Bias-Variance Trade-off and Model Selection for Proton Radius Extractions

Higinbotham, D. W.; Giuliani, Pablo; McClellan, Randall Evan; Širca, S.; Yan, X.

doi:10.48550/arxiv.1812.05706

Cited by 3 publications

(10 citation statements)

References 5 publications

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“…As in Refs [37,53], we evaluate the performance of each of the seven models using a bias-variance trade-off criterion. Bias is understood as the discrepancy between the true value of m (coming from one of the generators n F true ) and the extracted value.…”

Section: Bayesian Approachmentioning

confidence: 99%

“…In our particular case, we define the transfer function in terms of coefficients that encode the linear part of the response of the fitted model parameters to small changes in the data inputs. We have already implemented an early version of these ideas to estimate the bias and variance of models within the proton puzzle context [37] and in Ref. [38] to estimate the effect of dispersive corrections on the 12 C elastic cross section.…”

Section: Introductionmentioning

confidence: 99%

“…The performance of the seven models is evaluated in terms of bias and variance [39], similar to the approach implemented in [40,41] to extract the charge radius of the proton from electron scattering data. The "bias-variance trade-off" is an important concept in statistics and machine learning that addresses the complexity of a model.…”

Section: Introductionmentioning

confidence: 99%

“…Tables III and IV show for the 48 Ca and 208 Pb examples, respectively, the original locations q 0 as well as the locations q m that minimize the FOM defined in Eq. (37). Tables III and IV also show the numerical values of the TF times the respective error σ j for the density at r = 0 fm and the radius for both data sets q 0 and q m .…”

mentioning

confidence: 99%

“…( 36)), what goes in the FOM Eq. (37). We are using ρ(0) as a representative of the total interior density.…”

mentioning

confidence: 99%

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From noise to information: The transfer function formalism for uncertainty quantification in reconstructing the nuclear density

Giuliani

Piekarewicz

2021

Phys. Rev. C

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Background: The neutron distribution of neutron-rich nuclei provides critical information on the structure of finite nuclei and neutron stars. Parity violating experiments -such as PREX and CREX -provide a clean and largely model-independent determination of neutron densities. Such experiments, however, are challenging and expensive which is why sound statistical arguments are required to maximize the information gained.Purpose: To introduce a new framework, "the transfer function formalism", aimed at uncertainty quantification, model selection, and experimental design in the context of neutron densities.Methods: The transfer functions (TFs) are built analytically by expressing the linear response of the objective function (e.g., χ 2 ) to small perturbations of the data. Using the TF formalism, we are able to analyze the expected overall uncertainty -quantified in terms of bias and varianceof the mean square radius and interior density of 48 Ca and 208 Pb.Results: Using relativistic mean field models as a proxy for the weak-charge density -and assuming that a total of five measurements could be performed on the weak form factor of 48 Ca and 208 Pb -we identify the optimal models and experimental locations that minimize the uncertainty in the extraction of the radius and interior density. We also explore the use of the TF formalism to understand the influence of prior distributions for the model parameters, as well as the optimization of model hyperparameters not constrained by the data.Conclusions: We establish how the choice of experimental locations and the model that is used can have a significant impact on the final uncertainties of the extracted quantities of interest. For challenging experiments such as CREX and PREX, a proper quantification of such uncertainties is critical. We have demonstrated how the TF formalism provides several advantages for this type of analysis.

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Section: Bayesian Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…( 36)), what goes in the FOM Eq. (37). We are using ρ(0) as a representative of the total interior density.…”

mentioning

confidence: 99%

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From noise to information: The transfer function formalism for uncertainty quantification in reconstructing the nuclear density

Giuliani

Piekarewicz

2021

Phys. Rev. C

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How analytic choices can affect the extraction of electromagnetic form factors from elastic electron scattering cross section data

2020

Self Cite

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Scientists often try to incorporate prior knowledge into their regression algorithms, such as a particular analytic behavior or a known value at a kinematic endpoint. Unfortunately, there is often no unique way to make use of this prior knowledge, and thus, different analytic choices can lead to very different regression results from the same set of data. To illustrate this point in the context of the proton electromagnetic form factors, we use the Mainz elastic data with its 1422 cross section points and 31 normalization parameters. Starting with a complex unbound nonlinear regression, we will show how the addition of a single theory-motivated constraint removes an oscillation from the magnetic form factor and shifts the extracted proton charge radius. We then repeat both regressions using the same algorithm, but with a rebinned version of the Mainz dataset. These examples illustrate how analytic choices, such as the function that is being used or even the binning of the data, can dramatically affect the results of a complex regression. These results also demonstrate why it is critical when using regression algorithms to have either a physical model in mind or a firm mathematical basis to avoid confirmation bias.

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Proton charge radius extraction from electron scattering data using dispersively improved chiral effective field theory

et al. 2019

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We extract the proton charge radius from the elastic form factor (FF) data using a novel theoretical framework combining chiral effective field theory and dispersion analysis. Complex analyticity in the momentum transfer correlates the behavior of the spacelike FF at finite Q 2 with the derivative at Q 2 = 0. The FF calculated in the predictive theory contains the radius as a free parameter. We determine its value by comparing the predictions with a descriptive global fit of the spacelike FF data, taking into account the theoretical and experimental uncertainties. Our method allows us to use the finite-Q 2 FF data for constraining the radius (up to Q 2 ∼ 0.5 GeV 2 and larger) and avoids the difficulties arising in methods relying on the Q 2 → 0 extrapolation. We obtain a radius of 0.844(7) fm, consistent with the high-precision muonic hydrogen results.

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Bias-Variance Trade-off and Model Selection for Proton Radius Extractions

Cited by 3 publications

References 5 publications

From noise to information: The transfer function formalism for uncertainty quantification in reconstructing the nuclear density

From noise to information: The transfer function formalism for uncertainty quantification in reconstructing the nuclear density

How analytic choices can affect the extraction of electromagnetic form factors from elastic electron scattering cross section data

Proton charge radius extraction from electron scattering data using dispersively improved chiral effective field theory

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