Parameter Inference

EOS can infer parameters based on a database of experimental or theoretical constraints and its built-in observables. The examples in this notebook illustrate how to find a specific constraint from the list of all built-in observables, construct an eos.Analysis object that represents the statistical analysis, and infer mean value and standard deviation of a list of parameters through optimization or Monte Carlo methods.

Listing the built-in Constraints

The full list of built-in constraints for the most-recent EOS release is available online here. You can also show this list using the eos.Constraints class. Searching for a specific constraint is possible by filtering for specific strings in the constraint name’s prefix, name, or suffix parts. The following example only shows constraints that contain a '->D' in the prefix part:

[1]:
import eos
eos.Constraints(prefix='->D')
[1]:
qualified name observables type reference
B->D::f_++f_0@FNAL+MILC:2015B ... MultivariateGaussian FNAL+MILC:2015B
B->D::f_++f_0@HPQCD:2015A ... MultivariateGaussian HPQCD:2015A
B->D::f_+@FKKM:2008A ... Gaussian FKKM:2008A
B->D^(*)::FormFactors[f_+,f_0,A_0,A_1,A_2,V,T_1,T_2,T_23]@GKvD:2018A ... MultivariateGaussian(Covariance) GKvD:2018A
B->D^(*)::FormFactors[f_+,f_0,f_T,A_0,A_1,A_2,V,T_1,T_2,T_23]@GKvD:2018A ... MultivariateGaussian(Covariance) GKvD:2018A
B->D^(*)::FormFactors[parametric,LCSRLattice]@GKvD:2018A ... MultivariateGaussian(Covariance) GKvD:2018A
B->D^(*)::FormFactors[parametric,LCSR]@GKvD:2018A ... MultivariateGaussian(Covariance) GKvD:2018A
B->D^(*)lnu::R_D^(*)@HFLAV:2019A ... MultivariateGaussian(Covariance) HFLAV:2019A
B->D^*::A_1@FKKM:2008A ... Gaussian FKKM:2008A
B->D^*::A_1[s_max]@FNAL+MILC:2014A ... Gaussian FNAL+MILC:2014A
B->D^*::A_1[s_max]@HFLAV:2019A ... Gaussian HFLAV:2019A
B->D^*::A_1[s_max]@HPQCD:2017A ... Gaussian HPQCD:2017A
B->D^*::A_2@FKKM:2008A ... Gaussian FKKM:2008A
B->D^*::FormFactors@FNAL+MILC:2021A ... MultivariateGaussian(Covariance) FNAL+MILC:2021A
B->D^*::FormFactors@HPQCD:2023A ... MultivariateGaussian(Covariance) HPQCD:2023A
B->D^*::FormFactors@JLQCD:2023A ... MultivariateGaussian(Covariance) JLQCD:2023A
B->D^*::V@FKKM:2008A ... Gaussian FKKM:2008A
B^-->D^*0l^-nu::BR@HFLAV:2021A ... Gaussian HFLAV:2021A
B^0->D^*+l^-nu::BR@HFLAV:2021A ... Gaussian HFLAV:2021A
B^0->D^*+l^-nu::KinematicDistribution[w]@Belle-II:2023C ... MultivariateGaussian(Covariance) Belle-II:2023C
B^0->D^+e^-nu::BRs@Belle:2015A ... MultivariateGaussian(Covariance) Belle:2015A
B^0->D^+l^-nu::KinematicalDistribution[w]@Belle:2015A ... MultivariateGaussian(Covariance) Belle:2015A
B^0->D^+mu^-nu::BRs@Belle:2015A ... MultivariateGaussian(Covariance) Belle:2015A
B_(s)->D_(s)^(*)::BR@MvDV:2024A ... MultivariateGaussian(Covariance) MvDV:2024A
B_(s)->D_(s)^(*)::FormFactors[f_0(Mpi2),f_0(MK2),A_0(Mpi2),A_0(MK2)]@BGJvD:2019A ... MultivariateGaussian(Covariance) BGJvD:2019A
B_s->D_s::f_++f_0@HPQCD:2019A ... MultivariateGaussian(Covariance) HPQCD:2019A
B_s->D_s^(*)::FormFactors[f_+,f_0,A_0,A_1,A_2,V,T_1,T_2,T_23]@BGJvD:2019A ... MultivariateGaussian(Covariance) BGJvD:2019A
B_s->D_s^(*)::FormFactors[f_+,f_0,f_T,A_0,A_1,A_2,V,T_1,T_2,T_23]@BGJvD:2019A ... MultivariateGaussian(Covariance) BGJvD:2019A
B_s->D_s^(*)::FormFactors[parametric,LCSRLattice]@BGJvD:2019A ... MultivariateGaussian(Covariance) BGJvD:2019A
B_s->D_s^(*)::FormFactors[parametric,LCSR]@BGJvD:2019A ... MultivariateGaussian(Covariance) BGJvD:2019A
B_s->D_s^*::A_1[s_max]@HPQCD:2017A ... Gaussian HPQCD:2017A
B_s->D_s^*::A_1[s_max]@HPQCD:2019A ... Gaussian HPQCD:2019A
e^+e^-->D^+D^-::sigma@BES:2008A ... MultivariateGaussian(Covariance) BES:2008A
e^+e^-->D^+D^-::sigma@BES:2017A ... MultivariateGaussian(Covariance) BES:2017A
e^+e^-->D^+D^-::sigma@BaBar:2007B ... MultivariateGaussian(Covariance) BaBar:2007B
e^+e^-->D^+D^-::sigma@Belle:2008B ... MultivariateGaussian(Covariance) Belle:2008B
e^+e^-->D^0Dbar^0::sigma@BES:2008A ... MultivariateGaussian(Covariance) BES:2008A
e^+e^-->D^0Dbar^0::sigma@BES:2017A ... MultivariateGaussian(Covariance) BES:2017A
e^+e^-->D^0Dbar^0::sigma@BaBar:2007B ... MultivariateGaussian(Covariance) BaBar:2007B
e^+e^-->D^0Dbar^0::sigma@Belle:2008B ... MultivariateGaussian(Covariance) Belle:2008B

Visualizing the built-in Constraints

For what follows we will use the two experimental constraints B^0->D^+e^-nu::BRs@Belle:2015A and B^0->D^+mu^-nu::BRs@Belle:2015A, to infer the CKM matrix element \(|V_{cb}|\). We can readily display these two constraints, along with the default theory prediction (without any uncertainties), using the following code:

[2]:
plot_args = {
    'plot': {
        'x': { 'label': r'$q^2$', 'unit': r'$\textnormal{GeV}^2$', 'range': [0.0, 11.63] },
        'y': { 'label': r'$d\mathcal{B}/dq^2$',                    'range': [0.0,  5e-3] },
        'legend': { 'location': 'lower left' }
    },
    'contents': [
        {
            'label': r'$\ell=e$',
            'type': 'observable',
            'observable': 'B->Dlnu::dBR/dq2;l=e,q=d',
            'variable': 'q2',
            'color': 'black',
            'range': [0.02, 11.63],
        },
        {
            'label': r'Belle 2015 $\ell=e,\, q=d$',
            'type': 'constraint',
            'color': 'C0',
            'constraints': 'B^0->D^+e^-nu::BRs@Belle:2015A',
            'observable': 'B->Dlnu::BR',
            'variable': 'q2',
            'rescale-by-width': True
        },
        {
            'label': r'Belle 2015 $\ell=\mu,\,q=d$',
            'type': 'constraint',
            'color': 'C1',
            'constraints': 'B^0->D^+mu^-nu::BRs@Belle:2015A',
            'observable': 'B->Dlnu::BR',
            'variable': 'q2',
            'rescale-by-width': True
        },
    ]
}
eos.plot.Plotter(plot_args).plot()
[2]:
(<Figure size 640x480 with 1 Axes>,
 <Axes: xlabel='$q^2$\\,[$\\textnormal{GeV}^2$]', ylabel='$d\\mathcal{B}/dq^2$'>)
../_images/user-guide_inference_7_1.png

Defining the Statistical Analysis

To define our statistical analysis for the inference of \(|V_{cb}|\) from measurements of the \(\bar{B}\to D\ell^-\bar\nu\) branching ratios, we must decide how to parametrize the hadronic form factors that emerge in semileptonic \(\bar{B}\to D\) transitions and how to constraint them. For what follows we will use the parameterization and constraints as in the example notebook on theory predictions and uncertainties.

We then create an eos.Analysis object as follows:

[3]:
analysis_args = {
    'global_options': { 'form-factors': 'BSZ2015', 'model': 'CKM' },
    'priors': [
        { 'parameter': 'CKM::abs(V_cb)',           'min':  38e-3, 'max':  45e-3 , 'type': 'uniform'},
        { 'parameter': 'B->D::alpha^f+_0@BSZ2015', 'min':  0.0,   'max':  1.0   , 'type': 'uniform'},
        { 'parameter': 'B->D::alpha^f+_1@BSZ2015', 'min': -4.0,   'max': -1.0   , 'type': 'uniform'},
        { 'parameter': 'B->D::alpha^f+_2@BSZ2015', 'min': +4.0,   'max': +6.0   , 'type': 'uniform'},
        { 'parameter': 'B->D::alpha^f0_1@BSZ2015', 'min': -1.0,   'max': +2.0   , 'type': 'uniform'},
        { 'parameter': 'B->D::alpha^f0_2@BSZ2015', 'min': -2.0,   'max':  0.0   , 'type': 'uniform'}
    ],
    'likelihood': [
        'B->D::f_++f_0@HPQCD:2015A',
        'B->D::f_++f_0@FNAL+MILC:2015B',
        'B^0->D^+e^-nu::BRs@Belle:2015A',
        'B^0->D^+mu^-nu::BRs@Belle:2015A'
    ]
}
analysis = eos.Analysis(**analysis_args)
analysis.parameters['CKM::abs(V_cb)'].set(42.0e-3)

In the above, the global options ensure that our choice of form factor parametrization is used throughout, and that for CKM matrix elements the CKM model is used. The latter provides parametric access to the \(V_{cb}\) matrix element through two parameters: the absolute value CKM::abs(V_cb) and the complex phase CKM::arg(V_cb). The latter is not accessible from \(b\to c\ell\bar\nu\). We provide the parameters in our analysis through the specifications of the Bayesian priors. In the above, each prior is a uniform prior that covers the range from min to max. The likelihood is defined through a list constraints, which in the above includes both the experimental measurements by the Belle collaboration as well as the theoretical lattice QCD results. Finally, we set the starting value of CKM::abs(V_cb) to a sensible value of \(42\cdot 10^{-3}\).

We can now proceed to optimize the log(posterior) through a call to analysis.optimize. In a Jupyter notebook, it is useful to display the return value of this method, which illustrates the best-fit point. We can further display a summary of fit quality using the analysis.goodness_of_fit method.

[4]:
bfp = analysis.optimize()
display(bfp)
display(analysis.goodness_of_fit())
parametervalue
$|V_{cb}|$0.0419
$\alpha_{+,0}^{B \to D,\mathrm{BSZ2015}}$0.6668
$\alpha_{+,1}^{B \to D,\mathrm{BSZ2015}}$-2.5411
$\alpha_{+,2}^{B \to D,\mathrm{BSZ2015}}$4.7844
$\alpha_{0,1}^{B \to D,\mathrm{BSZ2015}}$0.2558
$\alpha_{0,2}^{B \to D,\mathrm{BSZ2015}}$-0.9263
constraintχ2±χd.o.f.local p-value
B->D::f_++f_0@FNAL+MILC:2015B3.5102783.4146%
B->D::f_++f_0@HPQCD:2015A3.0279569.5678%
B^0->D^+e^-nu::BRs@Belle:2015A11.85141029.5126%
B^0->D^+mu^-nu::BRs@Belle:2015A5.24171087.4456%

total χ223.6312
total degrees of freedom26
p-value59.7042%

Sampling from the Posterior

To sample from the posterior, EOS provides the analysis.sample method. Optionally, this can also produce posterior-predictive samples for a list of observables. We can use these samples to illustrate the results of our fit in comparison to the experimental constraints.

For this example, we produce such posterior-predictive samples for the differential \(\bar{B}\to D^+e^-\bar\nu\) branching ratio in 40 points in the kinematical variable \(q^2\); the square of the momentum transfer to the \(e^-\bar\nu\) pair. Due to the strong dependence of the branching ratio on \(q^2\), we do not distribute the points equally across the full phase space. Instead, we equally distribute half of the points in the interval \([0.02\,\text{GeV}^2, 1.00\,\text{GeV}^2]\) and the other half in the remainder of the phase space.

We produce N \(= 20000\) samples with a thinning factor (or stride) of \(5\). This means that stride * N \(= 100000\) samples are produced, but only every \(5\)th sample is returned. This improves the quality of the samples by reducing the autocorrelation. Before the samples are produced, the Markov Chain self-adapts in a series of preruns, the number of which is governed by the preprun argument. In each prerun, pre_N samples are drawn before the adaptation step. The samples obtained as part of the preruns are discarded. To ensure efficient sampling, the chain is started in the best-fit point obtained earlier through optimization.

[5]:
import numpy as np
e_q2values  = np.unique(np.concatenate((np.linspace(0.02,  1.00, 20), np.linspace(1.00, 11.60, 20))))
e_obs       = [eos.Observable.make(
                  'B->Dlnu::dBR/dq2', analysis.parameters, eos.Kinematics(q2=q2),
                  eos.Options({'form-factors': 'BSZ2015', 'l': 'e', 'q': 'd'}))
              for q2 in e_q2values]
parameter_samples, log_posterior, e_samples  = analysis.sample(N=20000, stride=5, pre_N=3000, preruns=5, start_point=bfp.point, observables=e_obs)

The values of the log(posterior) are stored in log_posterior. The posterior-preditive samples for the observables are stored in e_samples, and are only returned if the observables keyword argument is provided.

We can plot our result using the uncertainty plot type. It expects a data item that contains the samples as well as the x-axis values used in the production of the samples. This plot type will interpolate between neighboring x-axis values, and display the \(68\%\) probability envelope along the median curve.

[6]:
plot_args = {
    'plot': {
        'x': { 'label': r'$q^2$', 'unit': r'$\textnormal{GeV}^2$', 'range': [0.0, 11.63] },
        'y': { 'label': r'$d\mathcal{B}/dq^2$',                    'range': [0.0,  5e-3] },
        'legend': { 'location': 'lower left' }
    },
    'contents': [
        {
          'label': r'$\ell=\mu$', 'type': 'uncertainty', 'range': [0.02, 11.60],
          'data': { 'samples': e_samples, 'xvalues': e_q2values }
        },
        {
            'label': r'Belle 2015 $\ell=e,\, q=d$',
            'type': 'constraint',
            'color': 'C0',
            'constraints': 'B^0->D^+e^-nu::BRs@Belle:2015A',
            'observable': 'B->Dlnu::BR',
            'variable': 'q2',
            'rescale-by-width': True
        },
        {
            'label': r'Belle 2015 $\ell=\mu,\,q=d$',
            'type': 'constraint',
            'color': 'C1',
            'constraints': 'B^0->D^+mu^-nu::BRs@Belle:2015A',
            'observable': 'B->Dlnu::BR',
            'variable': 'q2',
            'rescale-by-width': True
        },
    ]
}
eos.plot.Plotter(plot_args).plot()
[6]:
(<Figure size 640x480 with 1 Axes>,
 <Axes: xlabel='$q^2$\\,[$\\textnormal{GeV}^2$]', ylabel='$d\\mathcal{B}/dq^2$'>)
../_images/user-guide_inference_19_1.png

The distribution of the parameter samples, here using \(|V_{cb}|\) as an example, can be inspected using regular histograms or a smooth histogram based on a kernel density estimate (KDE). For the latter, the parameter bandwidth regulates the smoothing. EOS applies a relative bandwidth factor with respect to SciPy’s best bandwidth estimate, i.e., specifying 'bandwidth': 2 double SciPy’s estimate for the bandwidth.

[7]:
plot_args = {
    'plot': {
        'x': { 'label': r'$|V_{cb}|$', 'range': [38e-3, 47e-3] },
        'legend': { 'location': 'upper left' }
    },
    'contents': [
        {
            'type': 'histogram',
            'data': { 'samples': parameter_samples[:, 0] }
        },
        {
            'type': 'kde', 'color': 'C0', 'label': 'posterior', 'bandwidth': 2,
            'range': [40e-3, 45e-3],
            'data': { 'samples': parameter_samples[:, 0] }
        }
    ]
}
eos.plot.Plotter(plot_args).plot()
[7]:
(<Figure size 640x480 with 1 Axes>, <Axes: xlabel='$|V_{cb}|$'>)
../_images/user-guide_inference_21_1.png

We can also illustrate the correlation between \(|V_{cb}|\) and any form factor parameter. Her, we use the normalization of the form factors at \(q^2 = 0\) as an example. Contours of equal probability at the \(68\%\) and \(95\%\) levels can be generated using a KDE as follows:

[8]:
plot_args = {
    'plot': {
        'x': { 'label': r'$|V_{cb}|$', 'range': [38e-3, 47e-3] },
        'y': { 'label': r'$f_+(0)$',   'range': [0.6, 0.75] },
    },
    'contents': [
        {
            'type': 'kde2D', 'color': 'C1', 'label': 'posterior',
            'levels': [68, 95], 'contours': ['lines','areas'], 'bandwidth':3,
            'data': { 'samples': parameter_samples[:, (0,1)] }
        }
    ]
}
eos.plot.Plotter(plot_args).plot()
[8]:
(<Figure size 640x480 with 1 Axes>,
 <Axes: xlabel='$|V_{cb}|$', ylabel='$f_+(0)$'>)
../_images/user-guide_inference_23_1.png

Here the bandwidth parameter takes the same role as in the 1D histogram.

We can compute the mean value and its standard deviation using numpy methods

[9]:
print('$|V_{{cb}}|$ = {mean:.4f} +/- {std:.4f}'.format(
    mean=np.average(parameter_samples[:,0]),
    std=np.std(parameter_samples[:, 0])
))
$|V_{cb}|$ = 0.0420 +/- 0.0009