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Some aspects of statistics at LHC

Some aspects of statistics at LHC. N.V.Krasnikov INR, Moscow. Outline. 1. Introduction 2. Parameters estimation 3. Confidence intervals 4. Systematics 5. Upper limits at LHC 6. Conclusion. 1. Introduction The statistical model of an analysis

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Some aspects of statistics at LHC

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  1. Some aspects of statistics at LHC N.V.Krasnikov INR, Moscow October 2013

  2. Outline 1. Introduction 2. Parameters estimation 3. Confidence intervals 4. Systematics 5. Upper limits at LHC 6. Conclusion October 2013

  3. 1. Introduction The statistical model of an analysis provides the complete description of that analysis The main problem – from the known probability density and x = xobs to extract some information on θ parameter Two approaches 1.Frequentist method 2. Bayesian method Also very important – the notion of the likelihood October 2013

  4. Likelihood - the probability density evaluated at the observed value x=xobs October 2013

  5. Frequents statistics – general philosophy In frequentist statistics, probabilities are associated only with data, i.e. outcomes of repeatable observations. The preferred models are those for which our observations have non small probabilities October 2013

  6. October 2013

  7. Bayesian statistics – general philosophy In Bayesian statistics , interpretation of probability is extended to degree of belief(subjective probability). Bayesian methods can provide more natural treatment of non repeatable phenomena : systematic uncertainties October 2013

  8. Parameters estimation Maximum likelihood principle October 2013

  9. Normal distribution October 2013

  10. Bayesian method • In Bayes approach For flat prior π(θ) = const Bayes and likelihood coincide October 2013

  11. Confidence intervals Suppose we measure x = xobs • What are possible values of θ parameter? • Frequentist answer: Neyman belt construction Alternative: Bayes credible interval October 2013

  12. Neyman belt construction (1-α) – confidence level. The choice of x1 and x2 is not unique October 2013

  13. Neyman belt construction October 2013

  14. Neyman belt construction October 2013

  15. Neyman belt construction • For normal distribution Neyman belt equations for lower limit lead to October 2013

  16. Neyman belt equations October 2013

  17. Maximal likelihood Approximate estimate For normal distribution October 2013

  18. Bayes approach Bayes theorem P(A|B)*P(B) = P(B|A)*P(B) P(A|B) – conditional probability October 2013

  19. Bayes approach • Due to Bayes formula the statistics problem is reduced to the probability problem October 2013

  20. Bayes approach • The main problem – prior function π(θ) is not known • For what prior frequentist and Bayes approaches coincide? October 2013

  21. The relation between Bayes and frequentist approaches • Two examples 1.Example A October 2013

  22. The relation between Bayes and frequentist approaches 2.Example B October 2013

  23. Parameter determination with additional constraint • Consider the case of normal distribution with additional constraint Maximum likelihood method gives October 2013

  24. Parameter determination with additional constraint • How to construct the confidence interval for the μ parameter? Cousins, Feldman method • Maximum of October 2013

  25. Neyman belt construction • The ordering principle on • As a consequence we find October 2013

  26. Likelihood method • For x0 < 0 • or October 2013

  27. Likelihood principle • For x0 > 0 • or October 2013

  28. Bayes approach • We choose π(μ) = θ(μ) So prior function is zero for negative μ automatically • The equation for the credible interval determination October 2013

  29. Confidence intervals for Poisson distribution • The generalization of Neyman belt construction is Klopper-Pearson interval October 2013

  30. Poisson distribution • The Kloper-Pearson interval is conservative and it does not have the coverage property. Coverage is the probability that interval covers true value with the probability Besides for So we have negative probability - contradiction October 2013

  31. Stevens interval • To overcome these problems Stevens (1952) suggested to introduce new random variable U. Modified equations are October 2013

  32. Stevens equations • One can derive Stevens equations using the regularization of discrete Poisson distribution(S.Bityukov,N.V.K). Namely let us introduce Poisson generalization • The integral • is not well defined October 2013

  33. Stevens interval Let us introduce the regularization October 2013

  34. Stevens interval • We can use Neyman belt construction for regularized distribution and we find October 2013

  35. Stevens interval In the limit of the regularization removement we find October 2013

  36. Likelihood method • The use of likelihood method gives • The solution is October 2013

  37. Likelihood method October 2013

  38. Bayes approach • The basic equations are • Due to identities October 2013

  39. Bayesapproach Upper Klopper-Pearson limit coincides with Bayesian limit for flat prior and lower limit corresponds to prior The Stevens equations for non dependent on λ are equivalent to Bayes approach with prior function October 2013

  40. Uncertainties in extraction of an upper limit October 2013

  41. Modified frequentist definition • We require(S.Bityukov,N.V.K.,2012) that • Our definition is equivalent to Bayes • approach with prior function October 2013

  42. Signal extraction for nonzero background • Consider the case • Cousins-Feldman method October 2013

  43. Nonzero background • Likelihood ordering Plus Neyman construction October 2013

  44. CLS method(T.Junk,A.Read) • Upper bound • CLS method • In Bayes approach it corresponds to the replacement October 2013

  45. Bayes method • For flat prior • We can interprete this formula in terms of conditional probability October 2013

  46. Bayes method • Namely the probability that parameter λ lies in the interval [λ, λ+dλ] provided λ≥λb is determined by the formula that coincides with the previous Bayes formula October 2013

  47. Systematics • 1. Systematics that can be eliminated by the measurement of some variables in other kinematic region • 2. Uncertainties related with nonexact accuracy in determination of particle momenta, misidentification... • 3. Uncertainties related with nonexact knowledge of theoretical cross sections October 2013

  48. Systematics • 3 methods to deal with systematics(at least) 1. Suppose we measure some events in two kinematic regions with distribution functions + , The random variable Z = X-Y obeys normal distribution As a consequence we find October 2013

  49. Systematics • For Poisson distributions and due to identity October 2013

  50. Systematics • The problem is reduced to the determination of the ρ parameter from experimental data October 2013

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