Bayesian parameter estimation for heavy-ion collisions: inferring properties of the quark-gluon plasma J. Scott Moreland—Duke U. XLVII International Symposium on Multiparticle Dynamics September 14, 2017
Lattice predicts existence of a quark-gluon plasma Lattice QCD calculations find a pseudo-critical phase transition temperature T ≈ 155 MeV, where hadrons melt to form a deconfined soup of quarks and gluons dubbed a quark-gluon plasma (QGP) early universe Temperature T [MeV] quark-gluon plasma critical point? n u c l e a r T ~ 155 MeV c o l l i s i o n s hadron gas Baryon Density μ [GeV] J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 1 / 24
What are the quark-gluon plasma bulk properties? How and under what Equation of state? conditions is it formed Relations between in a nuclear collision? thermal quantities, e.g. P = P ( ǫ ) How does it recombine Transport properties? to form colorless hadrons? shear/bulk viscosity, probe energy loss, etc J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 2 / 24
Formulating an inverse problem M ODEL - TO - DATA COMPARISON ( IN AN IDEAL WORLD ) � Model A Model B Model C Model D Model E Exp Data J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 3 / 24
Formulating an inverse problem R EALISTIC MODEL - TO - DATA COMPARISON ? ? Model A Model B Model C Model D Model E Exp Data J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 3 / 24
I) B AYESIAN PARAMETER ESTIMATION
Formulating an inverse problem P ARAMETRIZE T HEORY LANDSCAPE Model A Model B Model C Model D Model E Exp Data J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 4 / 24
Formulating an inverse problem B AYESIAN PARAMETER ESTIMATION continuous model parameter: x P ( x ⋆ | model , data ) Exp Data B AYES ’ T HEOREM : P ( x ⋆ | model , data ) ∝ P ( model , data | x ⋆ ) P ( x ⋆ ) � �� � � �� � � �� � posterior likelihood prior J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 4 / 24
Formulating an inverse problem B AYESIAN PARAMETER ESTIMATION continuous model parameter: x P ( x ⋆ | model , y exp ) Exp Data B AYES ’ T HEOREM : P ( x ⋆ | model , data ) ∝ P ( model , data | x ⋆ ) P ( x ⋆ ) � �� � � �� � � �� � posterior likelihood prior J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 4 / 24
Formulating an inverse problem B AYESIAN PARAMETER ESTIMATION continuous model parameter: x P ( x ⋆ | model , y exp ) Exp Data B AYES ’ T HEOREM : P ( x ⋆ | model , data ) ∝ P ( model , data | x ⋆ ) P ( x ⋆ ) � �� � � �� � � �� � posterior likelihood prior J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 4 / 24
Formulating an inverse problem Y IELDS POSTERIOR DISTRIBUTION ON x ⋆ P ( x |model, data) 4 2 0 2 4 x Includes uncertainty in “best-fit value” J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 4 / 24
Multiple observables posterior = likelihood × prior More than one observable f : x �→ ( y 1 , ..., y n ) ? No problem, calculate likelihood using multivariate Gaussian Log-likelihood ln( L ) = − 1 2 (ln( | Σ | ) + ( y − y exp ) T Σ − 1 ( y − y exp ) + k ln( 2 π )) Σ = Σ model + Σ stat exp + Σ sys exp J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 5 / 24
Multiple model parameters posterior = likelihood × prior Likelihood function L ( x ) → L ( x 1 , ..., x n ) Curse of dimensionality MCMC importance sampling: Typically interested in 1. large number of walkers in marginalized probabilities { x 1 , ..., x n } space L ( x 1 , ..., x n ) easy to calculate, 2. update walker positions hard to integrate. 3. accept new x with prob P ∼ L new / L old Solution Marginalize by histogramming Monte Carlo integration, e.g. over flattened dimensions importance sampling J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 6 / 24
MCMC and evaluating the likelihood Number of likelihood samples needed for MCMC varies greatly not enough better better still Several of the published results in this talk use N sample > 10 6 If model is slow, e.g. 1 CPU hour per likelihood evaluation � ...good luck J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 7 / 24
Training an emulator 2 Random functions 1 Gaussian process: • Stochastic function: maps inputs Output 0 to normally-distributed outputs • Specified by mean and −1 covariance functions −2 2 Conditioned on data As a model emulator: • Non-parametric interpolation 1 • Predicts probability distributions Output 0 • Narrow near training points, wide in gaps Mean prediction −1 Uncertainty • Fast surrogate to actual model Training data −2 0 1 2 3 4 5 Input J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 8 / 24
Workflow Physics model update walkers y = f ( x ) Emulated model { x } → { x ′ } L ( y , y exp ) MCMC update ...after many steps Bayesian posterior histogram { x } to visualize J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 9 / 24
Bayesian parameter estimation in physics LIGO E XPERIMENT • P LANCK C OLLABORATION 2015: Average 40 Effective Precession Full Precession constraints on inflation 30 GW150914 Astron. Astrophys. 594 (2016) m 2 (M ⊙ ) 20 GW170104 • CKM parameters 10 LVT151012 Eur. Phys. J. C21 (2001) GW151226 0 • G ALAXY FORMATION 10 20 30 40 50 60 Astron. Astrophys. 409 (2003) m 1 (M ⊙ ) est. black hole masses ...and many more examples not listed here PRL 118.221101 Adapt machinery to relativistic heavy-ion collisions? J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 10 / 24
II) B AYESIAN PARAMETER ESTIMATION APPLIED TO HEAVY - ION PHYSICS
Bayesian methodology for heavy-ion collisions e u g o l a n T RUSTED FRAMEWORK E XPERIMENTAL DATA F REE PARAMETER ( S ) A General relativity gravitational waves black hole masses � Relativistic hydro particle yields & corr. transport coefficients time: 0 fm/c 20 fm/c Hydro framework imposes local energy and momentum conservation. Clearly breaks in dilute limit. Should apply with care. J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 11 / 24
Bayesian methodology for heavy-ion collisions e u g o l a n T RUSTED FRAMEWORK E XPERIMENTAL DATA F REE PARAMETER ( S ) A General relativity gravitational waves black hole masses � Relativistic hydro particle yields & corr. transport coefficients � Hydro for heavy-ion collisions not trusted on same level as e.g. GR for gravitational waves • Posterior results always subject to framework credibility J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 11 / 24
Seminal Bayesian works in heavy-ion physics Rel. Probability • Event-averaged hydro energy norm. • Parametric pre-flow 50 σ sat (mb) σ sat (mb) • Parametric initial state 30 1 W.N./Sat. frac. W.N./Sat. frac. • First Bayesian posterior on ( η/ s )( T ) 0 1.25 Init. Flow Init. Flow • Omits bulk viscosity 0.25 0.5 • Two centrality bins η /s η /s 0.02 5 Rel. Probability T dep. of η 0 0.85 1.025 1.2 30 40 50 0 0.5 1 0.25 0.75 1.25 0.02 0.26 0.5 0 2.5 5 σ sat (mb) η /s T dep. of η energy norm. W.N./Sat. frac. Init. Flow Determining Fundamental Properties of Matter Created in Ultrarelativistic Heavy-Ion Collisions , Novak, Novak, Pratt, Vredevoogd, Coleman-Smith, Wolpert PRC 89 (2014) 034917 J. S. Moreland (Duke U.) Bayesian parameter estimation for HIC 12 / 24
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