G. Cowan RHUL Physics Bayesian Higgs combination page 1 Bayesian Higgs combination using shapes ATLAS Statistics Meeting CERN, 19 December, 2007 Glen Cowan Physics Department Royal Holloway, University of London

G. Cowan RHUL Physics Bayesian Higgs combination page 2 Outline, etc. 4 Sep 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→  supplied as sample inputs for Higgs combination exercise (Bruce Mellado, Yaquan Fang, Leonardo Carminati)

G. Cowan RHUL Physics Bayesian Higgs combination page 3 Inputs 10 fb  sisi bibi i = bin index m H = 130 GeV

G. Cowan RHUL Physics Bayesian Higgs combination page 4 Expected signal/background for bin i Expected numbers of signal/background events in bin i = total number  probability to be in bin i.  = "Global strength parameter", SM is  = 1. Shape pdfs f b, f s from MC, in general have some uncertainty. Vary shapes by parameterizing and varying the parameters according to appropriate priors.

G. Cowan RHUL Physics Bayesian Higgs combination page 5 Likelihood, Bayes factor Assume data in each bin is n i ~ Poisson (  i + b i ) The Bayes Factor (evidence for discovery) is where the integrals are over the internal parameters of the model(s).

G. Cowan RHUL Physics Bayesian Higgs combination page 6 Calculating the Bayes factor Bayes factors need marginalized likelihoods for numerator and denominator, both of form Compute these using importance sampling where f(  ) = multivariate Gaussian with mean, covariance determined from L(  )  (  ) by MINUIT.

G. Cowan RHUL Physics Bayesian Higgs combination page 7 Normalization of signal/background In principle SM + MC provides absolute predictions for s i, b i. But systematic uncertainty in background probably large compared to stat. fluctuation in number of background events, √ b tot So take a broad prior for b tot, e.g., Uniform[0,∞]. 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 √ b tot. For signal use e.g. Gaussian centred about MC prediction with best estimate for error.

G. Cowan RHUL Physics Bayesian Higgs combination page 8 Most optimistic scenario No uncertainty in predicted signal, background Create test data set -- nearest integers to SM expectation For  = 1, B 10 = 625

G. Cowan RHUL Physics Bayesian Higgs combination page 9 Uncertainties in total rates Flat prior for b tot and varying uncertainties for total signal rate: For  = 1, B 10 ~ 307 Similar to study based on event counts (see Stat Forum 4 Sep 07)

G. Cowan RHUL Physics Bayesian Higgs combination page 10 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.

G. Cowan RHUL Physics Bayesian Higgs combination page 11 Also for next time... Straightforward extension to get limits write down (flatish) prior for  sample full posterior with MCMC to find pdf of  and solve Exclude the tested m H if  up < 1, repeat for all m H. Also straightforward to extend to multiple channels, and to include subsidiary measurements that help constrain background. Investigate sampling distribution of B 10 (calibration relative to p-value).