Bayesian techniques and applications to QCD · 2018. 11. 22. · ALEXANDER ROTHKOPF - UIS Bayesian techniques and applications to QCD XIIITH QUARK CONFINEMENT AND THE HADRON SPECTRUM

ALEXANDER ROTHKOPF - UIS

Bayesian techniques and

applications to QCD

XIIITH QUARK CONFINEMENT AND THE HADRON SPECTRUM CONFERENCE 2018 – 01/08 – MAYNOOTH, IRELAND

Alexander RothkopfFaculty of Science and Technology

Department of Mathematics and Physics

University of Stavanger

Selected textbooks on Bayesian techniques:

Bayesian Data Analysis, A. Gelman et. al. CRC Press

Statistical Rethinking, R. McElreath CRC Press

Gaussian Processes for Machine Learning C. Rasmussen, C.Williams, MIT Press


Physics motivation

XIIIth Quark Confinement and the Hadron Spectrum Conference 2018

BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD

Matter in extreme

conditionsExperimental data

[ALICE collaboration]

arXiv:1805.04390

[CMS collaboration]

PRL120 (2018)142301

Phenomenological

Models

W. Zhao et. al.EPJ.C77 (2017) 645

S. McDonald et.al. PRC95 (2017) 064913

B. Krouppa et.al.

PRD97 (2018) 016017

1st principles

theory computations

N. Astrakhantsev et.al.

JHEP 1704 (2017) 101

S.Kim, P.Petreczky,

A.R. in preparation

Need robust statis-

tical tools to shed

light on the underly-

ing properties of

strongly interacting

matter

Shear viscosity

of the QGP

Melting tempera-

tures of quarkonia


Outline



Introduction: Bayesian inference

Applications to QCD

1. Lattice QCD spectral function reconstruction à la Bayes

2. QGP model parameter estimation with Bayes

Conclusion


Introduction: Statistical Inference



“To draw conclusions about unobserved quantities, based on empirical data”

Unobservable: What are the true parameters i governing the unknown process?

Potentially observable: Future observations 𝒚made from the unknown process?

Unknown

Process

y1

y2

y3

y4

𝒚

generates

Data (empirical/virtual)

(parameters i)


Domain

knowledge

(I)

Bayesian inference I



Bayes uses generalized concept of probability: a measure of uncertainty

What is the chance of the new Ariane 6 rocket launch to succeed?

generates

Unknown

Process

(parameters i)

y1

y2

y3

y4

𝒚


Starting point: joint probability distribution of involved quantities.

Requires knowledge about: data generation process and scientific problem itself

posterior likelihood prior

Bayes theorem

cannot answer by large # of trials, model informed by domain knowledge, model parameters carry uncertainty


Domain

knowledge

(I)

Bayesian inference II



generates

Unknown

Process

(parameters i)

y1

y2

y3

y4

𝒚


Hierarchical models: parameters of prior can carry uncertainty too p(m), p()

Parameters of prior often referred to as default model m and weight

m: most probable

in absence of data “hyperparameter”

Dependence on prior information explicit in joint probability


Domain

knowledge

(I)

Bayesian inference III



generates

Unknown

Process

(parameters i)

y1

y2

y3

y4

𝒚

Goal: Compute (simulate) the posterior distribution, gives access to

marginal posterior: Bayesian probability for parameter j

posterior predictive distribution: valuable for validating the analysis

Once new data y’ becomes available, easily incorporated: old posterior as prior


No issue with “subjectivity”: prior information = domain knowledge with uncertainty


Towards modern Bayesian inference



Improvements in applied Bayesian statistics over the past two decades:

Prior based on domain knowledge not due to computational convenience

Uncertainty in prior parameters can be self-consistently included

Simulate the full posterior via Monte Carlo Methods, not just its maximum

In 2018 full fledged Bayesian analysis is only one download away

With funding by Google, MPI, DOE, NSF:

open source STAN framework

Samples the posterior of multilevel hierarchical models

using an efficient Hamiltonian (hybrid) Monte Carlo algorithm

(thanks to Nicolas Wink from HD for bringing it to my attention)mc-stan.org


Outline




Applications to QCD



Conclusion


Inverse problems a la Bayes



Setting up the likelihood probability: how is the observed data generated

y observed data

K detector imperfection

n measurement noise

y simulated correlator

K Euclidean Kernel

n simulation uncertainty

From detector simulation OR

from QCD spectral representation

On the lattice often approximately

Gaussian(0,σ), may differ substantially

Experiments: unfolding detector data - Lattice QCD: spectral function reconstruction

2 fit in the language of Bayes: P[x]=1 Maximum Posterior = Maximum Likelihood

inverse-problems often ill-posed, i.e. maximum likelihood not unique: needs regularization

Mario Kruger Wed 14:20, Mikael Kuusela Fri. 14:00 Olaf Kaczmarek Thu 18:00, T>0 quarkonium Sun. 14:00,

posters by Nikita Astrakhantsev and Ryan Quinn


Bayesian spectral reconstruction



ρ(ω)

ω

J/ψ ψ‘

D/D thresh.

T~0

Log[D(τ)]

τ

τ∈[0

,1/T

]

x,y,z

T>0

Lattice QCD is an highly distorting detector for spectral functions

Domain knowledge as prior information and regulator: e.g. QCD spectra > 0

Based on the concept of scale invariance,

positivity and smoothness: BR prior Y. Burnier, A.R. PRL111 (2013) 182003

Other regularizations: Tikhonov MEM

Spectral reconstruction = Inversion of Laplace transform : highly ill-posed


Posterior

Prior

Inverse Laplace transform (MC-Stan)



Likelihood (N data)

Hyperprior

Self consistent

regularization


What are the central challenges?



Information content of the lattice simulation:

How to setup simulations to improve relevant information content?

(anisotropic lattices, multi-level algorithms, etc.)

Reconstructions at T>0 limited by finite Euclidean extent (cont. limit not a solution)see e.g. T>0 quarkonium talk Sun. 14:00

How to encode analytic properties of the spectrum in a Bayesian fashion

Can we learn appropriate regularization from prior knowledge (neural networks)

Analytic structure of correlators encoded in restricted functional space for see e.g. A. Cyrol et. al. arXiv:1804.00945

Currently regulators use concepts unspecific to QCD (e.g. smoothness)

Log[D

(τ)]

τ

Log[D

(τ)]

τ1/2T0 1/2T1

T0 < T1Resolution of

reconstruction

degrades

(ringing etc.)


Outline




Applications to QCD



Conclusion


Bayesian modelling of QGP parameters



Phenomenological success in previous years based on combination of

Fluctuating initial

condition models

Viscous relativistic

hydrodynamicsHadronization

models

Hadronic phase

transport models

Input with uncertainty (e.g. lQCD E.O.S) & parametrization choice (e.g. /S(T) )

Modeling: fix some parameters as input, leave subset of parameters to estimate

related to

initial conditions

p entropy deposition

k shape fluctuation

w Gaussian Nucleon width

N Normalization

related to QGP

Tswitch QGP -> Hadronic

/Shrg spec. visc. T<Tswitch

/S min spec. visc. T= Tswitch

/S slope spec. visc. T> Tswitch

/Shrg mag. bulk visc.

Given values for the 9 free parameters ()

The models produce data ysim : e.g.

dN/dy, v2, … for each centrality

Learn about QCD from systematic comparison with experimental dataRecent Bayesian studies: J. Bernhard et.al. PRC94 (2016) 024907, see also J. Auvinen PRC97 (2018) 044905


ysim evaluation very costly, approximate via Gaussian distribution

Gaussian processes



1. Full model runs ytrain for a small number of sets of O(100) parameter sets

2. Estimate from training set a Gaussian distribution mean and variance for ysim

3. Conditional probability gives:

Model correlations as Gaussian – hyperparameters via fit to training data:

Central ingredient likelihood: comparison between experiment and models


The combined Bayesian analysis



Crosscheck the approximation: generating

ytst both via GP and full models

J. B

ern

hard

et.

al. P

RC

94 (

2016)

024907

Sample posterior with uniform priors on

see also the talk by Vladimir Kovalenko Wed. 16:50


Outline




Applications to QCD



Conclusion


Conclusions



Bayesian inference: a flexible approach to extract insight from empirical data

Availability of dedicated MC libraries: no more hurdles to full posterior estimation

In its modern form: prior information = domain knowledge with uncertainties

posterior likelihood prior

Bayes theorem

Go raibh maith agat as do aird

Thank you for your attention

For highly complex models: Gaussian processes may offer efficient approximation

marginal posterior posterior predictive distribution

Challenge: Design of priors from physics & information content of (lattice) input


How much code for all of this?



Prepare the input data (covariance matrix etc.)

Define the MC-Stan Model

Compile the model

Run the simulation of the posterior (Nτ=32, Nω=1000, 50 chains on corei7 notebook ~3h)

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Bayesian techniques and applications to QCD · 2018. 11. 22. · ALEXANDER ROTHKOPF - UIS Bayesian techniques and applications to QCD XIIITH QUARK CONFINEMENT AND THE HADRON SPECTRUM