5. Mixture of logit models: normal distribution

Example of a normal mixture of logit models, using Bayesian inference.

Michel Bierlaire, EPFL Thu Nov 20 2025, 11:26:01

import biogeme.biogeme_logging as blog
from IPython.core.display_functions import display
from biogeme.bayesian_estimation import BayesianResults, get_pandas_estimated_parameters
from biogeme.biogeme import BIOGEME
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 b05_normal_mixtures.py')

Example b05_normal_mixtures.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, 0, 0)

Define a random parameter, normally distributed, designed to be used for Monte-Carlo simulation.

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

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

b_time_s = Beta('b_time_s', 10, POSITIVE_LOWER_BOUND, None, 0)
b_time_eps = Draws('b_time_eps', 'NORMAL')

The purpose of the DistributedParameter operator is to explicitly store the simulated individual-level parameters in the output file.

b_time_rnd = DistributedParameter('b_time_rnd', b_time + b_time_s * b_time_eps)

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}

When performing maximum likelihood estimation, in order to obtain the loglikelihood, we would first calculate the kernel conditional on b_time_rnd, and then integrate over b_time_rnd using Monte-Carlo. However, when performing Bayesian estimation, the random parameters will be explicitly simulated. Therefore, what the algorithm needs is the conditional log likelihood, which is simply a (log) logit here. This is one of the most important advantage of this estimation method: it does not require to calculate the complicated integrals.

conditional_log_likelihood = loglogit(v, av, CHOICE)

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

USER_NOTES = 'Example of Bayesian estimation of a mixture of logit models with three alternatives'

Create the Biogeme object.

the_biogeme = BIOGEME(
    database,
    conditional_log_likelihood,
    user_notes=USER_NOTES,
)
the_biogeme.model_name = 'b05_normal_mixture'

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 9243 ms (9.24 s)

Get the results in a pandas table

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

Diagnostics computation took 69.8 seconds (cached).
        Name  Value (mean)  Value (median)  ...     R hat   ESS (bulk)   ESS (tail)
0  asc_train     -0.400545       -0.399224  ...  1.000274  4764.255073  5518.165472
1    asc_car      0.139390        0.139290  ...  1.001596  2727.944492  5227.745062
2     b_time     -2.272711       -2.269103  ...  1.004406  1225.734728  2834.903303
3   b_time_s      1.675588        1.672388  ...  1.006639   868.962343  1806.335511
4     b_cost     -1.288904       -1.287596  ...  1.000337  4855.099327  5827.038372

[5 rows x 12 columns]