Bayesian techniques and applications to QCD · 2018. 11. 22. · ALEXANDER ROTHKOPF - UIS Bayesian...
Transcript of Bayesian techniques and applications to QCD · 2018. 11. 22. · ALEXANDER ROTHKOPF - UIS Bayesian...
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
ALEXANDER ROTHKOPF - UIS
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
ALEXANDER ROTHKOPF - UIS
Outline
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
Introduction: Bayesian inference
Applications to QCD
1. Lattice QCD spectral function reconstruction à la Bayes
2. QGP model parameter estimation with Bayes
Conclusion
ALEXANDER ROTHKOPF - UIS
Introduction: Statistical Inference
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
“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)
ALEXANDER ROTHKOPF - UIS
Domain
knowledge
(I)
Bayesian inference I
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
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
Data (empirical/virtual)
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
ALEXANDER ROTHKOPF - UIS
Domain
knowledge
(I)
Bayesian inference II
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
generates
Unknown
Process
(parameters i)
y1
y2
y3
y4
𝒚
Data (empirical/virtual)
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
ALEXANDER ROTHKOPF - UIS
Domain
knowledge
(I)
Bayesian inference III
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
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
Data (empirical/virtual)
No issue with “subjectivity”: prior information = domain knowledge with uncertainty
ALEXANDER ROTHKOPF - UIS
Towards modern Bayesian inference
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
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
ALEXANDER ROTHKOPF - UIS
Outline
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
Introduction: Bayesian inference
Applications to QCD
1. Lattice QCD spectral function reconstruction à la Bayes
2. QGP model parameter estimation with Bayes
Conclusion
ALEXANDER ROTHKOPF - UIS
Inverse problems a la Bayes
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
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
ALEXANDER ROTHKOPF - UIS
Bayesian spectral reconstruction
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
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
ALEXANDER ROTHKOPF - UIS
Posterior
Prior
Inverse Laplace transform (MC-Stan)
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
Likelihood (N data)
Hyperprior
Self consistent
regularization
ALEXANDER ROTHKOPF - UIS
What are the central challenges?
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
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.)
ALEXANDER ROTHKOPF - UIS
Outline
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
Introduction: Bayesian inference
Applications to QCD
1. Lattice QCD spectral function reconstruction à la Bayes
2. QGP model parameter estimation with Bayes
Conclusion
ALEXANDER ROTHKOPF - UIS
Bayesian modelling of QGP parameters
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
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
ALEXANDER ROTHKOPF - UIS
ysim evaluation very costly, approximate via Gaussian distribution
Gaussian processes
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
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
ALEXANDER ROTHKOPF - UIS
The combined Bayesian analysis
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
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
ALEXANDER ROTHKOPF - UIS
Outline
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
Introduction: Bayesian inference
Applications to QCD
1. Lattice QCD spectral function reconstruction à la Bayes
2. QGP model parameter estimation with Bayes
Conclusion
ALEXANDER ROTHKOPF - UIS
Conclusions
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
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
ALEXANDER ROTHKOPF - UIS
How much code for all of this?
XIIIth Quark Confinement and the Hadron Spectrum Conference 2018
BAYESIAN TECHNIQUES AND APPLICATIONS TO QCD
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)