Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 258.882 LHC PhysicsExperimental Methods and MeasurementsLikelihoods and Selections[Lecture 20, April 22, 2009]C.Paus, LHC Physics: Likelihoods and Selections 2Organization Project 2 ...●Matthew handed in, not yet corrected Project 3●looks like people have no particular issues●recitation in Friday was rather quiet Conference Schedule●Tuesday May 19 at 12:00 Kolker RoomC.Paus, LHC Physics: B Physics Trigger Strategies 3Final Conference Project LHC Physics: “Experimental Methods and Measurements” Plenary Session (12:00–13:30, May 19, Kolker Room)●Welcome and LHC Overview (C.Paus)●Search for Standard Model Higgs Boson: Overview (?)●Search for Higgs in H→ZZ* (Mattew Chan)●Search for Higgs in H→WW* (?)●Search for Higgs in qqH→qqWW* (?)‘09The Physics Colloquium SeriesThursday, April 23 at 4:15 pm in room 10-250Alain AspectInstitut d'Optique, Palaiseau, France"Wave particle duality for a single photon: from Einstein's LichtQuanten to Wheeler's Delayed Choice Experiment" SpringFor a full listing of this semester’s colloquia, please visit our website at web.mit.edu/physicsColloquium SeriesPhysicsC.Paus, LHC Physics: Likelihoods and Selections 5Lecture Outline Likelihoods and Selections●likelihoods and fits●statistical uncertainties●full likelihood for lifetimes●checking whether it makes sense●goodness of fits●projections Sophisticate Selections●likelihoods●neural networksC.Paus, LHC Physics: Likelihoods and Selections 6Maximum Likelihood Estimator Taylor expansion around minimum, pfit Consider this as a PDF for true value of parameter p●PDF is a Gaussian with mean value pfit●variance is given asC.Paus, LHC Physics: Likelihoods and Selections 7Maximum Likelihood Estimator Again Taylor expansion around minimum, pfit, but using definition of the variance σ Values of the likelihood for 1, 2, n σ (standard deviations) from the central value areC.Paus, LHC Physics: Likelihoods and Selections 8Picture of Uncertaintiesτ corresponds to our parameter pτ* corresponds to pfitC.Paus, LHC Physics: Likelihoods and Selections 9Correspondence to χ2 For Gaussian PDF we know One standard deviation is interval which includes 68%●change in minimum χ2 by 12 = 1●two standard deviations correspond to Δχ2 = 22 = 4 (95%)●or n standard deviations correspond to Δχ2 = n2C.Paus, LHC Physics: Likelihoods and Selections 10Correspondence to χ2 Confidence level intervals for Gaussian n sigma●in root: α = 1.0 – TMath::Erf(1.0*n/sqrt(2))●or: P = TMath::Erf(1.0*n/sqrt(2))●probability for 5 standard deviations is astonishingly small well, it should beC.Paus, LHC Physics: Likelihoods and Selections 11Analytical Estimate of Variance Lifetime likelihood and variance:C.Paus, LHC Physics: Likelihoods and Selections 12Full Likelihood for our Analysis So far Likelihood for 1 measurement type as input, ti●but we are using more measurement types ....●mass, (uncertainty of mass,) uncertainty of proper time How to account for these additional dimensions?●very simple for likelihood: multiply PDFs (should be independent): P(t,m) = P(t) P(m)●also treat signal and background separatelysignal: has a lifetimebackground: no lifetime, t = 0Gaussian massdistributionflat mass distributionC.Paus, LHC Physics: Likelihoods and Selections 13Full Likelihood for our Analysis Including detector imperfections●proper time has uncertainty attached to it●smears out the signal exponential distribution as well as the δ distribution of the background In principle P(Δt) and P(Δm) to be included●if not we implicitly assume them to be the same●mass is fine but proper time is not: looks different for signal and background●add Psig(Δt) and Pbg(Δt) factors to the two components, need a template for this (not needed for your assignment)C.Paus, LHC Physics: Likelihoods and Selections 14Full Likelihood for our Analysis Including detector imperfectionsC.Paus, LHC Physics: Likelihoods and Selections 15Goodness of Fit Least square fit tells explicitly probability of fit●P = TMath::Prob(Chi2Min, nDoF)●comes out very close to zero or one? something is wrong! Minimum likelihood does not work like thatAverage lifetime of samples the same → max. likelihood the sameC.Paus, LHC Physics: Likelihoods and Selections 16Goodness of Fit How to get a goodness of fit from max. likelihood?●general answer: statisticians are still writing papers about it! → no unique and fully accepted answer Let's take a physicist's approach●need a chi2 like quantity for all observables●make sure that they have reasonable probabilities●need: histograms with data and theory curve●data looks simple, we got that but need to find binning so we have enough events to apply Gaussian statistics●not easy: each event has potentially different theory curve!●sum up full theory curve for all events●take χ2 value to determine probability for the picture●nDoF= (number of bins – number of parameters in picture)C.Paus, LHC Physics: Likelihoods and Selections 17Testing for Biases Likelihood fits often are complex and very difficult to implement correctly●test for fitting bias is absolutely essential●in some cases biases cannot be completely avoided How to safe yourself from trouble?●toy Monte Carlo is the answer●implement a toy in which you generate data exactly according to your implemented model●generating a large number of toy experiments should give you on average exactly the correct answer (the lifetime you put in)●uncertainties should be as expected from statistics●this means, the pulls ((p-pfit,i)/Δpfit,i)2 are Gaussian with●mean equals 0, within uncertainties●width equals 1, within uncertaintiesC.Paus, LHC Physics: Likelihoods and Selections 18Testing for Biases Example (updated: fitCTauBuJpsiK.C) provided at●~paus/8.882/614/MixFit/scripts/fitCTauBuJpsiK.C Sequence to perform pulls●perform the fit and store the results to re-initialize your fit with them when you start (iMode = iModeFit)●edit the initialization values by hand:●now generate data for about 100 experiments, setting the number of events equal to the events in your input file (iMode = iModePulls), check iModePulls in the script... most of the parameters are self