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• Constraining Effective Field Theories with Machine Learning

ATLAS ML workshop, October 15-17 2018

Gilles Louppe g.louppe@uliege.be

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 Bayesian computation.

Rely on summary statistics.

Machine learning algorithms

Density estimation, Cᴀʀʟ, autoregressive models, normalizing �ows, etc.

Use latent structure

Matrix Element Method, Optimal Observables, Shower deconstruction, Event Deconstruction.

Neglect or approximate shower + detector, explicitly calculate integral.

*new* Mining gold from the simulator.

Leverage matrix-element information + Machine Learning.

Likelihood-free inference methods

z

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• Mining gold from simulators

is usually intractable.

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

• 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 Field Theories, effectively

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• LHC processes

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Credits: Johann Brehmer 14 / 24

• LHC processes

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Credits: Johann Brehmer 14 / 24

• LHC processes

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Credits: Johann Brehmer 14 / 24

• LHC processes

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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) = ,p σ(θ)

1 dz 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

2 ig μ † a ν

μν a

Λ2 f WW

4 g2 †

μν a μν a

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Credits: Johann Brehmer 17 / 24

• Precise likelihood ratio estimates

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Credits: Johann Brehmer 18 / 24

• Increased data ef�ciency

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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

• Summary Many LHC analysis (and much of modern science) are based on "likelihood- free" simulations.

New inference algorithms:

Combine with the power of machine learning

First application to LHC physics: stronger EFT constraints with less simulations.

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• Collaborators

Johann Brehmer, Kyle Cranmer and Juan Pavez

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• References Stoye, M., Brehmer, J., Louppe, G., Pavez, J., & Cranmer, K. (2018). Likelihood- free inference with an improved cross-entropy estimator. arXiv preprint arXiv:1808.00973.

Brehmer, J., Louppe, G., Pavez, J., & Cranmer, K. (2018). Mining gold from implicit models to improve likelihood-free inference. arXiv preprint arXiv:1805.12244.

Brehmer, J., Cranmer, K., Louppe, G., & Pavez, J. (2018). Constraining Effective Field Theories with Machine Learning. arXiv preprint arXiv:1805.00013.

Brehmer, J., Cranmer, K., Louppe, G., & Pavez, J. (2018). A Guide to Constraining Effective Field Theories with Machine Learning. arXiv preprint arXiv:1805.00020.

Cranmer, K., Pavez, J., & Louppe, G. (2015). Approximating likelihood ratios with calibrated discriminative classi�ers. arXiv preprint arXiv:1506.02169.

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• The end.

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