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Constraining Effective FieldTheories with Machine Learning
ATLAS ML workshop, October 15-17 2018
Gilles Louppe [email protected]
with Johann Brehmer, Kyle Cranmer and Juan Pavez.
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―――
Refs: GaltonBoard.com 1 / 24
The probability of ending in bin corresponds to the total probability of all the
paths from start to .
x
z x
p(x∣θ) = p(x, z∣θ)dz = θ (1 − θ)∫ (n
x) x n−x
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What if we shift or remove some of the pins?
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The probability of ending in bin still corresponds to the total probability of all the
paths from start to :
But this integral can no longer be simpli�ed analytically!
As grows larger, evaluating becomes intractable since the number of
paths grows combinatorially.
Generating observations remains easy: drop balls.
x
z x
p(x∣θ) = p(x, z∣θ)dz∫
n p(x∣θ)
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The Galton board is a metaphore for the simulator-based scienti�c method:
the Galton board device is the equivalent of the scienti�c simulator
are parameters of interest
are stochastic execution traces through the simulator
are observables
Inference in this context requires likelihood-free algorithms.
θ
z
x
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―――
Credits: Johann Brehmer 6 / 24
The Galton board of particle physics
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Treat the simulator as a black box
Histograms of observables,Approximate Bayesiancomputation.
Rely on summary statistics.
Machine learning algorithms
Density estimation, Cᴀʀʟ,autoregressive models, normalizing�ows, etc.
Use latent structure
Matrix Element Method, OptimalObservables, Showerdeconstruction, EventDeconstruction.
Neglect or approximate shower +detector, explicitly calculate integral.
*new* Mining gold from thesimulator.
Leverage matrix-element information +Machine Learning.
Likelihood-free inference methods
z
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Mining gold from simulators
is usually intractable.
What about ?
p(x∣θ)
p(x, z∣θ)
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This can be computed as the ball falls down the board!
p(x, z∣θ) = p(z ∣θ)p(z ∣z , θ) … p(z ∣z , θ)p(x∣z , θ)1 2 1 T <T ≤T
= p(z ∣θ)p(z ∣θ) … p(z ∣θ)p(x∣z )1 2 T T
= p(x∣z ) θ (1 − θ)T
t
∏ z t 1−z t
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As the trajectory and the observable are emitted, it is often
possible:
to calculate the joint likelihood ;
to calculate the joint likelihood ratio ;
to calculate the joint score .
We call this process mining gold from your simulator!
z = z , ..., z 1 T x
p(x, z∣θ)
r(x, z∣θ , θ )0 1
t(x, z∣θ ) = ∇ log p(x, z∣θ)0 θ ∣∣θ 0
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Rᴀsᴄᴀʟ
―――
Refs: Brehmer et al, 2018 (arXiv:1805.12244) 12 / 24
Constraining Effective FieldTheories, effectively
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LHC processes
―――
Credits: Johann Brehmer 14 / 24
LHC processes
―――
Credits: Johann Brehmer 14 / 24
LHC processes
―――
Credits: Johann Brehmer 14 / 24
LHC processes
―――
Credits: Johann Brehmer 14 / 24
p(x∣θ) = p(z ∣θ)p(z ∣z )p(z ∣z )p(x∣z )dz dz dz
intractable
∭ p s p d s d p s d
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Key insights:
The distribution of parton-level momenta
where and are the total and differential cross sections, is tractable.
Downstream processes , and do not depend on .
This implies that both and can be mined. E.g.,
p(z ∣θ) = ,pσ(θ)
1dz p
dσ(θ)
σ(θ) dz p
dσ(θ)
p(z ∣z )s p p(z ∣z )d s p(x∣z )d θ
⇒ r(x, z∣θ , θ )0 1 t(x, z∣θ )0
r(x, z∣θ , θ )0 1 = =
p(z ∣θ )p 1
p(z ∣θ )p 0
p(z ∣z )s p
p(z ∣z )s p
p(z ∣z )d s
p(z ∣z )d s
p(x∣z )d
p(x∣z )d
p(z ∣θ )p 1
p(z ∣θ )p 0
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Proof of concept
Higgs production in weak boson fusion
Goal: Constraints on two theory parameters:
L = L + (D ϕ) σ D ϕ W − (ϕ ϕ) W WSM
Λ2
f W
2ig μ † a ν
μνa
Λ2
f WW
4g2
†μνa μν a
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Credits: Johann Brehmer 17 / 24
Precise likelihood ratio estimates
―――
Credits: Johann Brehmer 18 / 24
Increased data ef�ciency
―――
Credits: Johann Brehmer 19 / 24
Better sensitivity
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Credits: Johann Brehmer 20 / 24
Stronger bounds
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Credits: Johann Brehmer 21 / 24
SummaryMany LHC analysis (and much of modern science) are based on "likelihood-free" simulations.
New inference algorithms:
Leverage more information from the simulator
Combine with the power of machine learning
First application to LHC physics: stronger EFT constraints with lesssimulations.
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Collaborators
Johann Brehmer, Kyle Cranmer and Juan Pavez
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ReferencesStoye, M., Brehmer, J., Louppe, G., Pavez, J., & Cranmer, K. (2018). Likelihood-free inference with an improved cross-entropy estimator. arXiv preprintarXiv:1808.00973.
Brehmer, J., Louppe, G., Pavez, J., & Cranmer, K. (2018). Mining gold fromimplicit models to improve likelihood-free inference. arXiv preprintarXiv:1805.12244.
Brehmer, J., Cranmer, K., Louppe, G., & Pavez, J. (2018). Constraining EffectiveField Theories with Machine Learning. arXiv preprint arXiv:1805.00013.
Brehmer, J., Cranmer, K., Louppe, G., & Pavez, J. (2018). A Guide to ConstrainingEffective Field Theories with Machine Learning. arXiv preprintarXiv:1805.00020.
Cranmer, K., Pavez, J., & Louppe, G. (2015). Approximating likelihood ratioswith calibrated discriminative classi�ers. arXiv preprint arXiv:1506.02169.
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The end.
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