26. Triangular mixture with panel data

Bayesian estimation of a mixture of logit models. The mixing distribution is user-defined (triangular, here). The datafile is organized as panel data.

Michel Bierlaire, EPFL Tue Nov 18 2025, 18:31:04

from functools import partial
from pathlib import Path

import pymc as pm
from IPython.core.display_functions import display

See the data processing script swissmetro_panel.py.

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

import biogeme.biogeme_logging as blog
from biogeme.bayesian_estimation import (
    BayesianResults,
    BayesianResultsSummary,
    get_pandas_estimated_parameters,
)
from biogeme.biogeme import BIOGEME
from biogeme.expressions import (
    Beta,
    DistributedParameter,
    Draws,
)
from biogeme.models import loglogit

logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b26triangular_panel_mixture.py')

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

POSITIVE_LOWER_BOUND = 1.0e-5

Define a random parameter with a triangular distribution. The triangular distribution is not directly available from Biogeme. It has to be generated by a function provided by the user, based on PyMC available distributions.

See the PyMC documentation: https://www.pymc.io/projects/docs/en/stable/api/distributions.html

Mean of the distribution.

b_time = Beta('b_time', 0, None, None, 0)

Scale of the distribution. It is advised not to use 0 as starting value for the following parameter.

b_time_s = Beta('b_time_s', 1, POSITIVE_LOWER_BOUND, None, 0)

Distribution of the draws

TriangularFactory = partial(
    pm.Triangular,
    lower=-1.0,
    c=0.0,
    upper=1.0,
)

Associate the function with a name

DISTRIBUTIONS = {'TRIANGULAR': TriangularFactory}

Define a random parameter with a triangular distribution, designed to be used for Monte-Carlo simulation.

b_time_rnd = DistributedParameter(
    'b_time_rnd',
    b_time
    + b_time_s
    * Draws('b_time_rnd_err_term', 'TRIANGULAR', dict_of_distributions=DISTRIBUTIONS),
)

Parameters to be estimated.

b_cost = Beta('b_cost', 0, None, None, 0)

The constants are distributed across individuals, to address serial correlation. In a panel setting, the corresponding draws are generated at the individual level. Wrapping them in DistributedParameter ensures they are expanded consistently when combined with observation-level variables.

asc_car = Beta('asc_car', 0, None, None, 0)
asc_car_s = Beta('asc_car_s', 1, None, None, 0)
asc_car_rnd = DistributedParameter(
    'asc_car_rnd',
    asc_car
    + asc_car_s
    * Draws('asc_car_eps', 'TRIANGULAR', dict_of_distributions=DISTRIBUTIONS),
)

asc_train = Beta('asc_train', 0, None, None, 0)
asc_train_s = Beta('asc_train_s', 1, None, None, 0)
asc_train_rnd = DistributedParameter(
    'asc_train_rnd',
    asc_train
    + asc_train_s
    * Draws('asc_train_eps', 'TRIANGULAR', dict_of_distributions=DISTRIBUTIONS),
)

asc_sm = Beta('asc_sm', 0, None, None, 1)
asc_sm_s = Beta('asc_sm_s', 1, None, None, 0)
asc_sm_rnd = DistributedParameter(
    'asc_sm_rnd',
    asc_sm
    + asc_sm_s * Draws('asc_sm_eps', 'TRIANGULAR', dict_of_distributions=DISTRIBUTIONS),
)

Definition of the utility functions.

v_train = asc_train_rnd + b_time_rnd * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
v_swissmetro = asc_sm_rnd + b_time_rnd * SM_TT_SCALED + b_cost * SM_COST_SCALED
v_car = asc_car_rnd + b_time_rnd * 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}

Conditional to the random parameters, the likelihood of one observation is given by the logit model (called the kernel).

conditional_log_probability = loglogit(v, av, CHOICE)

Create the Biogeme object.

the_biogeme = BIOGEME(
    database,
    conditional_log_probability,
)
the_biogeme.model_name = 'b26triangular_panel'

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())