15. Discrete mixture with panel data

Bayesian estimation of a discrete mixture of logit models, also called latent class model. The datafile is organized as panel data.

Michel Bierlaire, EPFL Sat Nov 15 2025, 17:39:13

from pathlib import Path

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,
    exp,
    log,
)
from biogeme.models import logit

logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b15_panel_discrete.py')

Example b15_panel_discrete.py

Parameters to be estimated. One version for each latent_old class.

NUMBER_OF_CLASSES = 2
b_cost = [Beta(f'b_cost_class{i}', 0, None, None, 0) for i in range(NUMBER_OF_CLASSES)]

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

b_time = [Beta(f'b_time_class{i}', 0, None, None, 0) for i in range(NUMBER_OF_CLASSES)]

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

b_time_s = [
    Beta(f'b_time_s_class{i}', 1, None, None, 0) for i in range(NUMBER_OF_CLASSES)
]
b_time_rnd = [
    DistributedParameter(
        f'b_time_rnd_class{i}',
        b_time[i] + b_time_s[i] * Draws(f'b_time_eps_class{i}', 'NORMAL'),
    )
    for i in range(NUMBER_OF_CLASSES)
]

We do the same for the constants, to address serial correlation.

asc_car = [
    Beta(f'asc_car_class{i}', 0, None, None, 0) for i in range(NUMBER_OF_CLASSES)
]
asc_car_s = [
    Beta(f'asc_car_s_class{i}', 1, None, None, 0) for i in range(NUMBER_OF_CLASSES)
]
asc_car_rnd = [
    DistributedParameter(
        f'asc_car_rnd_class{i}',
        asc_car[i] + asc_car_s[i] * Draws(f'asc_car_eps_class{i}', 'NORMAL'),
    )
    for i in range(NUMBER_OF_CLASSES)
]

asc_train = [
    Beta(f'asc_train_class{i}', 0, None, None, 0) for i in range(NUMBER_OF_CLASSES)
]
asc_train_s = [
    Beta(f'asc_train_s_class{i}', 1, None, None, 0) for i in range(NUMBER_OF_CLASSES)
]
asc_train_rnd = [
    DistributedParameter(
        f'asc_train_rnd_class{i}',
        asc_train[i] + asc_train_s[i] * Draws(f'asc_train_eps_class{i}', 'NORMAL'),
    )
    for i in range(NUMBER_OF_CLASSES)
]

asc_sm = [Beta(f'asc_sm_class{i}', 0, None, None, 1) for i in range(NUMBER_OF_CLASSES)]
asc_sm_s = [
    Beta(f'asc_sm_s_class{i}', 1, None, None, 0) for i in range(NUMBER_OF_CLASSES)
]
asc_sm_rnd = [
    DistributedParameter(
        f'asc_sm_rnd_class{i}',
        asc_sm[i] + asc_sm_s[i] * Draws(f'asc_sm_eps_class{i}', 'NORMAL'),
    )
    for i in range(NUMBER_OF_CLASSES)
]

Class membership probability. Note: for Bayesian estimation, this should not call the logit model.

score_class_0 = Beta('score_class_0', -1.7, None, None, 0)
probability_class_1 = 1 / (1 + exp(score_class_0))
probability_class_0 = 1 - probability_class_1

In class 0, it is assumed that the time coefficient is zero.

b_time_rnd[0] = 0

Utility functions.

v_train_per_class = [
    asc_train_rnd[i] + b_time_rnd[i] * TRAIN_TT_SCALED + b_cost[i] * TRAIN_COST_SCALED
    for i in range(NUMBER_OF_CLASSES)
]
v_swissmetro_per_class = [
    asc_sm_rnd[i] + b_time_rnd[i] * SM_TT_SCALED + b_cost[i] * SM_COST_SCALED
    for i in range(NUMBER_OF_CLASSES)
]
v_car_per_class = [
    asc_car_rnd[i] + b_time_rnd[i] * CAR_TT_SCALED + b_cost[i] * CAR_CO_SCALED
    for i in range(NUMBER_OF_CLASSES)
]
v_per_class = [
    {1: v_train_per_class[i], 2: v_swissmetro_per_class[i], 3: v_car_per_class[i]}
    for i in range(NUMBER_OF_CLASSES)
]

Associate the availability conditions with the alternatives.

av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP}

The choice model is a discrete mixture of logit, with availability conditions We calculate the conditional probability for each class.

conditional_probability_per_class = [
    logit(v_per_class[i], av, CHOICE) for i in range(NUMBER_OF_CLASSES)
]

Conditional to the random variables, likelihood for the individual.

conditional_choice_probability = (
    probability_class_0 * conditional_probability_per_class[0]
    + probability_class_1 * conditional_probability_per_class[1]
)

We need the log probability per observation

conditional_log_probability = log(conditional_choice_probability)

the_biogeme = BIOGEME(
    database,
    conditional_log_probability,
    warmup=4000,
    bayesian_draws=4000,
    chains=4,
)
the_biogeme.model_name = 'b15_panel_discrete'

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                                4000
Total number of draws                                    16000
Acceptance rate target                                   0.9
Run time                                                 0:27:27.347566
Posterior predictive log-likelihood (sum of log mean p)  -2156.93
Expected log-likelihood E[log L(Y|θ)]                    -2342.30
Best-draw log-likelihood (posterior upper bound)         -2232.19
LOO (Leave-One-Out Cross-Validation)                     -3043.55
LOO Standard Error                                       78.39
Effective number of parameters (p_LOO)                   886.62

Present the parameter estimates in a pandas table.

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

                  Name  Value (mean)  ...   ESS (bulk)   ESS (tail)
0        score_class_0     -5.269729  ...  1107.881335  1279.186401
1     asc_train_class0     -2.572247  ...  4185.700594  8295.082582
2   asc_train_s_class0      0.745081  ...  6338.755023  7886.597741
3        b_cost_class0      2.733313  ...  2353.080701  2203.302640
4      asc_sm_s_class0      1.367400  ...  2252.533260  2750.966721
5       asc_car_class0     -0.729003  ...  3928.727344  5681.970166
6     asc_car_s_class0      0.995962  ...  5495.069755  8217.012967
7     asc_train_class1     -0.315026  ...  1456.703040  2634.804557
8   asc_train_s_class1      2.271266  ...   559.042723   869.542749
9        b_time_class1     -6.518591  ...  1623.510543  2064.187908
10     b_time_s_class1     -4.018999  ...  2015.845568  3133.353383
11       b_cost_class1     -4.218585  ...  2141.955596  3048.286362
12     asc_sm_s_class1     -0.166371  ...    36.034480   217.660331
13      asc_car_class1      0.500710  ...  3180.093599  5338.435005
14    asc_car_s_class1     -2.014075  ...     7.172490    10.306197

[15 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]
..            ...            ...            ...        ...
87   sample_stats         energy  [chain, draw]  [4, 4000]
88   sample_stats             lp  [chain, draw]  [4, 4000]
89   sample_stats        n_steps  [chain, draw]  [4, 4000]
90   sample_stats      step_size  [chain, draw]  [4, 4000]
91   sample_stats     tree_depth  [chain, draw]  [4, 4000]

[92 rows x 4 columns]