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Moving forward with Bayesian parameter estimation for chiral EFT
Sarah WesolowskiProgress in ab initio techniques in nuclear physics, TRIUMF 2019
Dick Furnstahl (OSU)Jordan Melendez (OSU)
Matt Pratola (OSU statistics)Daniel Phillips (OU)
a0!
a1!
0!
BUQEYE Collaboration!
Prior!Posterior!True value!
Supported in part by NSF, DOE, and the SciDAC NUCLEI project
Precision calculations go to the zoo
Hergert et al., Journal of Physics: Conference Series (2018) Ekstrom et al. PRC 91 051301 (2015)
• A zoo of modern chiral effective field theory (EFT) interactions is now available (cf. Robert’s talk)• Progress continues on two major fronts
• chiral interactions• many-body calculations
• What causes differences when using different interactions?• regulator dependence• renormalizability/ power counting? (J. Yang later today)• missing uncertainty quantification• 3NFs: are they overfit? inconsistent? need Delta’s?• you can probably think of more…
Zoo of precise many-body methods Zoo of interactions (maybe only 2 really...)
Recent progress in fitting interactions
• Two-body: nucleon-nucleon (NN) sector• LENPIC few/many-body NN calculations
Binder et al. PRC 98, 014002 (2018)
• Ongoing work at Chalmers group beyond N2LO sim/sepCarlsson et al., PRX 6 011019 (2016)
• Local chiral potentials with explicit DeltasPiarulli at al., PRC 91, 024003 (2015)
• BUQEYE: recent Bayesian exploration (in partial waves)Wesolowski et al., JPG 46, 045102 (2019)
• !N sector: coefficients from Roy-Steiner analysisHoferichter at al., Phys. Rev. Lett. 115, 192301 (2015)
• Three-body: NNN sector• LENPIC fits of cD and cE + few/many-body calculations
Kai, Hermann, and Robert’s talks;, Epelbaum et al., arxiv:1807.02848• More work with light nuclei in QMC
Baroni et al., PRC 98 (2018) no.4, 044003Piarulli et al., PRL 120 (2018) no.5, 052503Lynn et al., PRL 116 (2016) 062501
• BUQEYE+Chalmers to contrain cD and cE (Bayesian methods)
Case studies in NN parameter estimation
• Question: what is the value added of using Bayesian methods, especially in the NN sector?• Answer: can explore issues in a controlled setting• Prior assumptions such as naturalness constraints• Comparison of Bayes vs. traditional fitting/regression• Presentation of “sloppy directions” and redundancies• Impact and inclusion of truncation errors
• Observable calculations are quick, as opposed to few-body sector, so understand in NN sector first• Ultimate products: making progress on UQ +
understanding and testing physics issues
For *all* the details see:Wesolowski et al., J Phys G 46, 045102 (2019)
Constraining and testing an EFT: the user’s guide1. Outline EFT and all sources of uncertainty2. Be as explicit (i.e. Bayesian) as possible
(That doesn’t technically mean changing your procedure!)
3. Check your results. Validation is key.
For even more details see:Wesolowski et al., J. Phys. G 43, 074001 (2016)
Setup: sources of uncertainty
• Method/numerical errors (often well-controlled)• Theoretical errors (don’t just exist in EFT)• EFTs: truncation uncertainty• EFTs: Correlations between observables (3NFs)
• Experimental errors, correlated or uncorrelated
yexp
= yth
+ �yth
+ �yexp
+ �ynum.
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Put together everything you know about each source: in Bayes language these are “priors”
Propaganda: Follow the Bayes Way
Why use Bayesian statistics? • Parameter estimation: conventional optimization recovered as special case• Update expectations using Bayes’ theorem when have more information• Assumptions are made explicit (e.g. naturalness of LECs)• Clear prescriptions for combining errors• Statistics as diagnostics for physics• Model checking: we can test if our UQ model works and study sensitivities• Model selection: Is the Δ needed? Pionless vs. pionful formulations, …• Particularly well suited for (any) EFT, but generally suited for theory errors
pr(A|B, I) =pr(B|A, I)pr(A|I)
pr(B|I) =) pr(x|data, I)| {z }posterior
/ pr(data|x, I)| {z }likelihood
⇥ pr(x|I)| {z }prior
Parameter estimation
• Framework for parameter estimationFurnstahl et al., JPG 42 (2015) no.3, 034028Wesolowski et al., JPG 43 (2016) no.7, 074001Wesolowski et al., JPG (2019) in press
• Describing observables with theory:
• Reduces to !" fitting without theory error
• But we know theoretical error exists, especially for effective field theories
yexp
= yth
+ �yth
+ �yexp
�yth = yref
1X
n=k+1
cnQn
yth = yref
kX
n=0
cnQn
Q = max{p,m⇡}/⇤bFor chiral EFT:
Parameter estimation
• What we want to calculate:
• Posterior pdf with naturalness and truncation error:
• Naturalness: encoded in “hyperparameter”• Truncation error: , theory error assumptions
pr(~ak |yexp
,⌃exp
,⌃th
, I)EFT low-energy constantsat order k in the theory
Experimental valuesand uncertainties
Theoryuncertainty
Any other backgrounde.g., EFT naturalness
pr(~ak |yexp
,⌃exp
,⌃th
) / pr(yexp
|yth
,⌃exp
,⌃th
) pr(~ak | a)Likelihood Prior
a
⌃th
Exploring projected posteriors
• Starting slow:
• Work in a regime where theory error is very small• High enough order where truncation error is small• Regular least-squares likelihood with Gaussian prior
yexp
= yth
+ �yth
+ �yexp
pr(~ak|yexp
,⌃exp
) / pr(yexp
|~ak,⌃exp
) pr(~ak)
/ e�1
2
rT⌃�1
exp
r ⇥ e�(~ak)2/2a2
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Exploring projected posteriors
• Projected posteriors give:• info. on correlations• ”best” parameter values• indication of normality
• Uncorrelated and Gaussian• Here: Comparison to EKM’s
values from optimization
• Should not necessarily match!
C3P0 = 1.92+0.01�0.01
1.89
1.92
1.95
C3P0
0.60
0.75
0.90
D3P
0
0.60
0.75
0.90
D3P0
D3P0 = 0.69+0.04�0.04
3P0 channel at N3LOnp sector
50 100 150 200 250 300Elab [MeV]
�0.8
0.0
0.8
1.6
2.4
�res 3P
0[d
eg]
Emax = 200 MeV
EKM
PWA93
�resp.w. = �predp.w. � �PWA93p.w.
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pr(~ak|yexp
,⌃exp
) / exp
"�1
2
NdX
i=1
r2i�2
i
#⇥ e�(~ak)
2/2a2
Semilocal interactions from: Epelbaum et al., EPJ A 51 (2015) 53, Epelbaum et al., PRL 115 (2015) 122301
Wesolowski et al., JPG 46, 045102 (2019)
Exploring projected posteriors
• Set to 0 and use only 3 parameters
• Description of data still good• Recent explorations have
confirmed this also
eC1S0 = 0.55+0.00�0.00
�0.2
�0.1
0.0
C1S
0
C1S0 = �0.09+0.04�0.04
0.55
20.55
6
eC1S0
3
4
D1 1S
0
�0.2
�0.1
0.0
C1S0
3 4
D11S0
D11S0 = 3.30+0.38
�0.36
1S0 channel at N3LOnp sector D2
1S0<latexit sha1_base64="RZ4vqLeD/gfg3xavG0umQNK/8yo=">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</latexit><latexit sha1_base64="RZ4vqLeD/gfg3xavG0umQNK/8yo=">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</latexit><latexit sha1_base64="RZ4vqLeD/gfg3xavG0umQNK/8yo=">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</latexit><latexit sha1_base64="RZ4vqLeD/gfg3xavG0umQNK/8yo=">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</latexit>
50 100 150 200 250 300Elab [MeV]
�4.5
�3.0
�1.5
0.0
1.5
3.0
�res 1S
0[d
eg]
Emax = 200 MeV
EKM
PWA93
pr(~ak|yexp
,⌃exp
) / exp
"�1
2
NdX
i=1
r2i�2
i
#⇥ e�(~ak)
2/2a2
Wesolowski et al., JPG 46, 045102 (2019)
Effect of the prior
D1P1a D1P1 D1P1 D1P1 D1P1 D1P1
C1P
1C
1P
1C
1P
1C
1P
1
1
75 100 125 150 175 200
2
5
10
Emax
Increase Emax (more data at higher energy)
Incr
ease
prio
r wid
th
Less data, prior has much more influence
Increase amount of data, prior has less influence
We enough data thata value of 5 is ok, but this should always betested and verified.
Repeat same problem, vary givenspr(~ak|y
exp
,⌃exp
) / exp
"�1
2
NdX
i=1
r2i�2
i
#⇥ e�(~ak)
2/2a2
Wesolowski et al., JPG 46, 045102 (2019)
Including truncation errorsyexp
= yth
+ �yth
+ �yexp
�yth = yref
1X
n=k+1
cnQn
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Theoretical uncertainty takes this form, and we can put a prior on the unknown higher-order coefficients cn
cn|c ⇠ N (0, c2)<latexit sha1_base64="SNFe9Vk4bkZB2mc1eXEd0dCxX3Q=">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</latexit><latexit sha1_base64="SNFe9Vk4bkZB2mc1eXEd0dCxX3Q=">AAAEsXicbVPdbtMwFPa2AKP8dXDJjUU1aUjVlGygcTmNSXCFBqPbUFOK4zitFduJbGe0Mn4OnoZbeAbeBuenbE1r5efkfN/5fM7JcZQzqrTv/93Y3PLu3L23fb/z4OGjx0+6O08vVFZITAY4Y5m8ipAijAoy0FQzcpVLgnjEyGWUvi3xy2siFc3EZz3PyYijiaAJxUg717gbmDBKILZjAX/AMELSYAtDRTkMOdJTjJj5YPf8fgN9PXg57vb8fb9acNUIGqMHmnU23tmahnGGC06ExgwpNQz8XI8MkppiRmwnLBTJEU7RhAydKRAnamSq2izcdZ4YJpl0t9Cw8t6OMIgrNeeRY5YJqzZWOtdhw0Inb0aGirzQROB6o6RgUGewbBSMqSRYs7kzEJbU5QrxFEmEtWtnZzcU5DvOOEciNiG6JtiaqmNRYpC1nU4Yk8T9kypfEyOZShJb8+ndiTXB0es+9N3lSl+j4p6VxBLmomvYhLVkKVdHtFU4khNbq41dIIeVY41eTVwILnJcCWxFRqw4vYlyX8SetjmKTjgis9w2ppPTZKZN6erTNjvlaGZNuiCVX21KLm2DOquFYTeaDq0HtJ0st8Ng5N4Zi8spyZjpBW3ShIgsUqUEN/P1oJuhhVm3xjnWEuuibzNLz0rz1dLArJSEbkpamYTzm96eN739v4s7m0H7JK4aFwf7gbM/vuodnzSndBs8By/AHgjAETgG78EZGAAMfoJf4Df44x16X7xvXlRTNzeamGdgaXnpP2NqqNo=</latexit><latexit sha1_base64="SNFe9Vk4bkZB2mc1eXEd0dCxX3Q=">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</latexit><latexit sha1_base64="SNFe9Vk4bkZB2mc1eXEd0dCxX3Q=">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</latexit>
Again, interested in sampling:pr(~ak |y
exp
,⌃exp
,⌃th
) / pr(yexp
|yth
,⌃exp
,⌃th
) pr(~ak | a)
Form of theory covariance is quite simple and is can be added to the experimental covariance!
Wesolowski et al., JPG 46, 045102 (2019)
(⌃th,uncorr.)ij = (y
ref
)2i c2
kmaxX
n=k+1
Q2ni �ij ������!
kmax
!1
(yref
)2i c2 Q2k+2
i
1�Q2
i
�ij
(⌃th,corr.)ij = (y
ref
)i(yref
)j c2
kmaxX
n=k+1
Qni Q
nj ������!
kmax
!1
(yref
)i(yref
)j c2 Qk+1
i Qk+1
j
1�QiQj
Effect of including truncation errorsIf we have absorbed effects of higher-order operators byincluding theory errors, LEC extractions should be independentof Emax, the highest-energy datum used:
50 100 150 200 250 300Emax [MeV]
0.4
0.8
1.2
1.6
C1P
1
c =0.9
1P1 N2LO
kmax = k + 1, uncorr.
kmax ! 1, uncorr.
�yth = 0
50 100 150 200 250 300Emax [MeV]
0.4
0.8
1.2
1.6
C1P
1c =0.9
1P1 N2LO
kmax = k + 1, corr.
kmax ! 1, corr.
�yth = 0
Uncorrelated assumption Correlated assumptionarxiv:1808:08211
Hypothesis: (Daniel’s talk)Coefficients can be modeled as Gaussian processes
Strategy and moving forward for 3N
• Nice, simple form of posterior(note: can marginalize over hyperparameters to avoid tight assumptions)
• Sample using Markov Chain Monte Carlo (MCMC)• Fast for NN, but few-body not as cheap to compute• Goal: understand impact of uncertainties from 3N• Strategy: fix NN and !N and their uncertainties first
Using NNLOsep (Carlsson et al., PRX 6 011019 (2016))
• Theory uncertainty from fixed LECs and truncation error
• Then estimate cD and cE , simpler 2D problem⌃th = ⌃k + ⌃LECs
Summary
• Important to understand full information content of data • Most channels look Gaussian, but do statistical test before
approximations!• Use of projected posterior plots as a physics diagnostic
illustrated by the fourth-order s-wave LECs èparameter degeneracy• What is optimal trade-off between more data to determine
LECs more precisely and fit contamination at higher energies of omitted higher-order EFT terms?• Sensitivity to Emax removed with Bayesian UQ èLECs should
be independent• Accounting for truncation errors; verify with Emax plots• Moving forward… 3NF
Moving forward
• Combined, full error propagation. Put truncation error framework together with LEC error propagation• cD and cE estimation, UQ for 3BFs, fuller analysis
and comparison with recent work• Incorporation of GP model for truncation errors
into parameter estimation framework• Additional open issues:• Testing power counting• Expansion parameter studies
Backup: Case studies in NN parameter estimation• Case study 1: Exploring projected posteriors
• Correlations between parameters• Are there multiple modes?• Parameter redundancy (occurs for three cases at N3LO)• Gaussianity: do covariance approximations work?
• Case study 2: “Emax plots”• An EFT at order Qk is missing terms ~Qk+1
• We account for these missing terms• Higher-order effects in data should not affect parameter
estimates at lower orders in the EFT• Emax plots visualize this expectation: should “level off”
• Use ”SCS” interactions of Epelbaum, Krebs, and MeißnerEpelbaum et al., EPJ A 51 (2015) no.5, 53Epelbaum et al., PRL 115 (2015) no.12, 122301
Backup: Full posterior pdf with theory error
• Using particular assumptions:see Wesolowski et al., JPG (2019) in press [arXiv:1808.08211] for details
• Theory and experimental errors add!• Naturalness prior imposes a penalty on large LECs• Form of theory error:
pr(~ak |yexp
,⌃exp
,⌃th
) / e�1
2
r|(⌃
exp
+⌃
th
)
�1r e(~ak)2/2a2
r ⌘ yexp
� yth
(⌃th,uncorr.)ij = (y
ref
)2i c2
kmaxX
n=k+1
Q2ni �ij ������!
kmax
!1
(yref
)2i c2 Q2k+2
i
1�Q2
i
�ij
(⌃th,corr.)ij = (y
ref
)i(yref
)j c2
kmaxX
n=k+1
Qni Q
nj ������!
kmax
!1
(yref
)i(yref
)j c2 Qk+1
i Qk+1
j
1�QiQj
1. Theory error is completelyIndependent at different kinematic points
2. Theory error is the sameat all kinematic points
Reality: somewhere in between these two assumptions. Use Gaussian processes to model correlations (Daniel’s talk next!)
Exploring projected posteriors
arxiv:1808:08211
eC1S0 = 1.06+0.08�0.09
�0.30
�0.15
C1S0
C1S0 = �0.26+0.06�0.07
�3
�2
�1
D1 1S0
D11S0 = �1.88+0.47
�0.46
1.05
1.20
eC1S0
2.5
3.0
D2 1S0
�0.30
�0.15
C1S0
�3 �2 �1
D11S0
2.5
3.0
D21S0
D21S0 = 2.67+0.26
�0.29
1S0 channel at N3LOnp sector
arxiv:1808:08211
• Irregular structure of posterior• Nothing necessarily wrong
• But here: a physics issues actually explains the structure
• Parameter redundancy at N3LO, one can be eliminated
• True in 3S1 channel as well
pr(~ak|yexp
,⌃exp
) / exp
"�1
2
NdX
i=1
r2i�2
i
#⇥ e�(~ak)
2/2a2
Effect of including truncation errors
eC1S0 = �0.58+0.01�0.01
�0.60
�0.58
�0.56
eC1S0
0.75
1.00
1.25
C1S0
0.75
1.00
1.25
C1S0
C1S0 = 1.05+0.12�0.11
eC1S0 = �0.58+0.01�0.01
�0.60
�0.58
�0.56
eC1S0
0.75
1.00
1.25
C1S0
0.75
1.00
1.25
C1S0
C1S0 = 1.02+0.10�0.09
Uncorrelated assumption Correlated assumption
Use partial-wave cross sections extracted from PWA phase shifts
arxiv:1808:08211
