Continuous simulation of Beyond-Standard-Model processes with multiple parameters

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

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Motivation

Many Beyond Standard Model (BSM) processes are defined by more than one free parameters

Masses of hypothetical particlesCoupling constants…

Grid ScanScan the parameter spacewith grid pointsSimulate a sample of events on each point

ACAT 2011, 5-9 September, London

Var1

Var2

Jiahang ZHONG

ACAT 2011, 5-9 September, London 3

Motivation

The difficulties of the grid-scan approach:Curse of dimensionality

Npoints~Nd

Hard to go beyond 2DCostly for finer granularity

Var1

Var2

Jiahang ZHONG


Motivation


Npoints~Nd


Large statistics requiredSamples at different points are treated independentlyConsiderable statistics neededwithin each sample

Var1

Var2

Pass

Fail~10k evts

Jiahang ZHONG


Motivation


Npoints~Nd


Large statistics requiredSamples at different points are treated independentlyConsiderable statistics neededwithin each sample

DiscretenessConsiderable space between pointsSmoothing/interpolation neededConsequent systematic uncertainties

Var1

Var2

~TeV

~100GeV

Jiahang ZHONG

6

Motivation

Grid-scan:Curse of dimensionality

Large statistics needed

Discreteness

The aim of Continuous MCCompetent for multivariate parameter space

Less events to be simulated

Continuous estimation of signal yield over the parameter space

ACAT 2011, 5-9 September, LondonJiahang 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 acceptance, offline efficiencyNeed large amount and expensive detector simulation

Therefore, our method is focused on easing the estimation of ε

Motivation

ACAT 2011, 5-9 September, London 7Jiahang ZHONG

The procedure

Event generation


Var1

Var2

Var1

Var2

Grid Scan Continuous MC

O(10d) 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 ZHONG ACAT 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


Demo

Production of right-handed W boson and Majorana neutrino

Di-lepton final state2 leptons (e,μ)pT>20GeV, |eta|<2.5

cone20/pT<0.1

Two free parametersWR mass [500GeV,1500GeV]

NR mass [0, M(WR)]

Affect both the cross-section and efficiency

11

Demo

Continuous SimulationGenerated 100k events, each with random { M(WR), M(NR) }

Put each event through the selection criteria, and assign target value 1/0 if it pass/failFeed all events to a BNN, with { M(WR), M(NR) } as the input variablesUse the trained BNN as a function to provide ε±σε

Reference grid-scanA grid with 100GeV step in M(WR) and 50GeV step in M(NR) (171 samples in total)Sufficient statistics in each sample to achieve precise reference values


Demo

The BNN fitted efficiency Reference from grid-scan


Demo

The difference between fitted values and reference values


Demo

Uncertainty estimated by the BNN.


Demo

The real deviations vs. estimated uncertainties (Nσ)


Summary

New approach to simulate multivariate BSM processesMore space points, less eventsUse BNN fitting to obtain smooth yield estimation

Performance tested byThe deviation between BNN and reference valuesThis deviation vs. BNN uncertainty

Limitation: the assumption of smooth distributionNot sensitive to local abrupt changesLess performance across physics boundary.

17ACAT 2011, 5-9 September, LondonJiahang ZHONG

完Thank you!

18ACAT 2011, 5-9 September, LondonJiahang ZHONG

Backup

More detailed documentation of this methodhttp://arxiv.org/abs/1107.0166

The Bayesian Neural Network in TMVA/ROOThttp://www.sciencedirect.com/science/article/pii/S0010465511002682

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Links


http://arxiv.org/abs/1107.0166

http://www.sciencedirect.com/science/article/pii/S0010465511002682

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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

BackupHow does BNN fitting work

i

tt yyL 1)1( l

ljlj xwa )1()1(

)1()1(jj afh

)tanh()()1( xxf

j

jiji xwa )2()2(

)2()2(ii afy

xxf )()2(


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

1,0y

i

iiii xytxytL ))(1log(*)1()(log*)log(

HistogramBNN


Backup: Bayesian implementation of NN (II)

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Uncertainty estimation

Training:

Most probable value wMP

P(w|D)Probability of other w

Prediction

Probability

Uncertainty of y

Avoid excessive extrapolation (non-trivial for multivariate analysis)

wDwwx'x' dPPP )|(*),|y()|y(

HistogramBNN


Backup: Bayesian implementation of NN (III)

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RegulatorOvertraining is possible due to excessive complexity of NN

Early stopUse half input sample as monitor

Manual decision of when to stop excessive fitting

RegulatorPrior knowledge that “simpler” model is preferred

Adaptive during training

Save the monitor sample!!!

Early stop

Regulator


Continuous simulation of Beyond-Standard-Model processes with multiple parameters

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