G' Cowan

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Bayesian Higgs combination using shapes
ATLAS Statistics Meeting CERN, 19 December, 2007
Glen Cowan Physics Department Royal Holloway,
University of London g.cowan_at_rhul.ac.uk www.pp.rhu
l.ac.uk/cowan
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Outline, etc.
4 Sep 07 -- Bayesian Higgs combination based on
event counts Background estimated using
subsidiary measurement (sideband)
Combination of several channels Systematics
in signal efficiency, background (size of
sideband) Today -- extend/modify this to use
distribution of a variable ("shape") measured
for each event (e.g. reconstructed Higgs
mass). Use signal/background histograms for H?gg
supplied as sample inputs for Higgs combination
exercise (Bruce Mellado, Yaquan Fang, Leonardo
Carminati)
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Inputs
mH 130 GeV
bi
10 fb-1
i bin index
si
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Expected signal/background for bin i
Expected numbers of signal/background events in
bin i total number ? probability to be in bin i.
m "Global strength parameter", SM is m
1. Shape pdfs fb, fs from MC, in general have
some uncertainty. Vary shapes by parameterizing
and varying the parameters according to
appropriate priors.
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Likelihood, Bayes factor
Assume data in each bin is ni Poisson (mfi bi)
The Bayes Factor (evidence for discovery) is
where the integrals are over the internal
parameters of the model(s).
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Calculating the Bayes factor
Bayes factors need marginalized likelihoods for
numerator and denominator, both of form
Compute these using importance sampling
where f(q) multivariate Gaussian with mean,
covariance determined from L(q) p(q) by MINUIT.
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Normalization of signal/background
In principle SM MC provides absolute
predictions for si, bi. But systematic
uncertainty in background probably large
compared to stat. fluctuation in number of
background events, v btot So take a broad prior
for btot, e.g., Uniform0,8. Alternatively, use
e.g. Gaussian centred about MC prediction with
best guess for error result will not depend
on this sensitively unless sys. error comparable
to v btot. For signal use e.g. Gaussian centred
about MC prediction with best estimate for error.
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Most optimistic scenario
No uncertainty in predicted signal,
background Create test data set -- nearest
integers to SM expectation
For m 1, B10 625
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Uncertainties in total rates
Flat prior for btot and varying uncertainties for
total signal rate
For m 1, B10 307
Similar to study based on event counts (see Stat
Forum 4 Sep 07)
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Uncertainties in shapes
.Parameterize shape pdfs and write down priors
that reflect estimated uncertainty in the
shapes. E.g. from signal shape, fit a Gaussian to
MC prediction, write down prior for the mean
and sigma marginalize over the nuisance
parameters Attempted "simple" example of e.g. 10
uncertainty in width of signal, some
computational glitches -- result next time.
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Also for next time...
Straightforward extension to get limits
write down (flatish) prior for m, sample
full posterior with MCMC to find pdf of m and
solve
Exclude the tested mH if mup lt 1, repeat for all
mH.
Also straightforward to extend to multiple
channels, and to include subsidiary measurements
that help constrain background. Investigate
sampling distribution of B10 (calibration
relative to p-value).