SLIDE 1 Precision nuclear physics
Observable calculations are becoming increasingly precise What are the theory errors? Hamiltonian Calculation Observable
Ground-state energies for even oxygen isotopes
Experiment
Hergert et al. PRL 110, 242501 (2013)
Chiral effective field theory (EFT) is used to generate microscopic nuclear Hamiltonians and currents (note: plural!). Many versions (scales/ schemes) on the market.
SLIDE 2
Sources of uncertainty in EFT predictions
Hamiltonian: truncation errors regulator artifacts Low-energy constants: error from fitting to data Numerics: many-body methods basis truncation anything else Full uncertainty on prediction
SLIDE 3
Sources of uncertainty in EFT predictions
Hamiltonian: truncation errors regulator artifacts Low-energy constants: error from fitting to data Numerics: many-body methods basis truncation anything else Full uncertainty on prediction Bayesian methods treat on equal footing
SLIDE 4 a0! a1!
0!
BUQEYE Collaboration!
Prior! Posterior! True value!
Goal: Full uncertainty quantification (UQ) for effective field theory (EFT) predictions using Bayesian statistics
Some BUQEYE publications on UQ for EFT
- “A recipe for EFT uncertainty quantification in nuclear physics”,
- J. Phys. G 42, 034028 (2015)
- “Quantifying truncation errors in effective field theory”,
- Phys. Rev. C 92, 024005 (2015)
- “Bayesian parameter estimation for effective field theories”,
- J. Phys. G 43, 074001 (2016)
- “Bayesian truncation errors in chiral EFT: nucleon-nucleon observables”,
- Phys. Rev. C 96, 024003 (2017) [Editors’ Suggestion]
Bayesian Uncertainty Quantification: Errors for Your EFT
SLIDE 5 Bayesian interpretation of probability
Unrepeatable situations:
Probability that it will rain in Washington, D.C. tomorrow
Great introduction for physicists: “Bayes in the Sky” [arXiv:0803.4089]
Properties of the Universe (we have exactly one sample!)
Formulation of probability as “degree of belief”
an
Probability of a parameter
Repeatable situations:
Rolling dice Repeatable measurements
Beta decay credit: The 2015 Long Range Plan for Nuclear Science
“Based on a large amount of observations
- f the event, here is the probability”
“From the best of knowledge and previous measurements: the probability lies in this range”
SLIDE 6 Why Bayes for theory errors?
Frequentist approach: long-run relative frequency
- Outcomes of experiments treated as random variables
- Predict probabilities of observing various outcomes
- Well adapted to quantities that fluctuate statistically
- But systematic errors are problematic
Bayesian probabilities: pdf is a measure of state of knowledge
- Ideal for systematic/ theory errors that do not behave stochastically
- Assumptions and expectations encoded in prior pdfs
- Make explicit what is usually implicit: assumptions may be applied
consistently, tested, and modified in light of new information
SLIDE 7 Why Bayes for theory errors?
Frequentist approach: long-run relative frequency
- Outcomes of experiments treated as random variables
- Predict probabilities of observing various outcomes
- Well adapted to quantities that fluctuate statistically
- But systematic errors are problematic
Bayesian probabilities: pdf is a measure of state of knowledge
- Ideal for systematic/ theory errors that do not behave stochastically
- Assumptions and expectations encoded in prior pdfs
- Make explicit what is usually implicit: assumptions may be applied
consistently, tested, and modified in light of new information pdf for uncertainty: different prior assumptions about higher-order corrections
Observable(x) x 68% level
SLIDE 8 Why Bayes for theory errors?
Frequentist approach: long-run relative frequency
- Outcomes of experiments treated as random variables
- Predict probabilities of observing various outcomes
- Well adapted to quantities that fluctuate statistically
- But systematic errors are problematic
Bayesian probabilities: pdf is a measure of state of knowledge
- Ideal for systematic/ theory errors that do not behave stochastically
- Assumptions and expectations encoded in prior pdfs
- Make explicit what is usually implicit: assumptions may be applied
consistently, tested, and modified in light of new information Widespread application of Bayesian approaches in theoretical physics
- Interpretation of dark-matter searches; structure determination in
condensed matter physics, constrained curve-fitting in lattice QCD
- Is supersymmetry a “natural” approach to the hierarchy problem?
- Estimating uncertainties in perturbative QCD (e.g., parton distributions)
SLIDE 9
Joint probability for theory parameters
Example: want to “fit” parameters is read: “The probability that x is true given y”
pr(x|y)
pr(a|D, k, kmax, I)
Vector of parameters
{a0, a1, …ak}
Data k : truncation order kmax : omitted orders I: any other information
Here
SLIDE 10 Bayesian rules of probability as principles of logic
1: Sum rule
If set {xi} is exhaustive and exclusive
X
i
pr(xi|I) = 1
Z dx pr(x|I) = 1 pr(x|I) = Z dy pr(x, y|I) pr(x|I) = X
j
pr(x, yj|I)
- cf. complete and orthonormal
- implies marginalization (cf. inserting complete set of states)
SLIDE 11 Bayesian rules of probability as principles of logic
1: Sum rule 2: Product rule
If set {xi} is exhaustive and exclusive
X
i
pr(xi|I) = 1
Z dx pr(x|I) = 1
- cf. complete and orthonormal
- implies marginalization (cf. inserting complete set of states)
Expanding a joint probability of x and y
pr(x, y|I) = pr(x|y, I) pr(y|I) = pr(y|x, I) pr(x|I)
- If x and y are mutually independent: pr(x|y, I) = pr(x|I)
- Rearrange rule equality to get Bayes Theorem
pr(x, y|I) → pr(x|I) × pr(y|I)
pr(x|y, I) = pr(y|x, I)pr(x|I) pr(y|I)
pr(x|I) = Z dy pr(x, y|I)
pr(x|I) = X
j
pr(x, yj|I)
SLIDE 12
pr(a|D, k, kmax) ∝
pr(D|a, k, kmax) × pr(a|k, kmax)
Posterior Likelihood Prior 1D projections of a1 and a3 for naturalness prior: Likelihood overwhelms prior Prior suppresses unconstrained likelihood
Interaction between data and prior
