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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(TheoryData)
• 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!

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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
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Back to our Problem
posterior
prior
likelihood
Yes, but how do we encode this prior information?
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Bayesian Analysis An ExampleSolar Neutrinos
C. Bhat, P.C. Bhat, M. Paterno, H.B. Prosper,
Phys. Rev. Lett. 81, 5056 (1998)
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Making Sunshine
0.862 MeV(90), 0.383 MeV(10)
14.06 MeV
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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
13
Solar Neutrino Problem 1998
SNU
SNU
SNU
http//www.sns.ias.edu/jnb/Snviewgraphs/threesnpr
oblems.html
14
Super-K Electron Recoil Spectrum
Super-Kamiokande Collaboration, Phys. Rev. Lett.
82, 2644 (1999)
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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
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The Model Event Rates
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The Model Electron Recoil Spectrum
T measured electron kinetic energy t true
electron kinetic energy R(Tt) Super-K resolution
function
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Spectral Sensitivity
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Bayesian Analysis - I
posterior
prior
likelihood
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Bayesian Analysis - II
marginalization
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Pr(pD) Active Neutrinos
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Pr(pD) Sterile Neutrinos
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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.