The Reverend Bayes and Solar Neutrinos
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27 March 2000 CL Workshop, Fermilab, Harrison B. Prosper 1 The Reverend Bayes The Reverend Bayes and and Solar Solar Neutrinos Neutrinos Harrison B. Prosper Florida State University 27 March, 2000 CL Workshop, Fermilab CL Workshop, Fermilab
description
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. - PowerPoint PPT Presentation
Transcript of The Reverend Bayes and Solar Neutrinos
27 March 2000CL Workshop, Fermilab, Harrison B.
Prosper
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The Reverend BayesThe Reverend Bayesand and
Solar Solar NeutrinosNeutrinosHarrison B. ProsperFlorida State University
27 March, 2000
CL Workshop, FermilabCL Workshop, Fermilab
27 March 2000CL Workshop, Fermilab, Harrison B.
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OutlineOutline
The High Energy Physicist’s Problem
Bayesian Analysis: An Example
Final Comments
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The ProblemThe 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?
0.95 Probwith UP
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The “Standard Model”The “Standard Model”
ModelModel
LikelihoodLikelihood
Prior informationPrior information
What do the uncertainties mean?
Are they statistical, systematic, theoretical or some complicated combination of all three?
!),,,|Pr(
N
nebLN
Nn
bbLL ˆˆ,ˆˆ,ˆˆ
bLn
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Statistical InferenceStatistical Inference
Currently, statistical inference is based on probabilityCurrently, 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.
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Frequentist Inference Frequentist Inference
The GoodThe 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 BadThe Bad No systematic method to incorporate prior information “Grosse Fuge” reasoning is difficult and unnatural
The UglyThe Ugly Difficult to teach Doesn’t do what we want: Prob(Theory|Data)
Grosse Fuge, Beethoven, 1825
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Bayesian Inference Bayesian Inference
The GoodThe 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 BadThe Bad Can be computationally demanding Until recently, no “goodness of fit” test
The UglyThe Ugly Choosing prior probabilities can be, well, a “Grosse
Fuge”!
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“A Frequentist uses impeccable logic to answerthe wrong question, while a Bayesian answersthe right question by making assumptions that nobody can fully believe in.”
P.G. Hamer
Frequentist Bayesian
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Back to our ProblemBack to our Problem
posteriorposteriorposteriorposterior priorpriorpriorprior
bL
bL
IbLIbLN
IbLIbLN
IN
,,,
,,
)|,,,Pr(),,,,|Pr(
)|,,,Pr(),,,,|Pr(
),|Pr(
bL
bL
IbLIbLN
IbLIbLN
IN
,,,
,,
)|,,,Pr(),,,,|Pr(
)|,,,Pr(),,,,|Pr(
),|Pr(
likelihoodlikelihoodlikelihoodlikelihood
)|Pr(),|,,Pr()|,,,Pr( IIbLIbL )|Pr(),|,,Pr()|,,,Pr( IIbLIbL
Yes, but how do we encode this prior information?
bbLL ˆˆ,ˆˆ,ˆˆ
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Bayesian Analysis: An ExampleBayesian Analysis: An ExampleSolar NeutrinosSolar Neutrinos
C. Bhat, P.C. Bhat, M. Paterno, H.B. Prosper, Phys. Rev. Lett. 81, 5056 (1998)
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p p H e
H p He
He He He p
2
2 3
3 3 4 2
p p H e
H p He
He He He p
2
2 3
3 3 4 2
0.420 MeV0.420 MeV
3 4 7
7 7
7 42
He He Be
Be e Li
Li p He
3 4 7
7 7
7 42
He He Be
Be e Li
Li p He
0.862 MeV(90%), 0.383 MeV(10%)0.862 MeV(90%), 0.383 MeV(10%) 7 8
8 8
8 42
Be p B
B B e
B He
*
*
7 8
8 8
8 42
Be p B
B B e
B He
*
*
14.06 MeV14.06 MeV
Making SunshineMaking Sunshine
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Solar Neutrino SpectrumSolar Neutrino Spectrum
Flux at Earthpp 6.07Be 0.498B 5.7x10-4
(1010 cm-2 s-1)
Flux at Earthpp 6.07Be 0.498B 5.7x10-4
(1010 cm-2 s-1)
John Bahcall
J.N.Bahcalll
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Solar Neutrino Problem 1998Solar Neutrino Problem 1998
http://www.sns.ias.edu/~jnb/Snviewgraphs/threesnproblems.html
SNUSNU
SNU
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Super-K Electron Recoil SpectrumSuper-K Electron Recoil Spectrum
Super-Kamiokande Collaboration, Phys. Rev. Lett. 82, 2644 (1999)
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The Model: Survival ProbabilityThe Model: Survival Probability
N
rnn EfaaEP
0
)(),|(
N
rnn EfaaEP
0
)(),|(
The neutrino survival probability is:
The probability that a solar neutrinoof a given energy E arrives at the Earth.
We shall model the probability as follows:
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The Model: Event RatesThe Model: Event Rates
j
Eth jijii
dEEEEPS )()()|( j
Eth jijii
dEEEEPS )()()|(
p
S
j
i
j
i
p
S
j
i
j
i
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
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The Model: Electron Recoil SpectrumThe Model: Electron Recoil Spectrum
2/])2([)(
)2/(2)(
)},()]|(1[),()|(){()|()(
min
2max
)(
)(
0
0
max
min
maxmax
e
e
E
tE
eB
Et
B
mttttE
mEEEt
EtEPEtEPEdEtTRdtnTN
2/])2([)(
)2/(2)(
)},()]|(1[),()|(){()|()(
min
2max
)(
)(
0
0
max
min
maxmax
e
e
E
tE
eB
Et
B
mttttE
mEEEt
EtEPEtEPEdEtTRdtnTN
T measured electron kinetic energyt true electron kinetic energyR(T|t) Super-K resolution function
T measured electron kinetic energyt true electron kinetic energyR(T|t) Super-K resolution function
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Spectral SensitivitySpectral Sensitivity
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Bayesian Analysis - IBayesian Analysis - I
posteriorposteriorposteriorposterior priorpriorpriorprior
,
)|,Pr(),,|Pr(
)|,Pr(),,|Pr(),|,Pr(
a
IaIaD
IaIaDIDa
,
)|,Pr(),,|Pr(
)|,Pr(),,|Pr(),|,Pr(
a
IaIaD
IaIaDIDa
likelihoodlikelihoodlikelihoodlikelihood
)|Pr(),|Pr()|,Pr( IIaIa )|Pr(),|Pr()|,Pr( IIaIa
daIa
dI
),|Pr(
),ˆ|(gaussian)|Pr(
daIa
dI
),|Pr(
),ˆ|(gaussian)|Pr(
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Bayesian Analysis - IIBayesian Analysis - II
),|,Pr(),|Pr( IDaIDa
),|,Pr(),|Pr( IDaIDa
marginalizationmarginalizationmarginalizationmarginalization
a
IDaaEPpIDEp ),|Pr()],|([),,|Pr( a
IDaaEPpIDEp ),|Pr()],|([),,|Pr(
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Pr(Pr(pp|D): Active Neutrinos|D): Active Neutrinos
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Pr(Pr(pp|D): Sterile Neutrinos|D): Sterile Neutrinos
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Final CommentsFinal 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.
