2. Logit and sample with weights (Bayesian)

Example of a logit model with a weighted sample

Michel Bierlaire, EPFL Thu Oct 30 2025, 10:15:17

from pathlib import Path

from IPython.core.display_functions import display

See the data processing script: Data preparation for Swissmetro.

from swissmetro_data import (
    CAR_AV_SP,
    CAR_CO_SCALED,
    CAR_TT_SCALED,
    CHOICE,
    GROUP,
    SM_AV,
    SM_COST_SCALED,
    SM_TT_SCALED,
    TRAIN_AV_SP,
    TRAIN_COST_SCALED,
    TRAIN_TT_SCALED,
    database,
)

from biogeme.bayesian_estimation import (
    BayesianResults,
    BayesianResultsSummary,
    get_pandas_estimated_parameters,
)
from biogeme.biogeme import BIOGEME
from biogeme.expressions import Beta
from biogeme.models import loglogit

Parameters to be estimated.

asc_car = Beta('asc_car', 0, None, None, 0)
asc_train = Beta('asc_train', 0, None, None, 0)
asc_sm = Beta('asc_sm', 0, None, None, 1)
b_time = Beta('b_time', 0, None, None, 0)
b_cost = Beta('b_cost', 0, None, None, 0)

Definition of the utility functions.

v_train = asc_train + b_time * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
v_swissmetro = asc_sm + b_time * SM_TT_SCALED + b_cost * SM_COST_SCALED
v_car = asc_car + b_time * CAR_TT_SCALED + b_cost * CAR_CO_SCALED

Associate utility functions with the numbering of alternatives.

v = {1: v_train, 2: v_swissmetro, 3: v_car}

Associate the availability conditions with the alternatives.

av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP}

Definition of the model.

This is the contribution of each observation to the log likelihood function.

logprob = loglogit(v, av, CHOICE)

Definition of the weight.

WEIGHT_GROUP_2 = 8.890991e-01
WEIGHT_GROUP_3 = 1.2
weight = WEIGHT_GROUP_2 * (GROUP == 2) + WEIGHT_GROUP_3 * (GROUP == 3)

These notes will be included as such in the report file.

USER_NOTES = (
    'Example of a logit model with three alternatives: '
    'Train, Car and Swissmetro.'
    ' Weighted Exogenous Sample Maximum Likelihood estimator (WESML)'
)

Create the Biogeme object. It is possible to control the generation of the HTML and the yaml files. Note that these parameters can also be modified in the .TOML configuration file.

formulas = {'log_like': logprob, 'weight': weight}
the_biogeme = BIOGEME(
    database,
    formulas,
    mcmc_sampling_strategy='pymc',
    user_notes=USER_NOTES,
    generate_html=False,
    generate_yaml=True,
)
the_biogeme.model_name = 'b02_weight'

Estimate the posterior distribution of the parameters, or read the summary if it is already available. The main guard is required because PyMC may use multiprocessing, and worker processes re-import this script.

def main() -> None:
    """Estimate the weighted Bayesian logit model and display the results."""

    yaml_file = Path('saved_results') / f'{the_biogeme.model_name}.yaml'

    try:
        summary_results = BayesianResultsSummary.from_yaml_file(filename=yaml_file)
    except FileNotFoundError:
        results: BayesianResults = the_biogeme.bayesian_estimation()
        summary_results = results.to_summary()

    # %%
    print(summary_results.short_summary())

    # %%
    # Get the results in a pandas table
    pandas_results = get_pandas_estimated_parameters(
        estimation_results=summary_results
    )
    display(pandas_results)


if __name__ == '__main__':
    main()

Sample size                                              6768
Sampler                                                  NUTS
Number of chains                                         4
Number of draws per chain                                2000
Total number of draws                                    8000
Acceptance rate target                                   0.9
Run time                                                 0:01:35.139312
Posterior predictive log-likelihood (sum of log mean p)  -5667.43
Expected log-likelihood E[log L(Y|θ)]                    -5671.08
Best-draw log-likelihood (posterior upper bound)         -5669.08
LOO (Leave-One-Out Cross-Validation)                     -5674.74
LOO Standard Error                                       63.96
Effective number of parameters (p_LOO)                   7.31
        Name  Value (mean)  Value (median)  ...     R hat   ESS (bulk)   ESS (tail)
0  asc_train     -0.796737       -0.796639  ...  1.000728  3780.234615  3895.343524
1     b_time     -1.346555       -1.346712  ...  1.001276  3926.582401  4491.312471
2     b_cost     -1.141470       -1.142105  ...  1.000225  5085.158962  4430.570411
3    asc_car     -0.092223       -0.091787  ...  1.000991  4305.358178  4922.538386

[4 rows x 12 columns]