Bayesian Methods in Particle Physics: From SmallN to Large

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Bayesian Methods in Particle Physics From
Small-N to Large

• Harrison B. Prosper
• Florida State University
• SCMA IV
• 12-15 June, 2006

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Outline

• Measuring Zero
• Bayesian Fit
• Finding Needles in Haystacks
• Summary

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Measuring Zero
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Measuring Zero 1
In the mid-1980s, an experiment at the Institut
Laue Langevin (Grenoble, France) searched for
evidence of neutron antineutron oscillations, a
characteristic prediction of certain Grand
Unified Theories.
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CRISP Experiment Institut Laue Langevin
Field-off -gt N
Field-on -gt B
Magnetic shield
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Measuring Zero 2
Count number of signal background events
N. Suppress putative signal and count background
events B, independently.
Results N 3 B 7
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Measuring Zero 3
Classic 2-Parameter Counting Experiment N
Poisson(sb) B Poisson(b) Infer a statement
of form Prs lt u(N,B) 0.9
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Measuring Zero 4
In 1984, no exact solution existed in the
particle physics literature! Moreover,
calculating exact confidence intervals is,
according to Kendal and Stuart, a matter of
very considerable difficulty
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Measuring Zero 5
Exact in what way? Over some ensemble of
statements of the form 0 lt s lt u(N,B) at
least 90 of them should be true whatever the
true values of s and b. Neyman (1937)
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Measuring Zero - 6

• Tried a Bayesian approach
• f(s, bN) f(Ns, b) p(s, b) / f(N)
• f(Ns, b) p(bs) p(s) / f(N)
• Step 1. Compute the marginal likelihood
• f(Ns) ?f(Ns, b) p(bs) db
• Step 2.
• f(sN) f(Ns) p(s) / ?f(Ns) p(s) ds

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But is there a signal?

• 1. Hypothesis testing (J. Neyman)
• H0 s 0
• H1 s gt 0
• 2. p-value (R.A. Fisher)
• H0 s 0
• 3. Decision theory (J.M. Bernardo, 1999)

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Bayesian Fit
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Bayesian Fit

• Problem
• Given counts
• Data N N1, N2,..,NM
• Signal model A A1, A2,..,AM
• Background model B B1, B2,..,BM
• where M is number of bins (or pixels)
• find
• the admixture of A and B that best matches the
  observations N.

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Problem (DØ, 2005)
Observations Background Signal
model model (M)
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Bayesian Fit - Details

• Assume model of the form
• Marginalize over a and b

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Bayesian Fit Pr(Model)

• Moreover,
• One can compute f(Npa, pb) for different signal
  models M, in particular, for models M that differ
  by the value of a single parameter.
• Then compute the probability of model M
• Pr(MN) ?dpa ?dpb f(Npa, pb, M) p(pa,pbM)
  p(M) / p(N)

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Bayesian Fit Results (DØ, 1997)
mass 173.5 4.5 GeV signal 33
8 background 50.8 8.3
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Finding Needles in Haystacks
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The Needles
single top quark events
0.88 pb 1.98 pb
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The Haystacks
W boson events
2700 pb
signal noise 1 1000
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The Needles and the Haystacks
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Finding Needles - 1

• The optimal solution is to compute
• p(Sx) p(xS) p(S) / p(xS) p(S) p(xB)
  p(B)
• Every signal/noise discrimination method is
  ultimately an algorithm to approximate p(Sx), or
  a function thereof.

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Finding Needles - 2

• Problem
• Given
• D x ( x1,xN), y ( y1,yN) of N labeled
  events. x are the data, y are the labels.
• Find
• A function f(x, w), with parameters w, that
  approximates p(Sx)
• p(wx, y) p(x, yw) p(w) / p(x, y)
• p(yx, w) p(xw) p(w) / p(yx) p(x)
• p(yx, w) p(w) / p(yx)
• assuming p(xw) p(x)

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Finding Needles - 3

• Likelihood for classification
• p(yx, w) Pi f(xi, w)y 1 f(xi, w)1-y
• where y 0 for background events
• y 1 for signal events
• If f(x, w) flexible enough, then maximizing
  p(yx, w) with respect to w yields f p(Sx),
  asymptotically.

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Finding Needles - 4

• However, in a Bayesian calculation it is more
  natural to average with respect to the posterior
  density
• f(xD) ? f(x, w) p(wD) dw
• Questions
• 1. Do suitably flexible functions f(x, w) exist?
• 2. Is there a feasible way to do the integral?

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Answer 1 Yes!
A neural network is an example of a Kolmogorov
function, that is, a function capable of
approximating arbitrary mappings fRn -gt R
The parameters w (u, a, v, b) are called weights
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Answer 2 Yes!

• Computational Method
• Generate a Markov Chain (MC) of K points w,
  whose stationary density is p(wD), and average
  over the stationary part of the chain.
• Map problem to that of a particle moving in a
  spatially-varying potential and use methods of
  statistical mechanics to generate states (p, w)
  with probability exp(-H),
• where H is the Hamiltonian
• H p2 log p(wD), with momentum p.

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Hybrid Markov Chain Monte Carlo

• Computational Method
• For a fixed H traverse space (p, w) using
  Hamiltons equations, which guarantees that all
  points consistent with H will be visited with
  equal probability.
• To allow exploration of states with differing
  values of H one introduces, periodically, random
  changes to the momentum p.
• Software
• Flexible Bayesian Modeling by Radford Neal
• http//www.cs.utoronto.ca/radford/fbm.software.h
  tml

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Example - Finding SUSY!
Transverse momentum spectra Signal black curve
SignalNoise 125,000
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Example - Finding SUSY!
Distribution of f(xD) beyond 0.9 Assuming L
10 fb-1 Cut S B S/vB 0.99 1x103 2x104
7.0 SignalNoise 120
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Summary

• Bayesian methods have been at the heart of
  several important results in particle physics.
• However, there is considerable room for expanding
  their domain of application.
• A couple of current issues
• Is there a signal? Is the Bernardo approach
  useful in particle physics?
• Fitting Is there a practical (Bayesian?) method
  to test whether or not an N-dimensional function
  fits an N-dimensional swarm of points?