Unfolding with MultiFold Algorithm

This content delves into the MultiFold algorithm, exploring its applications, workings, and conclusions presented by Youqi Song at the RHIC/AGS annual users meeting. The algorithm enables result comparison with theories and experiments, corrects detector effects, and unfolds data through various methods such as iterative Bayesian unfolding and machine learning-driven approaches.

• MultiFold Algorithm
• Data Unfolding
• Machine Learning
• Particle Physics
• Data Analysis

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1. MultiFold: A user s perspective Youqi Song (youqi.song@yale.edu) 2024 RHIC/AGS annual users meeting, BNL ML&AI workshop, 6/11/2024

2. Outline What is MultiFold? the bare minimum to get started What are some applications of MultiFold? proof that the algorithm works How does MultiFold work? peeking into the black box RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 2/15

3. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Unfolding Allows for result comparison with theories and other experiments Corrects for detector effects, due to inefficiency, finite resolution, Output Ingredients for unfolding Particle-level Detector-level 1/N dN/dx 1/N dN/dx Truth Data ? x x 1/N dN/dx 1/N dN/dx PYTHIA+GEANT PYTHIA Input x x RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 3/15

4. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Unfolding a < observable 3 < b Unfolding methods: Iterative Bayesian unfolding (D'Agostini. arXiv:1010.0632(2010)) MultiFold(Andreassen et al. PRL 124, 182001 (2020)) Machine learning driven Unbinned Simultaneously unfolds many observables Correlationinformation is retained! function of observables observable 1 3, 4 and 5 Variations of the MultiFold/OmniFold algorithm: Variation Input observable 2 UniFold One event observable MultiFold Many event observables OmniFold Full phase space of the event RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 4/15

5. What is MultiFold Applications of MultiFold How MultiFold works Conclusions To get started git clone git@github.com:ericmetodiev/OmniFold.git More updated repo at https://github.com/hep-lbdl/OmniFold Run OmniFold Demo.ipynb Replace example files with your own trees Snippet of Python notebook from https://github.com/ericmetodiev/OmniFold/blob /master/OmniFold%20Demo.ipynb RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 5/15

6. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Applications MultiFold has been applied to several measurements in e+p collisions in HERA 8 observables unfolded 6 observables unfolded in p+p and heavy-ion collisions at RHIC in p+p collisions at LHC 6 observables unfolded 7 observables unfolded 4 observables unfolded 24 observables unfolded 8 observables unfolded RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 6/15

7. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Applications Probing the correlation between perturbative and nonperturbative components within jets at STAR arxiv: 2307.07718 PYTHIA8 Detroit tune HERWIG7.2 Default Simultaneously correct for: ?T: transverse momentum ?g: groomed jet radius ?g: groomed jet mass ?g: shared momentum fraction Good agreement between MultiFold and IBU verified RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 7/15

8. How does MultiFold work? RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 8/15

9. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Iterative reweighting: Step 1, iteration 1 ?1? =?(?)/?(?) Weights: Detector-level Ok for the binned case 1/N dN/dx Data Using Bayes Theorem; See derivation in backup ?(?) where ?(?) is a neural network and trained with the binary cross-entropy loss function to distinguish events coming from data vs from PYTHIA+GEANT x ?1(?) 1/N dN/dx PYTHIA+GEANT ?(?) NN input NN output event observable 1 probability that event is from data probability event is from GEANT x ?( ?) = 1 ?( ?) = ? = event observable 6 Density estimation Classification! RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 9/15

10. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Iterative reweighting: Step 1, iteration 1 ?1? =?(?)/?(?) Weights: Detector-level Ok for the binned case 1/N dN/dx Data Using Bayes Theorem; See derivation in backup ?(?) where ?(?) is a neural network and trained with the binary cross-entropy loss function to distinguish events coming from data vs from PYTHIA+GEANT x ?1(?) 1/N dN/dx PYTHIA+GEANT loss ? ? = log(? ? ) log(1 ? ? ) ? ???? ? ????? ?(?) NN output Sanity checks: Correct classification minimizes the loss function Unlikely events get weighted down probability that event is from data probability event is from GEANT x ?( ?) = 1 ?( ?) = ????????? ? ????????? = 0 loss ? ????????? Density estimation Classification! = 0 RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 10/15

11. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Iterative reweighting: Step 1, iteration 1 Detector-level 1/N dN/dx Data ?(?) PYTHIA+GEANT, reweighted 1/N dN/dx x ?1(?) 1/N dN/dx PYTHIA+GEANT x ?(?) PYTHIA+GEANT, reweighted PYTHIA, reweighted PYTHIA+GEANT, reweighted x event matching event 1, ?1 ?1 event 2, ?1 ?2 event N, ?1 ?? event 1 = ?1 event 2 event N event 1, ?1 ?1 event 2, ?1 ?2 event N, ?1 ?? event 1, ?1 ?1 event 2, ?1 ?2 event N, ?1 ?? event 1 observable 1 reweight ? event 1 observable 6 RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 11/15

12. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Iterative reweighting: Step 2, iteration 1 Detector response is stochastic Two identical particle-level events might not get mapped to identical detector-level events (?)/(1 ? ), ?1(?) is a weighting function of detector-level events Want ?1? , a weighting function of particle-level events where ? is a neural network and trained with the binary cross-entropy loss function ?1(?) PYTHIA, reweighted PYTHIA+GEANT, reweighted PYTHIA event 1 = ?1 event 2 event N event 1, ?1 ?1 event 2, ?1 ?2 event N, ?1 ?? event 1, ?1 ?1 event 2, ?1 ?2 event N, ?1 ?? ? RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 12/15

13. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Iterative reweighting: Step 2, iteration 1 PYTHIA, w/ weights from step 1 event 1, ?1 ?1 event 2, ?1 ?2 event N, ?1 ?? PYTHIA, w/ proper weighting function ?1? used to reweight both particle and detector-level events in iteration 2 ?2? , ?2(?), event 1, ?1 ?1 event 2, ?1 ?2 event N, ?1 ?? ?1(?) PYTHIA event 1 = ?1 event 2 event N Unfolding result after 1 iteration RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 13/15

14. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Iterative reweighting: Result Result: Particle-level events, reweighted by ??(?) step 2 output of the last iteration Unfolding methods: Iterative Bayesian unfolding (D'Agostini. arXiv:1010.0632(2010)) MultiFold(Andreassen et al. PRL 124, 182001 (2020)) Machine learning driven Unbinned Simultaneously unfolds many observables (D'Agostini. arXiv:1010.0632(2010)) Andreassen et al. PRL 124, 182001 (2020) reweighting is done event-by-event can adjust the input dimension of neural networks arxiv: 2307.07718 Good agreement between MultiFold and RooUnfold verified with data. RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 14/15

15. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Conclusions MultiFold(Andreassen et al. PRL 124, 182001 (2020)) Machine learning driven Unbinned Simultaneously unfolds many observables reweighting is done event-by-event can adjust the input dimension of neural networks Resources readily available, e.g., https://github.com/ericmetodiev/OmniFold and https://github.com/hep-lbdl/OmniFold Successful applications in H1, STAR, LHCb, ATLAS and CMS Easy access to correlation information among observables Promising potential for multi-differential measurements RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 15/15

16. Backup RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 16

17. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Applications Probing transverse-momentum dependent (TMD) parton distribution functions at H1 Andreev et al. PRL 128, 132002 (2022) Simultaneously correct for: Jet ?T Jet ? Jet ? Electron ?T Electron ?z Electron-jet imbalance Electron-jet azimuthal angle correlation Probing TMD jet fragmentation functions at LHCb Aaij et al. Phys. Rev. D 108, L031103 (2023) Similar measurement ongoing at STAR, see talk by Hannah Harrison-Smith Simultaneously correct for: Jet ?T Jet ? Hadron in jet longitudinal momentum fraction Hadron momentum wrt jet axis Youqi Song (Yale) 17/15

18. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Iterative reweighting: Step 1, iteration 1 ?1? =?(?)/?(?) Detector-level 1/N dN/dx Data Using Bayes Theorem ?(?) x ?1(?) 1/N dN/dx Derivation from Chapter 4, Probabilistic classification, of M. Sugiyama, T. Suzuki, and T. Kanamori, Density Ratio Estimation in Machine Learning (Cambridge University Press, 2012). PYTHIA+GEANT Normalized to 1 ?(?) probability that ? is from data probability that ? is from GEANT = x where ?(?) is a neural network and trained with the binary cross-entropy loss function RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 18/15

19. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Iterative reweighting: Step 1, iteration 2 Data event 1 event 2 event N PYTHIA+GEANT, reweighted to data event 1, ?2 ?1 event 2, ?2 ?2 event N, ?2 ?? ?2(?) PYTHIA+GEANT, w/ weights from iteration 1 event 1, ?1 ?1 event 2, ?1 ?2 event N, ?1 ?? RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 19/15

20. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Iterative reweighting: Step 2, iteration 2 PYTHIA, w/ weights from step 1, iteration 2 event 1, ?2 ?1 event 2, ?2 ?2 event N, ?2 ?? PYTHIA, w/ proper weighting function event 1, ?2 ?1 event 2, ?2 ?2 event N, ?2 ?? ?2(?) PYTHIA, w/ weights from step 2, iteration 1 event 1, ?1 ?1 event 2, ?1 ?2 event N, ?1 ?? Unfolding result after 2 iterations RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 20/15

21. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Iterative reweighting: Iteration n Iteration n, Step 1: Iteration n, Step 2: * ** ** or taken as the final weights if this is the last iteration *: (With weights pulled from step 1 of iteration n) **: (With weights pushed from step 2 of iteration (n-1)) RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 21

22. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Iterative reweighting: Toy example Adapted from slides by Ben Nachman Initial RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 22

23. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Iterative reweighting: Toy example Adapted from slides by Ben Nachman Initial Result from iteration 2 RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 23

24. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Iterative reweighting: Toy example Adapted from slides by Ben Nachman Result from iteration 1 Result from iteration Result from iteration 2 Initial RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 24

25. What is MultiFold Applications of MultiFold How MultiFold works Conclusions Iterative reweighting Why do we iterate? RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 25

26. Challenges Computationally expensive How to publish an unbinned result? arxiv:2109.13243 RHIC/AGS Users Meeting, 6/11/2024 Youqi Song (Yale) 26