Continuous simulation of Beyond-Standard-Model processes with multiple parameters Jiahang Zhong (University of Oxford * ) Shih-Chang Lee (Academia Sinica) ACAT 2011, 5-9 September, London * Was in Academia Sinica and Nanjing University

Motivation Many Beyond Standard Model (BSM) processes are defined by more than one free parameters Masses of hypothetical particles Coupling constants … Grid Scan Scan the parameter space with grid points Simulate a sample of events on each point ACAT 2011, 5-9 September, London 2 Var1 Var2 Jiahang ZHONG

Motivation The difficulties of the grid-scan approach: Curse of dimensionality N points ~N d Hard to go beyond 2D Costly for finer granularity ACAT 2011, 5-9 September, London 3 Var1 Var2 Jiahang ZHONG

Motivation The difficulties of the grid-scan approach: Curse of dimensionality N points ~N d Hard to go beyond 2D Costly for finer granularity Large statistics required Samples at different points are treated independently Considerable statistics needed within each sample ACAT 2011, 5-9 September, London 4 Var1 Var2 Pass Fail ~10k evts Jiahang ZHONG

Motivation The difficulties of the grid-scan approach: Curse of dimensionality N points ~N d Hard to go beyond 2D Costly for finer granularity Large statistics required Samples at different points are treated independently Considerable statistics needed within each sample Discreteness Considerable space between points Smoothing/interpolation needed Consequent systematic uncertainties ACAT 2011, 5-9 September, London 5 Var1 Var2 ~TeV ~100GeV Jiahang ZHONG

Motivation Grid-scan: Curse of dimensionality Large statistics needed Discreteness The aim of Continuous MC Competent for multivariate parameter space Less events to be simulated Continuous estimation of signal yield over the parameter space ACAT 2011, 5-9 September, London 6 Jiahang ZHONG

The usage of multivariate BSM simulation is to estimate signal yields over the parameter space. Yields: N(x)=L* σ(x) * ε(x) L: Luminosity. Irrelevant to x (the free parameters) σ: Cross section, branching ratio. Easy to calculate with event generators ε: Detector a cceptance, offline efficiency Need large amount and expensive detector simulation Therefore, our method is focused on easing the estimation of ε Motivation ACAT 2011, 5-9 September, London 7 Jiahang ZHONG

The procedure Event generation ACAT 2011, 5-9 September, London 8 Var1 Var2 Grid ScanContinuous MC O(10 d ) space points O(100k) space points O(10k) events/point O(1) events/point Jiahang ZHONG

The procedure Bayesian Neural Network (BNN) is used to fit the efficiency ε Desirable features of NN fitting Non-parametric modeling Smooth over the parameter space Unbinned fitting Suffer less from dimensionality Correlation between the variables Jiahang ZHONGACAT 2011, 5-9 September, London 9 Unbinned fitting vs. Binned Histogram

The procedure Bayesian implementations of NN further provide Automatic complexity control of NN topology during training Probabilistic output Uncertainty estimation of the output Uncertainty of the output estimated based on the p.d.f. of the NN parameters. Statistical fluctuation of the training sample Choice of NN topology Impact of fitting goodness at certain space point x Jiahang ZHONGACAT 2011, 5-9 September, London 10

Demo Production of right-handed W boson and Majorana neutrino Di-lepton final state 2 leptons (e, μ ) p T >20GeV, |eta|<2.5 cone20/p T <0.1 Two free parameters W R mass [500GeV,1500GeV] N R mass [0, M(W R )] Affect both the cross-section and efficiency 11

Demo Continuous Simulation Generated 100k events, each with random { M(W R ), M(N R ) } Put each event through the selection criteria, and assign target value 1/0 if it pass/fail Feed all events to a BNN, with { M(W R ), M(N R ) } as the input variables Use the trained BNN as a function to provide ε ±σ ε Reference grid-scan A grid with 100GeV step in M(W R ) and 50GeV step in M(N R ) (171 samples in total) Sufficient statistics in each sample to achieve precise reference values Jiahang ZHONGACAT 2011, 5-9 September, London 12

Demo The BNN fitted efficiency Reference from grid-scan Jiahang ZHONGACAT 2011, 5-9 September, London 13

Demo The difference between fitted values and reference values Jiahang ZHONGACAT 2011, 5-9 September, London 14

Demo Uncertainty estimated by the BNN. Jiahang ZHONGACAT 2011, 5-9 September, London 15

Demo The real deviations vs. estimated uncertainties (N σ ) Jiahang ZHONGACAT 2011, 5-9 September, London 16

Summary New approach to simulate multivariate BSM processes More space points, less events Use BNN fitting to obtain smooth yield estimation Performance tested by The deviation between BNN and reference values This deviation vs. BNN uncertainty Limitation: the assumption of smooth distribution Not sensitive to local abrupt changes Less performance across physics boundary. 17 ACAT 2011, 5-9 September, LondonJiahang ZHONG

完 Thank you! 18 ACAT 2011, 5-9 September, LondonJiahang ZHONG

Backup More detailed documentation of this method The Bayesian Neural Network in TMVA/ROOT Links ACAT 2011, 5-9 September, LondonJiahang ZHONG

A black-box of discriminator A white-box of non-parametric fitting tool A multivariate function y(x) Generic function approximator (analog to polynomial in 1D) Training  unbinned MLE fitting y: NN output, a probability, [0,1] t: Target value, 1=pass, 0=fail 20 Backup How does BNN fitting work ACAT 2011, 5-9 September, LondonJiahang ZHONG

Backup: Bayesian implementation of NN(I) 21 Probability fitting Unbinned fitting Full usage of every event Extrapolation/Interpolation Fit y as probability function Bernoulli likelihood Histogram BNN ACAT 2011, 5-9 September, LondonJiahang ZHONG

Backup: Bayesian implementation of NN (II) 22 Uncertainty estimation Training: Most probable value w MP P(w|D) Probability of other w Prediction Probability Uncertainty of y Avoid excessive extrapolation (non-trivial for multivariate analysis) Histogram BNN ACAT 2011, 5-9 September, LondonJiahang ZHONG

Backup: Bayesian implementation of NN (III) 23 Regulator Overtraining is possible due to excessive complexity of NN Early stop Use half input sample as monitor Manual decision of when to stop excessive fitting Regulator Prior knowledge that “simpler” model is preferred Adaptive during training Save the monitor sample!!! Early stop Regulator ACAT 2011, 5-9 September, LondonJiahang ZHONG