Ivo van Vulpen Why does RooFit put asymmetric errors on data points ? 10 slides on a ‘too easy’ topic that I hope confuse, but do not irritate you

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Presentation transcript:

Ivo van Vulpen Why does RooFit put asymmetric errors on data points ? 10 slides on a ‘too easy’ topic that I hope confuse, but do not irritate you ATLAS H  4 lepton peak Ivo van Vulpen (UvA/Nikhef)

Ivo van Vulpen ATLAS H  4 lepton peak - Summarize measurement - Make statement on underlying true value I’ll present 5 options. You tell me which one you prefer Why put an error on a data-point anyway ? Why put an error on a data-point anyway ?

Ivo van Vulpen Known λ (Poisson) Poisson distribution Probability to observe n events when λ are expected λ=4.00 Number of observed events #observed λ hypothesis fixed varying Binomial with n  ∞, p  0 en np=λ P(N obs |λ=4.0) ! !

Ivo van Vulpen Known λ (Poisson) Poisson distribution Probability to observe n events when λ are expected λ=4.90 Number of observed events #observed λ hypothesis fixed varying Binomial with n  ∞, p  0 en np=λ P(N obs |λ=4.9)

Ivo van Vulpen λ=1.00 λ=4.90 properties the famous √N (1) Mean: (2) Variance: (3) Most likely: first integer ≤ λ Option 1: Poisson spread for fixed λ

Ivo van Vulpen Treating it like a normal measurement What you have: 1) construct Likelihood λ as free parameter 3) Determine error interval: Δ(-2Log(lik)) = +1 2) Find value of λ that maximizes Likelihood

Ivo van Vulpen λ (hypothesis) P(N obs =4|λ) Note: normally you use -2log(Lik) Likelihood (ratio) Probability to observe n events when λ are expected Likelihood #observed λ hypothesis fixed varying Likelihood ! !

Ivo van Vulpen λ (hypothesis) P(N obs =4|λ) -2Log(P(N obs =4|λ)) Log(Likelihood) -2Log(Lik) ΔL=+1 Option 2: likelihood (ratio) Likelihood: P(4|λ)

Ivo van Vulpen Bayesian: statement on true value of λ Likelihood: Poisson distribution “what can I say about the measurement (number of observed events) given a theory expectation ?” Posterior pdf for λ: “what can I say about the underlying theory (true value of λ) given that I have observed of 4 events ?” What you have: What you want:

Ivo van Vulpen Choice of prior P(λ): Assume all values for λ are equally likely (“I know nothing”) λ (hypothesis) LikelihoodProbability density function Posterior PDF for λ  Integrate to get confidence interval Bayesian: statement on true value of λ

Ivo van Vulpen Option 3 and 4: Bayesian 16% 68% 8.3% 23.5% 68% Bayes central Bayes equal prob. option 1 option 2 Equal likelihood Ordering rule

Ivo van Vulpen Option 4: Frequentist largest λ fow which P(n≥n obs |λ)= smallest λ for which P(n≤n obs |λ)= λ = % Poisson λ = % Poisson If λ < 7.16 then probability to observe 4 events (or less) <16% Note: also using data that you did not observe

Ivo van Vulpen The options Poisson Likelihood Bayesian central Bayesian equal prob Frequentist Δλ:

Ivo van Vulpen Think about it and discuss with your colleagues tonight I’ll give the answer tomorrow

Ivo van Vulpen The options Poisson Likelihood Bayesian central Bayesian equal prob Frequentist Δλ:

Ivo van Vulpen ATLAS H  4 lepton peak Conclusion: - Now you know what RooFit uses - Hope you are a bit confused BobCousins_Poisson.pdf  Paper with details PoissonError.C  Implementation options shown here Ivo_Analytic_Poisson.pdf  Analytic properties (inverted) Poisson