25. Triangular mixture of logit

Bayesian estimation of a mixture of logit models. The mixing distribution is specified by the user. Here, a triangular distribution.

Michel Bierlaire, EPFL Tue Nov 18 2025, 12:35:26

from functools import partial

import biogeme.biogeme_logging as blog
import pymc as pm
from IPython.core.display_functions import display
from biogeme.bayesian_estimation import BayesianResults, get_pandas_estimated_parameters
from biogeme.biogeme import BIOGEME
from biogeme.draws import PyMcDistributionFactory
from biogeme.expressions import Beta, DistributedParameter, Draws
from biogeme.models import loglogit

See the data processing script: Data preparation for Swissmetro.

from swissmetro_data 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,
)

logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b25_triangular_mixture.py')

Example b25_triangular_mixture.py

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

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. The user must define a function that takes a str as argument (corresponding to the name of the random variable) and return a pymc.distributions.Distribution

triangular_factory: PyMcDistributionFactory = partial(
    pm.Triangular,
    lower=-1.0,
    c=0.0,
    upper=1.0,
)

Associate the function with a name

DISTRIBUTIONS = {'TRIANGULAR': triangular_factory}

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_eps', 'TRIANGULAR', dict_of_distributions=DISTRIBUTIONS),
)

Definition of the utility functions.

v_train = asc_train + b_time_rnd * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
v_swissmetro = asc_sm + b_time_rnd * SM_TT_SCALED + b_cost * SM_COST_SCALED
v_car = asc_car + 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 b_time_rnd, we have a 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 = 'b25_triangular'

Biogeme parameters read from biogeme.toml.

Estimate the parameters.

try:
    bayesian_results = BayesianResults.from_netcdf(
        filename=f'saved_results/{the_biogeme.model_name}.nc'
    )
except FileNotFoundError:
    bayesian_results = the_biogeme.bayesian_estimation()

Loaded NetCDF file size: 1.8 GB
load finished in 9624 ms (9.62 s)

Get the results in a pandas table

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

Diagnostics computation took 75.8 seconds (cached).
        Name  Value (mean)  Value (median)  ...     R hat   ESS (bulk)   ESS (tail)
0  asc_train     -0.393645       -0.393730  ...  1.000042  6129.449492  6131.566807
1     b_time     -2.282075       -2.280289  ...  1.001155  1696.593924  3441.429131
2     b_cost     -1.284068       -1.282969  ...  1.000366  7710.204636  6241.910672
3    asc_car      0.141109        0.140517  ...  1.000087  3587.630483  5766.455070
4   b_time_s      4.013062        4.006510  ...  1.002735  1169.172079  2912.845359

[5 rows x 12 columns]