Bayesian parameter estimation for relativistic heavy-ion collisions

Ph.D. defense presentation, Duke University

Transcript

1. Bayesian parameter estimation for relativistic heavy-ion collisions Jonah E. Bernhard Defense · March 30, 2018 Physics
2. A question How can we characterize a physical system if we can only observe its final state? 1 / 40
3. A question How can we characterize a physical system if we can only observe its final state? A burned down building 1 / 40
4. A question How can we characterize a physical system if we can only observe its final state? What did it look like? 1 / 40
5. A question How can we characterize a physical system if we can only observe its final state? Burn down some buildings 1 / 40
6. A question How can we characterize a physical system if we can only observe its final state? Properties x Physical process f (x) Data / observables y 1 / 40
7. A question How can we characterize a physical system if we can only observe its final state? Parameters x Computer model f (x) Data / observables y 1 / 40
8. Quark-gluon plasma (QGP) • Fluid-like phase of deconfined quarks and gluons • Postulated that early universe was QGP (t 1 sec) • Created in ultra-relativistic collisions of heavy nuclei • Extremely: Hot: 1012 K (solar core ∼ 107 K) Tiny: 10 fm = 10−14 meters (∼size of nucleus) Transient: 10−23 seconds (10 fm/c) • Can observe only emitted particles → Estimate QGP properties through model-to-data comparison Hadrons T ∼ 150 MeV QGP * Not to scale 2 / 40
9. This work Goal Quantitatively estimate fundamental properties of the QGP, including • Temperature-dependent transport coefficients • Specific shear viscosity (η/s)(T) • Specific bulk viscosity (ζ/s)(T) • Characteristics of the initial state that leads to QGP formation Process • Develop a complete dynamical model of the spacetime evolution of heavy-ion collisions • Tailor Bayesian parameter estimation methods for heavy-ion collisions • Derive probability distributions for the likely values of the model parameters Š Պő ŠƷƐĺƓƳƏĺƒ ƏĺƑ 3 / 40
10. Overview Input parameters QGP properties Model Heavy-ion collision spacetime evolution Gaussian process emulator Surrogate model Bayesian calibration Infer model parameters from data Posterior distribution Quantitative estimates of each parameter Experimental data Heavy-ion collision observables 4 / 40
11. Overview Input parameters QGP properties Model Heavy-ion collision spacetime evolution Gaussian process emulator Surrogate model Bayesian calibration Infer model parameters from data Posterior distribution Quantitative estimates of each parameter Experimental data Heavy-ion collision observables 4 / 40
12. Relativistic heavy-ion collisions Time − − − − − − − − − − − − − → Collision → Pre-equilibrium → Hot and dense QGP → Hadron gas → Freeze-out RHIC LHC Au-Au et al., √ s = 7.7–200 GeV Pb-Pb, √ s = 2.76 & 5.02 TeV 5 / 40
13. Experimental observables Detect emitted particles Distill into observables • Particle multiplicities • Energy deposition • Transverse momentum (pT ) distributions • Momentum anisotropy • etc... 6 / 40
14. Collective behavior Converts initial spatial anisotropy → anisotropic particle emission b Ini�al geometry Final momentum x y z Key evidence for the strongly-interacting QGP ϕ 7 / 40
15. Collective behavior Converts initial spatial anisotropy → anisotropic particle emission b Ini�al geometry Final momentum x y z Key evidence for the strongly-interacting QGP ϕ Strongly-interacting Fermi gas 7 / 40
16. Anisotropic ow coe cients Decompose azimuthal particle distribution into Fourier series dN dφ ∝ 1 + 2 ∞ n=1 vn cos n(φ − Ψn) Ə ņƑ ƒ ņƑ Ƒ 7ņ7 ˆƑ ƷƏĺƐƏ ˆƒ ƷƏĺƏƔ ˆƓ ƷƏĺƏƑ vn = flow coefficients (Fourier coeffs.) • v2 = elliptic flow • v3 = triangular flow • v4, v5, ... Driven by overlap geometry and nucleon position fluctuations: 0 ƷƏ=l 0 ƷѶ=l 8 / 40
17. Ə ƑƏ ƓƏ ѵƏ ѶƏ ƐƏƐ ƐƏƑ ƐƏƒ ƐƏƓ ƐƏƔ 71_ ņ7 ķ 7ņ7‹ķ 7 $ ņ7 Œ;(œ 1_ $  r +b;Ѵ7v Ə ƑƏ ƓƏ ѵƏ ѶƏ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ƏĺƐƔ ˆm ŔƑŕ ˆƑ ˆƒ ˆƓ Ѵo‰1†l†Ѵ-m|v 0Ŋ0Ƒĺƕѵ$;( 0Ŋ0ƔĺƏƑ$;( Ə ƑƏ ƓƏ ѵƏ ѶƏ ;m|u-Ѵb|‹ѷ ƏĺƏ ƏĺƔ ƐĺƏ ƐĺƔ r$ Œ;(œ  r ;-mr$ Ə ƑƏ ƓƏ ѵƏ ѶƏ ;m|u-Ѵb|‹ѷ ƏĺƏƏ ƏĺƏƑ ƏĺƏƓ r$ ņ r$ ;-mr$ =Ѵ†1|†-|bomv 0% Centrality b ∼ 0 100% Centrality b ∼ 2R Calibration data Centrality dependence of: • Charged-particle yields Nch • Transverse energy ET • Identified particle yields for π, K, p • Identified particle mean pT for π, K, p • Mean pT fluctuations • Anisotropic flow coefficients v2, v3, v4 Data: LHC ALICE, Pb-Pb collisions at √ s = 2.76 and 5.02 TeV (where available) 9 / 40
18. Overview Input parameters QGP properties Model Heavy-ion collision spacetime evolution Gaussian process emulator Surrogate model Bayesian calibration Infer model parameters from data Posterior distribution Quantitative estimates of each parameter Experimental data Heavy-ion collision observables 9 / 40
19. Multistage model 1. Initial conditions Entropy deposition t 1 fm/c 2. Pre-equilibrium Early-time dynamics and thermalization t < 1 fm/c 3. Viscous relativistic hydrodynamics Hot and dense QGP 1 < t < 10 fm/c 4. Particlization Conversion to discrete particles 1 < t < 10 fm/c 5. Boltzmann transport Hadronic scatterings and decays 10 < t < 100 fm/c Highlighted = original contributions 10 / 40
20. Initial conditions • Initial condition model provides entropy density immediately after collision • Many theories / models • Significant source of uncertainty This work Parametric model TRENTo • Developed by myself and Scott Moreland • Designed to simultaneously estimate initial condition and QGP medium properties 11 / 40
21. TRENTo: Parametric initial condition model Original work Ansatz Entropy density proportional to generalized mean of local nuclear density s ∝ Tp A + Tp B 2 1/p TA,B = z-integrated density (“thickness”) p ∈ (−∞, ∞) = tunable parameter • Interpolates among physically reasonable entropy deposition scalings • Mimics other models • Previous work: p ∼ 0, s ∼ √ TATB Other degrees of freedom Effective nucleon width w ‰ƷƏĺƓ =l ‰ƷƏĺѶѶ =l ƐƏ=l More: multiplicity fluctuations, minimum inter-nucleon distance, ... Ultimately: Propagate initial condition uncertainty into medium parameters 12 / 40
22. Pre-equilibrium Original work Free-streaming approximation • Expanding, noninteracting gas of massless partons • Sudden thermalization and switch to hydrodynamics at tunable time τfs Smooths out initial density, increases radial flow velocity ƷƏĺƏƐ =lņ1 ƷƐ =lņ1 ƷƑ =lņ1 ƐƏ=l 13 / 40
23. Viscous relativistic hydrodynamics • Conservation equations ∂µTµν = 0, Tµν = e uµuν − (P + Π)∆µν + πµν • Relaxation equations for viscous pressures τππ µν + πµν = 2ησµν − δπππµνθ + φ7π µ α πν α − τπππ µ α σν α + λπΠΠσµν τΠΠ + Π = −ζθ − δΠΠΠθ + λΠππµνσµν • Equation of state P = P(e): Hadron resonance gas (HRG) up to T = 165 MeV connected to lattice calculation (HotQCD Collaboration) at higher T • Hydrodynamic model implemented by the Ohio State University group 14 / 40
24. Speci c shear viscosity η/s • Shear viscosity η = fluid’s resistance to shear stress • Specific shear viscosity η/s = dimensionless ratio to entropy density • Small η/s → strongly-interacting • Theoretical conjecture: η/s ≥ 1/4π 0.08 Determination of (η/s)(T) is a primary goal of heavy-ion physics QCD matter compared to common fluids ƏĺƏ ƏĺƔ ƐĺƏ ƐĺƔ ƑĺƏ ƑĺƔ ƒĺƏ $ņ$1 ƏĺƐ Ɛ ƐƏ ņv )-|;u 1 ņƑ 1 Ƒ1 ;Ѵb†l 1 ņƑ 1 Ƒ1 r  !  ƐņƓ Tc = critical temperature, Pc = critical pressure HRG = hadron resonance gas, pQCD = perturbative QCD Shapes and locations of colored areas are approximate 15 / 40
25. Impact of shear viscosity Decreases collective behavior and flow (smaller vn) → Flow is primary experimental handle for η/s ƐƏ=l 7;-Ѵ ņvƷƏĺƑ ƐƏƏ ƑƏƏ ƒƏƏ ƒƔƏ $;lr;u-|†u;Œ;(œ ƷƐ =lņ1 Ƒ =lņ1 ƒ =lņ1 Ɣ =lņ1 16 / 40
26. Temperature-dependent viscosity parametrizations Original work ƐƔƏ ƑƏƏ ƑƔƏ ƒƏƏ $;lr;u-|†u;Œ;(œ ƏĺƏ ƏĺƑ ƏĺƓ Əĺѵ ņv "_;-u ƐƔƏ ƑƏƏ ƑƔƏ ƒƏƏ $;lr;u-|†u;Œ;(œ ƏĺƏƏ ƏĺƏƑ ƏĺƏƓ ƏĺƏѵ ƏĺƏѶ ƏĺƐƏ ņv †Ѵh Modified linear ansatz Parameters: minimum, slope, curvature (η/s)(T) = (η/s)min +(η/s)slope ·(T −Tc)· T Tc (η/s)crv Tc = 154 MeV (from HotQCD lattice EoS) Cauchy distribution (unnormalized) Parameters: max, width, location (T0) (ζ/s)(T) = (ζ/s)max 1 + T − (ζ/s)T0 (ζ/s)width ) 2 17 / 40
27. Particlization Original work Convert hydrodynamic medium → particles at tunable temperature Tswitch • Particlization hadronization or freeze-out • Particle species and momenta sampled from thermal hadron resonance gas — Cooper-Frye formula E dN d3p = g (2π)3 ∫ σ f (p) pµ d3σµ • System must be continuous across transition → distribution functions f (p) modified to account for viscous pressures (πµν and Π) • Masses of unstable resonances sampled from Breit-Wigner distribution 18 / 40
28. Boltzmann transport • Solves Boltzmann equation dfi (x, p) dt = Ci (x, p) • Hadronic scatterings and decays • Binary collisions and 2 → n processes — system cannot be too dense • Realistic chemical and kinetic freeze-out Implementation: Ultra-relativistic Quantum Molecular Dynamics (UrQMD) 19 / 40
29. Scaling up Original work • Need O(104) events to compute centrality dependence of observables • Average < 1 hour per event — still thousands total → Run on high-performance computational systems • National Energy Research Scientific Computing Center (NERSC) (DOE Office of Science) • Open Science Grid (OSG) • Developed automated workflow for generating large quantities of events • Used by most of Duke group • 20–40 million CPU hours used over past few years Images from LBL 20 / 40
30. Overview Input parameters QGP properties Model Heavy-ion collision spacetime evolution Gaussian process emulator Surrogate model Bayesian calibration Infer model parameters from data Posterior distribution Quantitative estimates of each parameter Experimental data Heavy-ion collision observables 20 / 40
31. Bayesian parameter estimation Infer the model inputs (x = properties) using data for the outputs (y = data) Bayes’ theorem posterior P(x|y) ∝ likelihood P(y|x) × prior P(x) Construct posterior probability distribution P(x|y) by Markov chain Monte Carlo sampling (MCMC) (weighted random walk through parameter space) 21 / 40
32. Example application: Gravitational waves Data LIGO gravitational wave strain properties of space-time in the strong-field, high-velocity regime and confirm predictions of general relativity for the nonlinear dynamics of highly disturbed black holes. II. OBSERVATION On September 14, 2015 at 09:50:45 UTC, the LIGO Hanford, WA, and Livingston, LA, observatories detected the coincident signal GW150914 shown in Fig. 1. The initial detection was made by low-latency searches for generic gravitational-wave transients [41] and was reported within three minutes of data acquisition [43]. Subsequently, matched-filter analyses that use relativistic models of com- pact binary waveforms [44] recovered GW150914 as the most significant event from each detector for the observa- tions reported here. Occurring within the 10-ms intersite Posterior distribution Black hole masses therefore add 10 parameters per instrument to the model used in the analysis. For validation purposes we also considered an independent method that assumes frequency- independent calibration errors [87], and obtained consistent results. III. RESULTS The results of the analysis using binary coalescence waveforms are posterior PDFs for the parameters describ- ing the GW signal and the model evidence. A summary is provided in Table I. For the model evidence, we quote (the logarithm of) the Bayes factor Bs=n ¼ Z=Zn , which is the evidence for a coherent signal hypothesis divided by that for (Gaussian) noise [5]. At the leading order, the Bayes factor and the optimal SNR ρ ¼ ½ P k hhM k jhM k iŠ1=2 are related by ln Bs=n ≈ ρ2=2 [88]. Before discussing parameter estimates in detail, we consider how the inference is affected by the choice of the compact-binary waveform model. From Table I, we see that the posterior estimates for each parameter are broadly consistent across the two models, despite the fact that they are based on different analytical approaches and that they include different aspects of BBH spin dynamics. The models’ logarithms of the Bayes factors, 288.7 Æ 0.2 and 290.3 Æ 0.1, are also comparable for both models: the data do not allow us to conclusively prefer one model over the other [89]. Therefore, we use both for the Overall column in Table I. We combine the posterior samples of both distributions with equal weight, in effect marginalizing over our choice of waveform model. These averaged results SNR of ρ ¼ 25.1þ1.7 −1.7 . This value is higher than the one reported by the search [1,3] because it is obtained using a finer sampling of (a larger) parameter space. GW150914’s source corresponds to a stellar-mass BBH with individual source-frame masses msource 1 ¼ 36þ5 −4 M⊙ and msource 2 ¼ 29þ4 −4 M⊙, as shown in Table I and Fig. 1. The two BHs are nearly equal mass. We bound the mass ratio to the range 0.66 ≤ q ≤ 1 with 90% probability. For comparison, the highest observed neutron star mass is 2.01 Æ 0.04M⊙ [91], and the conservative upper-limit for 22 / 40
33. Overview Input parameters QGP properties Model Heavy-ion collision spacetime evolution Gaussian process emulator Surrogate model Bayesian calibration Infer model parameters from data Posterior distribution Quantitative estimates of each parameter Experimental data Heavy-ion collision observables 22 / 40
34. Computationally expensive models MCMC sampling requires (103 hours/model evaluation) × (106 evaluations) ∼ 109 total hours But even largest NERSC allocations are O(107) hours Use model emulator (fast surrogate) • Evaluate model at O(102) design points in parameter space • Interpolate input-output behavior with Gaussian process emulator This work • Tailored general method of Bayesian parameter estimation with Gaussian process emulators for heavy-ion collisions • Original computational implementation 23 / 40
35. Parameter design How to distribute points in parameter space for optimal emulator performance? Factorial (uniform grid) Required number of points grows exponentially — not viable in high dimensions Random Tends to create large gaps and tight clusters Latin hypercube • Semi-random, space-filling • Required number of points grows linearly • “Efficient scaffolding” for emulator • This work: 500 points in 14 dimensions 24 / 40
36. Gaussian processes Ə Ɛ Ƒ ƒ Ɠ mr†| Ƒ Ɛ Ə Ɛ Ƒ †|r†| !-m7ol=†m1|bomv Ə Ɛ Ƒ ƒ Ɠ mr†| om7b|bom;7om -=;‰mobv;Ѵ;vvrobm|v ;-mru;7b1|bom &m1;u|-bm|‹ $u-bmbm]7-|- Ə Ɛ Ƒ ƒ Ɠ mr†| om7b|bom;7om l-m‹mobv‹robm|v • Flexible, non-parametric interpolation or regression (n-dimensional) • Quantifies uncertainty of predictions → Ideal as a model emulator 25 / 40
37. Emulator validation: Example observable Original work ƑƏƏ ƓƏƏ ѵƏƏ ѶƏƏ ƐƏƏƏ ƐƑƏƏ l†Ѵ-|ouru;7b1|bom ƑƏƏ ƓƏƏ ѵƏƏ ѶƏƏ ƐƏƏƏ ƐƑƏƏ o7;Ѵ1-Ѵ1†Ѵ-|bom 71_ ņ7 ƑƏ ƒƏѷ ƒ Ƒ Ɛ Ə Ɛ Ƒ ƒ oul-ѴbŒ;7u;vb7†-Ѵv ƐƏ ƑƔ ƔƏ ƕƔ ƖƏ oul-Ѵt†-m|bѴ;v Ideally, normalized residuals have standard normal distribution ypred − ycalc σpred ∼ N(0, 1) Check visually using boxplots 26 / 40
38. Emulator validation: All observables Original work 1_ $   r r$ r $ rr $ r$ ņ r$ ˆƑ ˆƒ ˆƓ Ƒ Ɛ Ə Ɛ Ƒ oul-ѴbŒ;7u;vb7†-Ѵv Ə Ɣ ƐƏ ƐƔ !"ѷ;uuou ƐƏ ƑƔ ƔƏ ƕƔ ƖƏ oul-Ѵt†-m|bѴ;v Conclusion: Emulator faithfully captures its predictive uncertainty Acceptable as long as it is accounted for in final parameter estimates 27 / 40
39. Bayesian model calibration Bayes’ theorem posterior ∝ likelihood × prior Prior: flat in design space Likelihood (multivariate normal distribution): P(D|x) = 1 (2π)m det Σ exp − 1 2 [ym(x) − ye]TΣ−1[ym(x) − ye] (Notation: subscript e → experiment, m → model) Σ = covariance matrix = Σe + Σm • Experiment: Σe = Σstat e (diagonal) + Σsys e (non-diagonal) • Model: Σm from GP emulator (predictive uncertainty + stat. fluctuations) 28 / 40
40. Uncertainty correlation matrices Original work 1_ $   r r$ r $ rr $ r$ ņ r$ ˆƑ ˆƒ ˆƓ o7;ѴŐ;l†Ѵ-|ouő 1_ $   r r$ r $ rr $ r$ ņ r$ ˆƑ ˆƒ ˆƓ 1_ $   r r$ r $ rr $ r$ ņ r$ ˆƑ ˆƒ ˆƓ Šr;ubl;m| ƐĺƏ ƏĺƔ ƏĺƏ ƏĺƔ ƐĺƏ ouu;Ѵ-|bom From empirical model correlations Assumed systematic error correlation structure 29 / 40
41. Overview Input parameters QGP properties Model Heavy-ion collision spacetime evolution Gaussian process emulator Surrogate model Bayesian calibration Infer model parameters from data Posterior distribution Quantitative estimates of each parameter Experimental data Heavy-ion collision observables 29 / 40
42. Previous and current work Published Bayesian analyses • JEB et al., PRC 91 054910 (2015), arXiv:1502.00339 • JEB et al., PRC 94 024907 (2016), arXiv:1605.03954 This work New and improved — to be published 30 / 40
43. Ə ƑƏ ƓƏ ѵƏ ѶƏ ƐƏƏ ƐƏƐ ƐƏƑ ƐƏƒ ƐƏƓ ƐƏƔ 71_ ņ7 ķ 7ņ7‹ķ 7 $ ņ7 Œ;(œ 1_ $  r 0Ŋ0Ƒĺƕѵ$;( Ə ƑƏ ƓƏ ѵƏ ѶƏ ƐƏƏ ƐƏƐ ƐƏƑ ƐƏƒ ƐƏƓ ƐƏƔ 1_ $  r +b;Ѵ7v 0Ŋ0ƔĺƏƑ$;( Ə ƑƏ ƓƏ ѵƏ ѶƏ ƏĺƏ ƏĺƔ ƐĺƏ ƐĺƔ r$ Œ;(œ  r Ə ƑƏ ƓƏ ѵƏ ѶƏ ƏĺƏ ƏĺƔ ƐĺƏ ƐĺƔ  r ;-mr$ Ə ƑƏ ƓƏ ѵƏ ѶƏ ƏĺƏƏ ƏĺƏƐ ƏĺƏƑ ƏĺƏƒ ƏĺƏƓ r$ ņ r$ Ə ƑƏ ƓƏ ѵƏ ѶƏ ƏĺƏƏ ƏĺƏƐ ƏĺƏƑ ƏĺƏƒ ƏĺƏƓ ;-mr$ =Ѵ†1|†-|bomv Ə ƑƏ ƓƏ ѵƏ ѶƏ ;m|u-Ѵb|‹ѷ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ˆm ŔƑŕ ˆƑ ˆƒ ˆƓ Ə ƑƏ ƓƏ ѵƏ ѶƏ ;m|u-Ѵb|‹ѷ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ˆƑ ˆƒ ˆƓ Ѵo‰1†l†Ѵ-m|v Ə ƑƏ ƓƏ ѵƏ ѶƏ ƐƏƏ ƐƏƐ ƐƏƑ ƐƏƒ ƐƏƓ ƐƏƔ 71_ ņ7 ķ 7ņ7‹ķ 7 $ ņ7 Œ;(œ 1_ $  r 0Ŋ0Ƒĺƕѵ$;( Ə ƑƏ ƓƏ ѵƏ ѶƏ ƐƏƏ ƐƏƐ ƐƏƑ ƐƏƒ ƐƏƓ ƐƏƔ 1_ $  r +b;Ѵ7v 0Ŋ0ƔĺƏƑ$;( Ə ƑƏ ƓƏ ѵƏ ѶƏ ƏĺƏ ƏĺƔ ƐĺƏ ƐĺƔ r$ Œ;(œ  r Ə ƑƏ ƓƏ ѵƏ ѶƏ ƏĺƏ ƏĺƔ ƐĺƏ ƐĺƔ  r ;-mr$ Ə ƑƏ ƓƏ ѵƏ ѶƏ ƏĺƏƏ ƏĺƏƐ ƏĺƏƑ ƏĺƏƒ ƏĺƏƓ r$ ņ r$ Ə ƑƏ ƓƏ ѵƏ ѶƏ ƏĺƏƏ ƏĺƏƐ ƏĺƏƑ ƏĺƏƒ ƏĺƏƓ ;-mr$ =Ѵ†1|†-|bomv Ə ƑƏ ƓƏ ѵƏ ѶƏ ;m|u-Ѵb|‹ѷ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ˆm ŔƑŕ ˆƑ ˆƒ ˆƓ Ə ƑƏ ƓƏ ѵƏ ѶƏ ;m|u-Ѵb|‹ѷ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ˆƑ ˆƒ ˆƓ Ѵo‰1†l†Ѵ-m|v Prior (training data) Posterior (samples)
44. Ѷ ƐƓ ƑƏ oul Ƒĺƕѵ$;( ƐƒĺƖƳƐĺƑ ƐĺƐ ƐƏĺƏ ƐƕĺƔ ƑƔĺƏ oul ƔĺƏƑ$;( ƐѶĺƔƳƐĺѶ Ɛĺƕ ƏĺƔ ƏĺƏ ƏĺƔ r ƏĺƏƏѵƳƏĺƏƕѶ ƏĺƏƕѶ Ə Ɛ Ƒ =Ѵ†1| ƏĺƖƏƳƏĺƑƓ ƏĺƑƕ ƏĺƓ Əĺƕ ƐĺƏ ‰ Œ=lœ ƏĺƖѵƳƏĺƏƓ ƏĺƏƔ ƏĺƏ ƐĺƑ ƐĺƔ Ɛĺƕ 7 lbm Œ=lœ ƐĺƑѶƳƏĺƓƑ ƏĺƔƒ ƏĺƏƏ ƏĺƕƔ ƐĺƔƏ =v Œ=lņ1œ ƐĺƐѵƳƏĺƑƖ ƏĺƑƔ ƏĺƏ ƏĺƐ ƏĺƑ ņv lbm ƏĺƏѶƔƳƏĺƏƑѵ ƏĺƏƑƔ Ə Ɠ Ѷ ņv vѴor; Œ;( Ɛœ ƏĺѶƒƳƏĺѶƒ ƏĺѶƒ Ɛ Ə Ɛ ņv 1uˆ ƏĺƒƕƳƏĺƕƖ Əĺѵƒ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ņv l-Š ƏĺƏƒƕƳƏĺƏƓƏ ƏĺƏƑƑ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ņv ‰b7|_ Œ;(œ ƏĺƏƑƖƳƏĺƏƓƔ ƏĺƏƑѵ ƏĺƐƔƏ ƏĺƐƕƔ ƏĺƑƏƏ ņv $Ə Œ;(œ ƏĺƐƕƕƳƏĺƏƑƒ ƏĺƏƑƐ ƏĺƐƒƔ ƏĺƐƔƏ ƏĺƐѵƔ $ v‰b|1_ Œ;(œ ƏĺƐƔƑƳƏĺƏƏƒ ƏĺƏƏƒ Ѷ ƐƓ ƑƏ oul Ƒĺƕѵ$;( ƏĺƏ ƏĺƑ ƏĺƓ lo7;Ѵ v‹v ƐƏĺƏ ƐƕĺƔ ƑƔĺƏ oul ƔĺƏƑ$;( ƏĺƔ ƏĺƏ ƏĺƔ r Ə Ɛ Ƒ =Ѵ†1| ƏĺƓ Əĺƕ ƐĺƏ ‰ Œ=lœ ƏĺƏ ƐĺƑ ƐĺƔƐĺƕ 7 lbm Œ=lœ ƏĺƏƏ ƏĺƕƔ ƐĺƔƏ =v Œ=lņ1œ ƏĺƏ ƏĺƐ ƏĺƑ ņv lbm Ə Ɠ Ѷ ņv vѴor; Œ;( Ɛœ Ɛ Ə Ɛ ņv 1uˆ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ņv l-Š ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ņv ‰b7|_ Œ;(œ ƏĺƐƔƏ ƏĺƐƕƔ ƏĺƑƏƏ ņv $Ə Œ;(œ ƏĺƐƒƔ ƏĺƐƔƏ ƏĺƐѵƔ $ v‰b|1_ Œ;(œ ƏĺƏ ƏĺƑ ƏĺƓ lo7;Ѵ v‹v ƏĺƐƏƳƏĺƏƖ ƏĺƏѶ Posterior distribution Diagonal subplots Marginal distributions for each parameter Quantitative estimates Medians with 90% credible intervals Off-diagonals Joint marginal distributions between pairs of parameters (correlations) 32 / 40
45. Hot and Dense QCD Matter Unraveling the Mysteries of the Strongly Interacting Quark-Gluon-Plasma A Community White Paper on the Future of Relativistic Heavy-Ion Physics in the US “ The next 5–10 years of the US relativistic heavy-ion program will deliver...the quantitative determination of the transport coefficients of the Quark Gluon Plasma, such as the temperature de- pendent shear-viscosity to entropy-density ratio (η/s)(T) ... ” (Published 2012)
46. Posterior distribution: Speci c shear viscosity ƏĺƏ ƏĺƐ ƏĺƑ ņv lbm ƏĺƏѶƔƳƏĺƏƑѵ ƏĺƏƑƔ Ə Ɠ Ѷ ņv vѴor; Œ;( Ɛœ ƏĺѶƒƳƏĺѶƒ ƏĺѶƒ ƏĺƏ ƏĺƐ ƏĺƑ ņv lbm Ɛ Ə Ɛ ņv 1uˆ Ə Ɠ Ѷ ņv vѴor; Œ;( Ɛœ Ɛ Ə Ɛ ņv 1uˆ ƏĺƒƕƳƏĺƕƖ Əĺѵƒ (η/s)(T) = (η/s)min +(η/s)slope·(T −Tc)· T Tc (η/s)crv ƐƔƏ ƑƏƏ ƑƔƏ ƒƏƏ $;lr;u-|†u;Œ;(œ ƏĺƏ ƏĺƐ ƏĺƑ Əĺƒ ƏĺƓ ņv ƐņƓ ov|;uboul;7b-m ƖƏѷ1u;7b0Ѵ;u;]bom • (η/s)min ≈ 1/4π • Best constrained near T ∼ 175 MeV • Likely increases with T 34 / 40
47. Posterior distribution: Speci c bulk viscosity ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ņv l-Š ƏĺƏƒƕƳƏĺƏƓƏ ƏĺƏƑƑ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ņv ‰b7|_ Œ;(œ ƏĺƏƑƖƳƏĺƏƓƔ ƏĺƏƑѵ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ņv l-Š ƏĺƐƔƏ ƏĺƐƕƔ ƏĺƑƏƏ ņv $Ə Œ;(œ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ņv ‰b7|_ Œ;(œ ƏĺƐƔƏ ƏĺƐƕƔ ƏĺƑƏƏ ņv $Ə Œ;(œ ƏĺƐƕƕƳƏĺƏƑƒ ƏĺƏƑƐ (ζ/s)(T) = (ζ/s)max 1 + T − (ζ/s)T0 (ζ/s)width) 2 ƐƔƏ ƑƏƏ ƑƔƏ ƒƏƏ $;lr;u-|†u;Œ;(œ ƏĺƏƏ ƏĺƏƑ ƏĺƏƓ ƏĺƏѵ ƏĺƏѶ ņv ov|;uboul;7b-m ƖƏѷ1u;7b0Ѵ;u;]bom • Moderate peak — “tall” or “wide”, not both • Location not well-constrained within prior range 35 / 40
48. Posterior distribution: Initial entropy deposition ƐĺƏ ƏĺƔ ƏĺƏ ƏĺƔ ƐĺƏ r  !$ņ ŊѴ-vl- )o†m7;7 m†1Ѵ;om ƏĺƏƏѵƳƏĺƏƕѶ ƏĺƏƕѶ s ∝ Tp A + Tp B 2 1/p • Entropy deposition ∼ geometric mean of local nuclear density, s ∼ √ TATB • Determined simultaneously with QGP transport coefficients; mutual uncertainty accounted for • Corroborates behavior of successful models IP-Glasma and EKRT 36 / 40
49. Ѷ ƐƓ ƑƏ oul Ƒĺƕѵ$;( ƐƒĺƖƳƐĺƑ ƐĺƐ ƐƏĺƏ ƐƕĺƔ ƑƔĺƏ oul ƔĺƏƑ$;( ƐѶĺƔƳƐĺѶ Ɛĺƕ ƏĺƔ ƏĺƏ ƏĺƔ r ƏĺƏƏѵƳƏĺƏƕѶ ƏĺƏƕѶ Ə Ɛ Ƒ =Ѵ†1| ƏĺƖƏƳƏĺƑƓ ƏĺƑƕ ƏĺƓ Əĺƕ ƐĺƏ ‰ Œ=lœ ƏĺƖѵƳƏĺƏƓ ƏĺƏƔ ƏĺƏ ƐĺƑ ƐĺƔ Ɛĺƕ 7 lbm Œ=lœ ƐĺƑѶƳƏĺƓƑ ƏĺƔƒ ƏĺƏƏ ƏĺƕƔ ƐĺƔƏ =v Œ=lņ1œ ƐĺƐѵƳƏĺƑƖ ƏĺƑƔ ƏĺƏ ƏĺƐ ƏĺƑ ņv lbm ƏĺƏѶƔƳƏĺƏƑѵ ƏĺƏƑƔ Ə Ɠ Ѷ ņv vѴor; Œ;( Ɛœ ƏĺѶƒƳƏĺѶƒ ƏĺѶƒ Ɛ Ə Ɛ ņv 1uˆ ƏĺƒƕƳƏĺƕƖ Əĺѵƒ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ņv l-Š ƏĺƏƒƕƳƏĺƏƓƏ ƏĺƏƑƑ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ņv ‰b7|_ Œ;(œ ƏĺƏƑƖƳƏĺƏƓƔ ƏĺƏƑѵ ƏĺƐƔƏ ƏĺƐƕƔ ƏĺƑƏƏ ņv $Ə Œ;(œ ƏĺƐƕƕƳƏĺƏƑƒ ƏĺƏƑƐ ƏĺƐƒƔ ƏĺƐƔƏ ƏĺƐѵƔ $ v‰b|1_ Œ;(œ ƏĺƐƔƑƳƏĺƏƏƒ ƏĺƏƏƒ Ѷ ƐƓ ƑƏ oul Ƒĺƕѵ$;( ƏĺƏ ƏĺƑ ƏĺƓ lo7;Ѵ v‹v ƐƏĺƏ ƐƕĺƔ ƑƔĺƏ oul ƔĺƏƑ$;( ƏĺƔ ƏĺƏ ƏĺƔ r Ə Ɛ Ƒ =Ѵ†1| ƏĺƓ Əĺƕ ƐĺƏ ‰ Œ=lœ ƏĺƏ ƐĺƑ ƐĺƔƐĺƕ 7 lbm Œ=lœ ƏĺƏƏ ƏĺƕƔ ƐĺƔƏ =v Œ=lņ1œ ƏĺƏ ƏĺƐ ƏĺƑ ņv lbm Ə Ɠ Ѷ ņv vѴor; Œ;( Ɛœ Ɛ Ə Ɛ ņv 1uˆ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ņv l-Š ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ņv ‰b7|_ Œ;(œ ƏĺƐƔƏ ƏĺƐƕƔ ƏĺƑƏƏ ņv $Ə Œ;(œ ƏĺƐƒƔ ƏĺƐƔƏ ƏĺƐѵƔ $ v‰b|1_ Œ;(œ ƏĺƏ ƏĺƑ ƏĺƓ lo7;Ѵ v‹v ƏĺƐƏƳƏĺƏƖ ƏĺƏѶ Posterior distribution Much more information here! 37 / 40
50. Ѷ ƐƓ ƑƏ oul Ƒĺƕѵ$;( ƐƒĺƖƳƐĺƑ ƐĺƐ ƐƏĺƏ ƐƕĺƔ ƑƔĺƏ oul ƔĺƏƑ$;( ƐѶĺƔƳƐĺѶ Ɛĺƕ ƏĺƔ ƏĺƏ ƏĺƔ r ƏĺƏƏѵƳƏĺƏƕѶ ƏĺƏƕѶ Ə Ɛ Ƒ =Ѵ†1| ƏĺƖƏƳƏĺƑƓ ƏĺƑƕ ƏĺƓ Əĺƕ ƐĺƏ ‰ Œ=lœ ƏĺƖѵƳƏĺƏƓ ƏĺƏƔ ƏĺƏ ƐĺƑ ƐĺƔ Ɛĺƕ 7 lbm Œ=lœ ƐĺƑѶƳƏĺƓƑ ƏĺƔƒ ƏĺƏƏ ƏĺƕƔ ƐĺƔƏ =v Œ=lņ1œ ƐĺƐѵƳƏĺƑƖ ƏĺƑƔ ƏĺƏ ƏĺƐ ƏĺƑ ņv lbm ƏĺƏѶƔƳƏĺƏƑѵ ƏĺƏƑƔ Ə Ɠ Ѷ ņv vѴor; Œ;( Ɛœ ƏĺѶƒƳƏĺѶƒ ƏĺѶƒ Ɛ Ə Ɛ ņv 1uˆ ƏĺƒƕƳƏĺƕƖ Əĺѵƒ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ņv l-Š ƏĺƏƒƕƳƏĺƏƓƏ ƏĺƏƑƑ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ņv ‰b7|_ Œ;(œ ƏĺƏƑƖƳƏĺƏƓƔ ƏĺƏƑѵ ƏĺƐƔƏ ƏĺƐƕƔ ƏĺƑƏƏ ņv $Ə Œ;(œ ƏĺƐƕƕƳƏĺƏƑƒ ƏĺƏƑƐ ƏĺƐƒƔ ƏĺƐƔƏ ƏĺƐѵƔ $ v‰b|1_ Œ;(œ ƏĺƐƔƑƳƏĺƏƏƒ ƏĺƏƏƒ Ѷ ƐƓ ƑƏ oul Ƒĺƕѵ$;( ƏĺƏ ƏĺƑ ƏĺƓ lo7;Ѵ v‹v ƐƏĺƏ ƐƕĺƔ ƑƔĺƏ oul ƔĺƏƑ$;( ƏĺƔ ƏĺƏ ƏĺƔ r Ə Ɛ Ƒ =Ѵ†1| ƏĺƓ Əĺƕ ƐĺƏ ‰ Œ=lœ ƏĺƏ ƐĺƑ ƐĺƔƐĺƕ 7 lbm Œ=lœ ƏĺƏƏ ƏĺƕƔ ƐĺƔƏ =v Œ=lņ1œ ƏĺƏ ƏĺƐ ƏĺƑ ņv lbm Ə Ɠ Ѷ ņv vѴor; Œ;( Ɛœ Ɛ Ə Ɛ ņv 1uˆ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ņv l-Š ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ņv ‰b7|_ Œ;(œ ƏĺƐƔƏ ƏĺƐƕƔ ƏĺƑƏƏ ņv $Ə Œ;(œ ƏĺƐƒƔ ƏĺƐƔƏ ƏĺƐѵƔ $ v‰b|1_ Œ;(œ ƏĺƏ ƏĺƑ ƏĺƓ lo7;Ѵ v‹v ƏĺƐƏƳƏĺƏƖ ƏĺƏѶ Posterior distribution Much more information here! But what can the model do at the maximum posterior (“best-fit”) point? ↓ Should describe all calibration data + more observables 37 / 40
51. Ə ƑƏ ƓƏ ѵƏ ѶƏ ƐƏƐ ƐƏƑ ƐƏƒ ƐƏƓ ƐƏƔ 71_ ņ7 ķ 7ņ7‹ķ 7 $ ņ7 Œ;(œ 1_ $  r +b;Ѵ7v Ə ƑƏ ƓƏ ѵƏ ѶƏ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ƏĺƐƔ ˆm Ŕhŕ ˆƑ ŔƑŕ ˆƑ ŔƓŕ ˆƒ ŔƑŕ ˆƓ ŔƑŕ Ѵo‰1†l†Ѵ-m|v Ƒĺƕѵ$;( ƔĺƏƑ$;( ƼƐƏѷ Ə ƑƏ ƓƏ ѵƏ ѶƏ ƏĺƖ ƐĺƏ ƐĺƐ !-|bo ƼƐƏѷ Ə ƑƏ ƓƏ ѵƏ ѶƏ ƏĺƖ ƐĺƏ ƐĺƐ !-|bo Ə ƑƏ ƓƏ ѵƏ ѶƏ ƏĺƏ ƏĺƔ ƐĺƏ ƐĺƔ r$ Œ;(œ  r ;-mr$ Ə ƑƏ ƓƏ ѵƏ ѶƏ ƏĺƏƏ ƏĺƏƑ ƏĺƏƓ r$ ņ r$ ;-mr$ =Ѵ†1|†-|bomv ƼƐƏѷ Ə ƑƏ ƓƏ ѵƏ ѶƏ ;m|u-Ѵb|‹ѷ ƏĺƖ ƐĺƏ ƐĺƐ !-|bo ƼƐƏѷ Ə ƑƏ ƓƏ ѵƏ ѶƏ ;m|u-Ѵb|‹ѷ ƏĺƖ ƐĺƏ ƐĺƐ !-|bo Maximum a posteriori parameters Norm 13.94 (2.76 TeV) 18.38 (5.02 TeV) p 0.007 σfluct 0.918 w 0.956 fm dmin 1.27 fm τfs 1.16 fm/c η/s min 0.081 η/s slope 1.11 GeV−1 η/s crv −0.48 ζ/s max 0.052 ζ/s width 0.022 GeV ζ/s T0 183. MeV Tswitch 151. MeV 38 / 40
52. Cross-check observable: Flow correlations Correlation between event-by-event fluctuations of the magnitudes of flow harmonics m and n: SC(m, n) = v2 m v2 n − v2 m v2 n Empirically: Very sensitive to (η/s)(T) Model correctly describes signs and qualitative centrality dependence Ə Ƒ Ɠ ѵ Ѷ ƐƏ ƏĺƕƔ ƏĺƔƏ ƏĺƑƔ ƏĺƏƏ ƏĺƑƔ "Őlķmő Ɛ; ƕ ;m|u-Ѵ Ƒĺƕѵ$;( ƔĺƏƑ$;(Őru;7b1|bomő Ə ƑƏ ƓƏ ѵƏ Ɛ Ə Ɛ Ƒ Ɛ; ѵ bmbl†l0b-v "ŐƓķƑő "ŐƒķƑő Ə Ƒ Ɠ ѵ Ѷ ƐƏ ;m|u-Ѵb|‹ѷ ƏĺƏƔ ƏĺƏƏ ƏĺƏƔ ƏĺƐƏ ƏĺƐƔ "Őlķmőņ ˆƑ l ˆƑ m Ə ƑƏ ƓƏ ѵƏ ;m|u-Ѵb|‹ѷ ƏĺƑƔ ƏĺƏƏ ƏĺƑƔ ƏĺƔƏ ƏĺƕƔ ƐĺƏƏ "ŐƓķƑőņ ˆƑ Ɠ ˆƑ Ƒ "ŐƒķƑőņ ˆƑ ƒ ˆƑ Ƒ 39 / 40
53. Conclusion First estimates of fundamental QGP properties (and other related quantities) with well-defined uncertainties ƐƔƏ ƑƏƏ ƑƔƏ ƒƏƏ $;lr;u-|†u;Œ;(œ ƏĺƏ ƏĺƐ ƏĺƑ Əĺƒ ƏĺƓ ņv ƐņƓ "_;-uˆbv1ovb|‹ ov|;uboul;7b-m ƖƏѷ1u;7b0Ѵ;u;]bom ƐƔƏ ƑƏƏ ƑƔƏ ƒƏƏ $;lr;u-|†u;Œ;(œ ƏĺƏƏ ƏĺƏƑ ƏĺƏƓ ƏĺƏѵ ƏĺƏѶ ņv †Ѵhˆbv1ovb|‹ Method is general; tools are flexible and publicly available Has been and will be applied to other models and/or experimental data 40 / 40
54. TRENTo ansatz −2 −1 0 1 2 x [fm] 0.0 0.2 0.4 0.6 0.8 1.0 Thickness [fm−2] Arithmetic: p=1 Geometric: p=0 Harmonic: p= − 1 Participant × 0.3 Beam view x −8 −6 −4 −2 0 2 4 6 8 x [fm] 0 2 4 Thickness [fm¡2] Pb+Pb 2.76 TeV ~ Tmin < ~ T < ~ Tmax ¡1 < p < 1 p = 0 Generalized mean interpolates between min and max of local nuclear thickness Eikonal for p ≤ 0 s ∝ Tp A + Tp B 2 1/p =                    max(TA , TB) p → +∞ (TA + TB)/2 p = +1 (arithmetic) √ TATB p = 0 (geometric) 2 TATB/(TA + TB) p = −1 (harmonic) min(TA , TB) p → −∞ 1 / 10
55. TRENTo reproducing other models 0 1 2 3 4 ~ TA [fm¡2] 0 1 2 3 Entropy density [fm¡3] Gen. mean, p = ¡ 0: 67 KLN 0 1 2 3 4 ~ TA [fm¡2] Gen. mean, p = 0 EKRT 0 1 2 3 4 ~ TA [fm¡2] 1 fm¡2 2 fm¡2 ~ TB =3 fm¡2 Gen. mean, p = 1 Wounded nucleon Fit p to reproduce effective entropy deposition of other models s ∝ Tp A + Tp B 2 1/p 2 / 10
56. Equation of state ƐƏƏ ƑƏƏ ƒƏƏ ƓƏƏ $;lr;u-|†u;Œ;(œ Ə Ɛ Ƒ ƒ Ɠ Ɣ Ő; ƒőņ$Ɠ omm;1|bomu-m]; $u-1;-mol-Ѵ‹ ƐƏƏ ƑƏƏ ƒƏƏ ƓƏƏ $;lr;u-|†u;Œ;(œ ƏĺƐ ƏĺƑ Əĺƒ 1Ƒ v "r;;7o=vo†m7 ! -||b1; ‹0ub7 • Connect HRG trace anomaly to lattice (HotQCD) between 165–200 MeV • Compute pressure P(T) T4 = P0 T4 0 + ∫ T T0 dT Θµµ T 5 , T0 = 50 MeV 3 / 10
57. Resonance mass distributions ƏĺƔ ƐĺƏ ƐĺƔ ƑĺƏ lŒ;(œ uo0-0bѴb|‹ ŐƕƕƏő ŐƐƑƒƑő ŐƐƔƒƔő ƏĺƏ ƏĺƔ ƐĺƏ ƐĺƔ ƑĺƏ rŒ;(œ rƑ=ŐrőŒ-u0ĺ†mb|vœ ƵƐƏ ƵƑƔ $ƷƐƔƏ;( ŐƕƕƏőĹƳƐƑѷ ŐƐƑƒƑőĹ Ɩѷ ŐƐƔƒƔőĹƳƐѵѷ bmb|;‰b7|_ ,;uo‰b7|_ P(m) ∝ Γ(m) (m − m0)2 + Γ(m)2/4 , Γ(m) = Γ0 m − mmin m0 − mmin 4 / 10
58. Particlization: Viscous corrections ƏĺƏ ƏĺƐ ƏĺƑ Əĺƒ ƏĺƓ ƏĺƔ -Ɛ ņƏ Őbmr†|ő ƏĺƐ ƏĺƏ ƏĺƐ ƏĺƑ Əĺƒ ƏĺƓ ƏĺƔ †|r†|v "_;-uomѴ‹ -Ɛ ņƏ ņƏ ;ņ;Ə -Ƒ ņƏ ƏĺƏ ƏĺƐ ƏĺƑ Əĺƒ ƏĺƓ ƏĺƔ -Ɛ ņƏ Őbmr†|ő ƏĺƐ ƏĺƏ ƏĺƐ ƏĺƑ Əĺƒ ƏĺƓ ƏĺƔ $ƷƐƔƏ;( ѴѴ u;vom-m1;v "_;-u-m70†Ѵh Tµν = sp g ∫ d3p (2π)3 pµpν E f (p) = e uµuν − (P + Π)∆µν + πµν pi → pi = pi + j λij pj , λij = (λshear )ij + λbulkδij , (λshear )ij = τ 2η πij 5 / 10
59. Particlization: Parametric bulk viscous corrections ƐĺƏ ƏĺƔ ƏĺƏ ƏĺƔ ƐĺƏ ņƏ Ɛ Ə Ɛ Ƒ !;Ѵ-|bˆ;1_-m]; mņmƏ r ņ r Ə ņƏ ;ņ;Ə $ƷƐƔƏ;( ѴѴ u;vom-m1;v ƏĺƏ ƏĺƔ ƐĺƏ ƐĺƔ ƑĺƏ rŒ;(œ ƐƏ Ѷ ƐƏ ѵ ƐƏ Ɠ ƐƏ Ƒ ƐƏƏ =Őrő ƷƏ ƏĺƑƏ ƏĺƓƏ bomvķ$ƷƐƔƏ;( -u-l;|ub1 !$ Scale momentum and density to reproduce total kinetic pressure while holding energy density constant → two equations, two unknowns: P + Π = zbulk sp g ∫ d3p (2π)3 p2 3E f (p + λbulkp), e = zbulk sp g ∫ d3p (2π)3 E f (p + λbulkp) 6 / 10
60. Training a Gaussian process Ə Ɛ Ƒ ƒ Ɠ mr†| Ƒ Ɛ Ə Ɛ Ƒ †|r†| ˆ;u=b| ƷƏĺƏƒķ Ƒ m ƷƏĺƏƐ Ə Ɛ Ƒ ƒ Ɠ mr†| &m7;u=b| ƷƐƏķ Ƒ m ƷƏĺƑ Ə Ɛ Ƒ ƒ Ɠ mr†| -ŠѴbh;Ѵb_oo7 ƷƑĺƐƕķ Ƒ m ƷƏĺƏƖƓƒ  1|†-Ѵ=†m1|bom L(θ) = 1 (2π)d det Ktt exp − 1 2 yT t K−1 tt yt , k(xi , xj ) = σ2 f exp − 1 2 k xki − xkj k 2 + σ2 n δij 7 / 10
61. Principal component analysis (PCA) • Singular value decomposition: Y = UΣV T • Transform observables to linearly uncorrelated principal components: Z = YV Example for randomly generated data ƒ Ə ƒ ‹Ɛ ƒ Ə ƒ ‹Ƒ  ŒƐ ŐѶƏѷő ŒƑ ŐƑƏѷő ƒ Ə ƒ ŒƐ ƒ Ə ƒ ŒƑ 8 / 10
62. Realistic PCA vectors Original work 1_ $   r r$ r $ rr $ r$ ņ r$ ˆƑ ˆƒ ˆƓ ƏĺƐ ƏĺƏ ƏĺƐ ƏĺƑ Əĺƒ 1o;==b1b;m| ƐŐƓƖѷő ƑŐƑƒѷő ƒŐƐƔѷő Ɛ Ɠ ƕ ƐƏ †l0;uo= ƏĺƏ ƏĺƑ ƏĺƓ Əĺѵ ƏĺѶ ƐĺƏ †l†Ѵ-|bˆ;;ŠrѴ-bm;7ˆ-ub-m1;=u-1|bom 9 / 10
63. Uncertainty quanti cation details • Assume ye = ym(x) + , ∼ N(0, Σ) • Covariance matrix: Σ = Σe + Σm • Experimental covariance: Σe = Σstat e + Σsys e In general Σij = ρijσi σj, where ρij are correlation coefficients: ρstat ij = δij (definition), ρsys ij = exp − 1 2 ci − cj 2 (assumption) • Model covariance from Gaussian process predictive uncertainty + systematic error parameter: Σm = V ΣGP m,z + (σsys m )2I V T = ΣGP m + (σsys m )2V TV 10 / 10