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

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,
    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 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 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())

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                                                 0:03:39.208410
Posterior predictive log-likelihood (sum of log mean p)  -4162.13
Expected log-likelihood E[log L(Y|θ)]                    -4532.20
Best-draw log-likelihood (posterior upper bound)         -4231.22
LOO (Leave-One-Out Cross-Validation)                     -5196.91
LOO Standard Error                                       52.81
Effective number of parameters (p_LOO)                   1034.78

Present the parameter estimates in a pandas table.

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

        Name  Value (mean)  Value (median)  ...     R hat   ESS (bulk)   ESS (tail)
0  asc_train     -0.399395       -0.400166  ...  0.999962  3577.391788  4640.525909
1    asc_car      0.140966        0.139651  ...  1.000144  2218.946224  3267.833341
2     b_time     -2.276639       -2.274176  ...  1.001548  1082.820890  1784.924732
3   b_time_s      1.679823        1.677619  ...  1.002909   777.818456  1290.233632
4     b_cost     -1.289441       -1.288741  ...  1.000734  4419.290332  5276.551069

[5 rows x 12 columns]

Report the variables stored in the Bayesian estimation results.

display(summary_results.report_stored_variables())

             group           variable                dims            shape
0    constant_data          CAR_AV_SP               [obs]           [6768]
1    constant_data      CAR_CO_SCALED               [obs]           [6768]
2    constant_data      CAR_TT_SCALED               [obs]           [6768]
3    constant_data             CHOICE               [obs]           [6768]
4    constant_data              SM_AV               [obs]           [6768]
5    constant_data     SM_COST_SCALED               [obs]           [6768]
6    constant_data       SM_TT_SCALED               [obs]           [6768]
7    constant_data        TRAIN_AV_SP               [obs]           [6768]
8    constant_data  TRAIN_COST_SCALED               [obs]           [6768]
9    constant_data    TRAIN_TT_SCALED               [obs]           [6768]
10  log_likelihood            _choice  [chain, draw, obs]  [4, 2000, 6768]
11       posterior            asc_car       [chain, draw]        [4, 2000]
12       posterior          asc_train       [chain, draw]        [4, 2000]
13       posterior             b_cost       [chain, draw]        [4, 2000]
14       posterior             b_time       [chain, draw]        [4, 2000]
15       posterior         b_time_eps  [chain, draw, obs]  [4, 2000, 6768]
16       posterior         b_time_rnd  [chain, draw, obs]  [4, 2000, 6768]
17       posterior           b_time_s       [chain, draw]        [4, 2000]
18       posterior           log_like  [chain, draw, obs]  [4, 2000, 6768]
19           prior            asc_car       [chain, draw]        [1, 2000]
20           prior          asc_train       [chain, draw]        [1, 2000]
21           prior             b_cost       [chain, draw]        [1, 2000]
22           prior             b_time       [chain, draw]        [1, 2000]
23           prior         b_time_eps  [chain, draw, obs]  [1, 2000, 6768]
24           prior         b_time_rnd  [chain, draw, obs]  [1, 2000, 6768]
25           prior           b_time_s       [chain, draw]        [1, 2000]
26           prior           log_like  [chain, draw, obs]  [1, 2000, 6768]
27    sample_stats    acceptance_rate       [chain, draw]        [4, 2000]
28    sample_stats          diverging       [chain, draw]        [4, 2000]
29    sample_stats             energy       [chain, draw]        [4, 2000]
30    sample_stats                 lp       [chain, draw]        [4, 2000]
31    sample_stats            n_steps       [chain, draw]        [4, 2000]
32    sample_stats          step_size       [chain, draw]        [4, 2000]
33    sample_stats         tree_depth       [chain, draw]        [4, 2000]