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16. Latent class model with panel data¶
Bayesian estimation of a discrete mixture of logit models, also called latent class model. The class membership model includes socio-economic variables. The datafile is organized as panel data.
Michel Bierlaire, EPFL Sat Nov 15 2025, 18:12:05
from pathlib import Path
from IPython.core.display_functions import display
See the data processing script: Panel data preparation for Swissmetro.
from swissmetro_panel import (
CAR_AV_SP,
CAR_CO_SCALED,
CAR_TT_SCALED,
CHOICE,
INCOME,
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 b16_panel_discrete_socio_eco.py')
Example b16_panel_discrete_socio_eco.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)
]
Parameters for the class membership model.
class_cte = Beta('class_cte', 0, None, None, 0)
class_inc = Beta('class_inc', 0, None, None, 0)
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)
]
Class membership model.
score_class_0 = class_cte + class_inc * INCOME
probability_class_1 = 1 / (1 + exp(score_class_0))
probability_class_0 = 1 - probability_class_1
Conditional on 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=40,
bayesian_draws=40,
chains=1,
)
the_biogeme.model_name = 'b16_panel_discrete_socio_eco'
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 1
Number of draws per chain 40
Total number of draws 40
Acceptance rate target 0.9
Run time 0:01:03.347356
Posterior predictive log-likelihood (sum of log mean p) -2169.72
Expected log-likelihood E[log L(Y|θ)] -2351.29
Best-draw log-likelihood (posterior upper bound) -2306.25
LOO (Leave-One-Out Cross-Validation) -2648.12
LOO Standard Error 73.12
Effective number of parameters (p_LOO) 478.40
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 class_cte -4.988973 ... 20.070992 25.592163
1 class_inc -0.753073 ... 4.264195 20.215104
2 asc_train_class0 -3.020917 ... 33.364506 19.698549
3 asc_train_s_class0 -0.062239 ... 64.082400 28.842505
4 b_cost_class0 3.440974 ... 59.299171 36.086980
5 asc_sm_s_class0 0.665195 ... 21.246417 17.373155
6 asc_car_class0 -0.321605 ... 64.082400 61.172597
7 asc_car_s_class0 1.022808 ... 40.790573 37.630113
8 asc_train_class1 -0.425399 ... 3.898933 9.140108
9 asc_train_s_class1 2.494917 ... 3.649699 5.612999
10 b_time_class1 -6.293892 ... 38.394709 45.103858
11 b_time_s_class1 3.836986 ... 5.023141 32.148605
12 b_cost_class1 -4.039385 ... 4.166326 6.082433
13 asc_sm_s_class1 0.272630 ... 2.085751 5.692884
14 asc_car_class1 0.386951 ... 17.645090 27.094474
15 asc_car_s_class1 3.946871 ... 18.274849 15.748812
[16 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 INCOME [obs] [6768]
.. ... ... ... ...
90 sample_stats energy [chain, draw] [1, 40]
91 sample_stats lp [chain, draw] [1, 40]
92 sample_stats n_steps [chain, draw] [1, 40]
93 sample_stats step_size [chain, draw] [1, 40]
94 sample_stats tree_depth [chain, draw] [1, 40]
[95 rows x 4 columns]
Total running time of the script: (0 minutes 1.151 seconds)