3. Moneymetric and heteroscedastic specification

Although normalizing the scale to 1 is a common practice in random utility models, it is sometimes preferable to normalize another parameter. For instance, normalizing the cost coefficient to -1 sets the units of the utility function as currency units (CHF here), and the estimated coefficients are easily interpreted as willingness to pay. In that case, the scale must be estimated.

We also illustrate here a heteroscedastic specification, where a different scale is associated with different segments of the sample.

This example illustrates how to estimate such a specification with Bayesian inference.

Michel Bierlaire, EPFL Thu Nov 20 2025, 11:10:03

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

The scale parameters must stay away from zero. We define a small but positive lower bound

POSITIVE_LOWER_BOUND = 1.0e-5

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, 0, 0)
b_cost = Beta('b_cost', -1, None, None, 1)
scale_not_group3 = Beta('scale_not_group3', 1, POSITIVE_LOWER_BOUND, None, 0)
scale_group3 = Beta('scale_group3', 1, POSITIVE_LOWER_BOUND, 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

Scale associated with group 3 is estimated.

scale = (GROUP != 3) * scale_not_group3 + (GROUP == 3) * scale_group3

Scale the utility functions, and associate them with the numbering of alternatives.

v = {1: scale * v_train, 2: scale * v_swissmetro, 3: scale * 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)

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

USER_NOTES = (
    'Illustrates a moneymetric heteroscedastic specification. A different scale is'
    ' associated with different segments of the sample.'
)

Create the Biogeme object.

the_biogeme = BIOGEME(database, logprob, user_notes=USER_NOTES)
the_biogeme.model_name = 'b03_scale'

Estimate the posterior distribution of the parameters, or read the results if already available.

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())

Present the parameter estimates in a pandas table.

pandas_results = get_pandas_estimated_parameters(
    estimation_results=summary_results,
)
display(pandas_results)

Report the variables stored in the Bayesian estimation results.

display(summary_results.report_stored_variables())