Bayesian methods for effective field theories

Transcription

1 Bayesian methods for effective field theories Sarah Wesolowski The Ohio State University INT Bayes Workshop 2016 a1! Prior! Posterior! True value! Dick Furnstahl (OSU) Daniel Phillips (OU) 0! a0! BUQEYE Collaboration! Arbin Thapilaya (OU) Natalie Klco (OU -> UW) Harald Griesshammer (GWU)

2 Outline Introduction to effective field theories The parameter estimation problem Diagnostics for parameter estimation Model problems to the chiral EFT problem Conclusions and questions

3 Uncertainty quantification for modern nuclear calculations Era of precision observable calculations Ground-state energies for even oxygen isotopes Hamiltonian Calculation method uncertainty Experiment Observable Where are the theory errors? Hergert et al. PRL 110, (2013)

4 Uncertainty quantification for modern nuclear calculations Era of precision observable calculations Ground-state energies for even oxygen isotopes Include all sources of uncertainty Theory uncertainty from input interaction. Current efforts on interaction uncertainty: Experiment Navarro Pérez et al., JPG 42, (2015) Epelbaum et al., EPJA 51, 5 (2015) Carlsson et al., PRX 6, (2016) Focus on Bayesian approach: SW et al., JPG (2016) Hergert et al. PRL 110, (2013) RJF et al., PRC 92, (2015) RJF et al., JPG 42, (2015)

5 Effective field theories and you Relevant degrees of freedom depend on resolution Construct most general theory consistent with symmetries Model-independent predictions Scale separation leads to expansion parameter Natural-sized low-energy constants (LECs): controlled expansion

6 Effective field theories and you Relevant degrees of freedom depend on resolution Construct most general theory consistent with symmetries Model-independent predictions Scale separation leads to expansion parameter Natural-sized low-energy constants (LECs): controlled expansion Can estimate theoretical uncertainties!

7 Parameter estimation problem (schematic) Fit coefficients (LECs) of this function V (x) V 0 kx n=0 a n x n x = p b Small expansion parameter x

8 Parameter estimation problem (schematic) These are O(1) Fit coefficients (LECs) of this function V (x) V 0 kx a n x n x = p b n=0 Solution method: e.g., Lippman-Schwinger Small expansion parameter x Observable: match to experiment (x)

9 Parameter estimation problem (schematic) Uncertainty from fitting to data V (x) V 0 kx a n x n kx max + V 0 a n x n n=0 n=k+1 Solution method: e.g., Lippman-Schwinger All higher powers we left out of calculation Numerical errors in calculation method (x)

10 Parameter estimation problem (schematic) Order to which we calculate V (x) V 0 kx a n x n kx max + V 0 a n x n n=0 n=k+1 Solution method: e.g., Lippman-Schwinger Account for corrections above k with this: marginalization! (x)

11 Issues to address Uncertainty budget an s from fit to data truncation error calculation method kx V (x; a,k)=v 0 a n x n n=0 + k (x) x = p b What could possibly go wrong? Form of V at higher order may not be known V0 and Λb may not be properly identified Can have unnaturally large an s Regulator artifacts (won t go into detail here)

12 What was done before Vary piece of calculation: Cutoff regulator Λ Convergence with order unclear Underestimates uncertainty No statistical interpretation Fits of V (x; a,k) to one type of (x) NLO, N 2 LO, N 3 LO Issues with past analyses: What is the LEC uncertainty from fit to data? Optimization procedures opaque (priors?) Underfitting avoided by limiting range of data Epelbaum et al., Rev. Mod. Phys

13 What was done before Vary piece of calculation: Cutoff regulator Λ Convergence with order unclear Underestimates uncertainty Use V (x; a,k) to predict a (x) not in fit NLO, N 2 LO No statistical interpretation Issues with past analyses: What is the LEC uncertainty from fit to data? Optimization procedures opaque (priors?) Underfitting avoided by limiting range of data Epelbaum et al., Rev. Mod. Phys One proton-to-deuteron polarization transfer coefficient

14 More modern approaches: uncertainty from data Covariance analysis of LECs a from EFT fit to data Carlsson et al., Phys. Rev. X 6, (2016) Are they really Gaussian? Inclusion of truncation error in fits? (Birge factors) Simultaneous or separate fits to all types of data?

15 More modern approaches: truncation uncertainty (x; k) = 0 kx n=0 c n x n + k (x) Epelbaum, et al., Eur. Phys. E. J. Epelbaum, A 51, 53 H. (2015) Krebs, and and U.-G. PRL Meißner 115, [arxiv: ] (2015) Given these order-by-order calculations What is this? Estimate next-order correction c k+1 = max{c 0,c 1,...,c k } k(x) 0 c k+1 x k+1 Procedure for error estimates based on order-by-order convergence

16 More modern approaches: truncation uncertainty (x; k) = 0 kx n=0 c n x n + k (x) Furnstahl, Phillips, SW, Klco, Phys. Rev. C (2015) 95% DOB Given these order-by-order calculations What is this? 68% DOB Estimate next-order correction c k+1 = max{c 0,c 1,...,c k } k(x) 0 c k+1 x k+1 Bayesian derivation gives statistical interpretation and full pdf See talk of H. Griesshammer from last week for applications to other EFTs!

17 Set up Bayesian approach Goal: estimate coefficient posterior pr(a D,k,k max, I) I: any other information Vector of LECs {a 0, a 1, a k } Data k : truncation order k max : higher-order effects Known prior information: pr(a ā) pr(ā) Naturalness prior Prior for naturalness parameter

18 Set up Bayesian approach Goal: estimate coefficient posterior Integrate in higher-order coefficients {a k+1, a kmax } Z pr(a D,k,k max )= da marg pr(a, a marg D,k,k max ) Bayes theorem / Z da marg pr(d a, a marg,k,k max )pr(a, a marg k, k max )

19 Set up Bayesian approach Goal: estimate coefficient posterior Integrate in higher-order coefficients {a k+1, a kmax } Z pr(a D,k,k max )= da marg pr(a, a marg D,k,k max ) Bayes theorem / Z da marg pr(d a, a marg,k,k max )pr(a, a marg k, k max ) / Z Integrate in naturalness parameter da marg dā pr(d a, a marg,k,k max )pr(a, a marg ā,k,k max )pr(ā k, k max )

20 Diagnostic framework Set up Guidance Bayesian parameter estimation for effective field theories SW, Klco, Furnstahl, Phillips, Thapilaya J.Phys. G 43, (2016) Parameter estimation Validation Predictions

21 Diagnostic framework Set up Guidance Specify model to be fit: Prior information V (x) V 0 naturalness, symmetries, etc. kx a n x n n=0 Parameter estimation Validation functional form, hyperparameters pr(a ā) e a2 /2ā 2 ( 1 ā ln(ā pr(ā) = > /ā < ) when ā 2 [ā <, ā > ] 0 otherwise Predictions Likelihood to be used pr(d a,k,k max ) e 2 /2

22 Diagnostic framework Set up Guidance Real-life example: V V ~p 0 )= C e (1S0) + C (1S0) (p 2 + p 02 contact (~p, ~p 0 ) ) + D(1S0) 1 p2 p 02 + D(1S0) 2 (p4 + p 04 )+... Parameter estimation C n = scaled { e C (1S0),C (1S0),D 1 (1S0),D2 (1S0),...} scaled correct f, 4, b factored out Validation pr(c n ā, I) / e C2 n /2ā2 Predictions Added complication: we don t know b a priori

23 Diagnostic framework Set up Information from available data D Calculate Bayes factors for different orders: Guidance Parameter estimation Validation pr(m k D) pr(m k+1 D) = R da1 pr(d a 1,M k )pr(a 1 M k ) R da2 pr(d a 2,M k+1 )pr(a 2 M k+1 ) Naturalness prior Saturation Predictions Least-squares: uniform prior

24 Diagnostic framework Set up Guidance Information from available data D Calculate posterior for naturalness hyperparameter: Z pr(ā D,I)= da pr(d a,i)pr(a ā,i)pr(ā I) Parameter estimation Validation Predictions pr(ā D1 (5%),k=2,kmax = 2) Question for statisticians: how best to use this? ā

25 Diagnostic framework Set up Guidance Parameter estimation Detour to a model problem to explore generic features in EFT fitting Validation Predictions

26 Detour: parameter estimation in linear case Consider a model observable [Schindler/Phillips (2009)]: Real World : g(x) = 1 x tan Generate synthetic data 1.2 5% relative Gaussian error Expansion: g(x) = x +2.47x g(x) 0.8 Model EFT : g(x) = kx n=0 a n x n 0.4 Try to extract coefficients of model x Parameter estimation question: Are estimates consistent with underlying expansion?

27 Detour: parameter estimation in linear case Posterior Likelihood Prior pr(a D, k, k max ) / pr(d a,k,k max ) pr(a k, k max ) Quantity of interest Controlled by data ~ exp[-χ 2 /2] Prior knowledge

28 Detour: parameter estimation in linear case Posterior Likelihood Prior pr(a D, k, k max ) / pr(d a,k,k max ) pr(a k, k max ) Quantity of interest Controlled by data ~ exp[-χ 2 /2] Prior knowledge Compare results for different prior assumptions, take kmax = k pr(a D, k) / e 2 /2 1 pr(a D, k) / e 2 /2 e a2 /2ā 2

29 Detour: parameter estimation in linear case Posterior Likelihood Prior pr(a D, k, k max ) / pr(d a,k,k max ) pr(a k, k max ) Quantity of interest Controlled by data ~ exp[-χ 2 /2] Prior knowledge Compare results for different prior assumptions, take kmax = k pr(a D, k) / e 2 /2 1 pr(a D, k) / e 2 /2 e a2 /2ā 2 Highly unstable for k > 2 Errors large and unstable χ 2 /dof looks good! Stable with increasing k Errors are reasonable Saturates at k = 2

30 Detour: parameter estimation in linear case Posterior Likelihood Prior pr(a D, k, k max ) / pr(d a,k,k max ) pr(a k, k max ) Quantity of interest pr(a D, k) / e 2 /2 1 Controlled by data ~ exp[-χ 2 /2] pr(a D, k) / e Prior knowledge 2 /2 e a2 /2ā 2 Overfitting - fine-tuned to reproduce data in fit range

31 Detour: parameter estimation in linear case Posterior Likelihood Prior pr(a D, k, k max ) / pr(d a,k,k max ) pr(a k, k max ) Quantity of interest pr(a D, k) / e 2 /2 1 Controlled by data ~ exp[-χ 2 /2] pr(a D, k) / e Prior knowledge 2 /2 e a2 /2ā 2 Underfitting - model too simplistic for data Naturalness prior makes no difference

32 Detour: parameter estimation in linear case pr(a D, k) / e 2 /2 1 pr(a D, k) / e 2 /2 e a2 /2ā 2 Unconstrained higher orders distort lower orders Naturalness prior restricts correlations a a a3 a a a a a a a a a a a 3 10

33 Diagnostic framework Set up Guidance What about real life? Estimate posterior and associated diagnostics EKM N 2 LO 1 S0 channel Parameter estimation Validation Predictions Data used: synthetic phase shifts at N 2 LO

34 Diagnostic framework Set up EKM N 3 LO 1 S0 channel Guidance Large correlations Parameter estimation Validation Predictions Data used: synthetic phase shifts at N 3 LO

35 Why are the correlations so large? N 3 LO s-wave coefficients on-shell E.g., single linear combination of D 1 (1S0) and D 2 (1S0): D 1 (1S0) p2 p 02 + D 2 (1S0) (p4 + p 04 )= 1 4 D1 (1S0) +2D2 (1S0) (p2 + p 02 ) 2 N 3 LO NN contacts: p D1 (1S0) 2D 2 (1S0) (p2 p 02 ) 2 All s-wave LECs from EKM fits to Nijmegen phase shifts:

36 Why are the correlations so large? N 3 LO s-wave coefficients on-shell E.g., single linear combination of D 1 (1S0) and D 2 (1S0): D 1 (1S0) p2 p 02 + D 2 (1S0) (p4 + p 04 )= 1 4 D1 (1S0) +2D2 (1S0) (p2 + p 02 ) 2 N 3 LO NN contacts: p D1 (1S0) 2D 2 (1S0) (p2 p 02 ) 2 All s-wave LECs from EKM fits to Nijmegen phase shifts: enhancement of x4 could indicate overfitting: explore further!

37 Why are the correlations so large? N 3 LO s-wave coefficients on-shell Is there overfitting once we go to N 3 LO in the s-waves? N 3 LO: fine-tuned, large correlations Freeze D 2 (1S0), well-behaved

38 Diagnostic framework Set up Guidance Fluctuations between subdivided data sets Accumulate data, likelihood-prior competition Check power-law behavior (Lepage plots) Parameter estimation Validation Parameter estimates in each dimension for different data on same mesh in x Predictions

39 Diagnostic framework Set up Guidance Parameter estimation Validation gth(x) d(x) Fluctuations between subdivided data sets Accumulate data, likelihood-prior competition Check power-law behavior (Lepage plots) k =0, k max =3 k =1, k max =3 k =2, k max =3 Data error Theory error Predictions x Increasing slope in error as we increase order: good sign!

40 Diagnostic framework Set up Propagate errors to predictions! Guidance Parameter estimation Validation Predictions Also can calculate truncation uncertainty

41 Summary Recently completed diagnostic framework for generic EFTs S. Wesolowski et al., "Bayesian parameter estimation for effective field theories", J. Phys G 43, (2016) Parameter estimation for EKM NN interactions in progress Explore fitting to phase shifts vs. full cross sections Nearing computational wall for evidence integrals Exploring possible redundancy in s-waves Find Bayesian interpretations for current EFT fitting methods Naive least-squares is never used by practitioners- what are assumptions/priors? Parallelization of MCMC calculations Nested sampling? Next steps Mixture models as alternative to evidence

42 Questions Alternatives to Bayes factors? Relationship to common practices in EFTs What if form of 2 = How could we determine? NX (xi ; a,k) d i 2 i + (C xx k+1 ) 2 Model selection in pionless EFT sandbox. Where would GP emulators be useful? Should we be orthogonalizing the posterior? i=0 V (x; a,k)=v 0 b kx n=0 a n x n 2 Fit Cx so χ2/dof = 1: Birge factor? is not known at higher k?

43 Backup: Prototype EFT V (x) =V 0 kx A n (x) x n with x = p b n=0 Natural-sized coefficients Small expansion parameter x A n (x) =a n (µ)+f n (x, µ) a n, f n 1 when µ b f n (x, µ) encodes IR physics at order n

44 Backup: Marginalize higher-order corrections Marginalization over higher-order effects k =0, k max =3 g(x) =a 0 + a 1 x + a 2 x 2 + a 3 x 3 pr(a D, I) = Z da marg pr(a, a marg D, I) pr(a D, I) / Z da marg pr(d a, a marg,i)pr(a, a marg I) Correlated higher-order errors, see [arxiv:hep-ph/ ] and [arxiv: ]