Analytical Fits

This section describes how to run an analytic fit with Colibri.

In an analytic fit, the posterior mean and covariance of the PDF model parameters are obtained via the analytic solution to a linear regression problem. This fitting methodology is therefore only appropriate for linear models, which, in general, can be described by an equation of the following form:

\[f(\theta) = W\,\theta,\]

where \(W\) is a matrix that maps the parameters, \(\theta\), to the theory prediction vector, \(f(\theta)\).

Effectively, the analytic fit is treated as a “bayesian fit”, where the priors of the parameters are used to compute the evidence analytically.

See this section for an overview of the theoretical background of the analytic fit methodology, as well as details on when it might be appropriate to run an analytic fit.

In this section, we will provide an example that was implemented in the work presented in Ref. CMMU25.

Runcard

The following runcard can be used to run an analytic fit. This has been used to fit the weight minimisation (wmin) model presented in Ref. CMMU25.

meta: 'An example Level 1 closure test analytic fit using Colibri'

#######################
# Data and theory specs
#######################

dataset_inputs:
    # DIS data
    # NMC experiment
    # - {dataset: NMC_NC_NOTFIXED_P_EM-SIGMARED, variant: legacy} # (out-of-sample)
    # NUCLEAR experiments
    - {dataset: CHORUS_CC_NOTFIXED_PB_NU-SIGMARED, variant: legacy_dw}
    - {dataset: CHORUS_CC_NOTFIXED_PB_NB-SIGMARED, variant: legacy_dw}
    - {dataset: NUTEV_CC_NOTFIXED_FE_NU-SIGMARED, cfac: [MAS], variant: legacy_dw}
    # - {dataset: NUTEV_CC_NOTFIXED_FE_NB-SIGMARED, cfac: [MAS], variant: legacy_dw} # (out-of-sample)

    # HERACOMB experiments
    - {dataset: HERA_NC_318GEV_EM-SIGMARED, variant: legacy}
    # - {dataset: HERA_NC_225GEV_EP-SIGMARED, variant: legacy} # (out-of-sample)
    - {dataset: HERA_NC_251GEV_EP-SIGMARED, variant: legacy}
    - {dataset: HERA_NC_300GEV_EP-SIGMARED, variant: legacy}
    - {dataset: HERA_NC_318GEV_EP-SIGMARED, variant: legacy}
    # - {dataset: HERA_CC_318GEV_EM-SIGMARED, variant: legacy} # (out-of-sample)
    - {dataset: HERA_CC_318GEV_EP-SIGMARED, variant: legacy}
    - {dataset: HERA_NC_318GEV_EAVG_CHARM-SIGMARED, variant: legacy}
    - {dataset: HERA_NC_318GEV_EAVG_BOTTOM-SIGMARED, variant: legacy}


theoryid: 40_000_000                          # The theory from which the predictions are drawn
use_cuts: internal                     # The kinematic cuts to be applied to the data

closure_test_level: 1
level_1_seed: 123456
closure_test_pdf: 250503_pod_basis_40k_underlying_law_40w_pos

## NNPDF settings, these are needed by validphys to run report
closuretest:
    fakedata: true
    filterseed: 123456 # should be the same as level_1_seed
    fakepdf: 250503_pod_basis_40k_underlying_law_40w_pos # should be the same as closure_test_pdf


#####################
# Loss function specs
#####################
use_fit_t0: true                    # Whether the t0 covariance is used in the chi2 loss
t0pdfset: 240701-02-rs-nnpdf40-baseline         # The t0 PDF used to build the t0 covariance matrix

#############
# Model specs
#############

# Weight minimisation settings
wmin_settings:
    wminpdfset: 250503_pod_basis_40k
    n_basis: 70

###################
# Methodology specs
###################

# Analytic settings
analytic_settings:
    n_posterior_samples: 1              # Number of posterior samples written to exportgrids, ready for evolution
    full_sample_size: 50000             # Number of samples to be drawn from the posterior
    sampling_seed: 91234                # Random seed used for reproducible sampling


prior_settings:
    prior_distribution: uniform_parameter_prior
    prior_distribution_specs:
            {
            "min_val": -2.65,
            "max_val": 2.64,
            }

actions_:
- run_analytic_fit

Note that the prior_settings are the same as in a bayesian fit, which you can find more details about in this tutorial.