Note
Go to the end to download the full example code.
12. Mixture of logit with panel data¶
Bayesian estimation of a mixture of logit models. The datafile is organized as panel data. Note that, with Bayesian estimation, there is no need to calculate a Monte-Carlo integration.
Michel Bierlaire, EPFL Mon Jun 08 2026, 16:45:17
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
from biogeme.models import loglogit
logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b12_panel.py')
Example b12_panel.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.
b_cost = Beta('b_cost', 0, None, 0, 0)
Define a random parameter, normally distributed across individuals, designed to be used for Monte-Carlo simulation.
b_time = Beta('b_time', 0, None, 0, 0)
b_time_s = Beta('b_time_s', 1, POSITIVE_LOWER_BOUND, None, 0)
b_time_eps = Draws('b_time_eps', 'NORMAL')
b_time_eps.set_draw_per_individual()
b_time_rnd = DistributedParameter('b_time_rnd', b_time + b_time_s * b_time_eps)
We do the same for the constants, to address serial correlation.
asc_car = Beta('asc_car', 0, None, None, 0)
asc_car_s = Beta('asc_car_s', 1, POSITIVE_LOWER_BOUND, None, 0)
asc_car_eps = Draws('asc_car_eps', 'NORMAL')
asc_car_eps.set_draw_per_individual()
asc_car_rnd = DistributedParameter('asc_car_rnd', asc_car + asc_car_s * asc_car_eps)
asc_train = Beta('asc_train', 0, None, None, 0)
asc_train_s = Beta('asc_train_s', 1, POSITIVE_LOWER_BOUND, None, 0)
asc_train_eps = Draws('asc_train_eps', 'NORMAL')
asc_car_eps.set_draw_per_individual()
asc_train_rnd = DistributedParameter(
'asc_train_rnd', asc_train + asc_train_s * asc_train_eps
)
asc_sm = Beta('asc_sm', 0, None, None, 0)
asc_sm_s = Beta('asc_sm_s', 1, POSITIVE_LOWER_BOUND, None, 0)
asc_sm_eps = Draws('asc_sm_eps', 'NORMAL')
asc_sm_eps.set_draw_per_individual()
asc_sm_rnd = DistributedParameter('asc_sm_rnd', asc_sm + asc_sm_s * asc_sm_eps)
Definition of the utility functions.
v_train = asc_train_rnd + b_time_rnd * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
v_swissmetro = asc_sm_rnd + b_time_rnd * SM_TT_SCALED + b_cost * SM_COST_SCALED
v_car = asc_car_rnd + 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 on the random parameters, the likelihood of one observation is given by the logit model (called the kernel).
log_probability_one_observation = loglogit(v, av, CHOICE)
As the objective is to illustrate the syntax, we calculate the Monte-Carlo approximation with a small number of draws.
the_biogeme = BIOGEME(
database,
log_probability_one_observation,
warmup=10,
bayesian_draws=10,
chains=4,
)
the_biogeme.model_name = 'b12_panel'
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 10
Total number of draws 40
Acceptance rate target 0.9
Run time 0:00:37.450653
Posterior predictive log-likelihood (sum of log mean p) -2642.14
Expected log-likelihood E[log L(Y|θ)] -3248.84
Best-draw log-likelihood (posterior upper bound) -2364.31
LOO (Leave-One-Out Cross-Validation) -3994.52
LOO Standard Error 118.86
Effective number of parameters (p_LOO) 1352.38
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 -1.096755 -1.446587 ... 1.800519 12.850226 14.950761
1 asc_sm -0.114012 -0.643452 ... 1.959028 12.244405 19.509276
2 asc_car 0.178364 -0.314985 ... 1.873658 12.628609 17.331207
3 asc_train_s 1.510174 1.403013 ... 2.793608 11.260119 15.688116
4 b_time -3.626670 -3.261197 ... 1.962347 12.521625 14.950761
5 b_time_s 2.383355 2.781433 ... 1.774714 12.960763 22.604423
6 b_cost -2.222127 -1.887257 ... 2.484453 11.468328 10.000000
7 asc_sm_s 0.601985 0.049868 ... 4.355404 10.395467 10.000000
8 asc_car_s 0.798178 0.000040 ... 3.677059 10.887780 10.000000
[9 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]
.. ... ... ... ...
57 sample_stats energy [chain, draw] [4, 10]
58 sample_stats lp [chain, draw] [4, 10]
59 sample_stats n_steps [chain, draw] [4, 10]
60 sample_stats step_size [chain, draw] [4, 10]
61 sample_stats tree_depth [chain, draw] [4, 10]
[62 rows x 4 columns]
Total running time of the script: (0 minutes 0.432 seconds)