Note
Go to the end to download the full example code.
26. Triangular mixture with panel data¶
Bayesian estimation of a mixture of logit models. The mixing distribution is user-defined (triangular, here). The datafile is organized as panel data.
Michel Bierlaire, EPFL Tue Nov 18 2025, 18:31:04
from functools import partial
from pathlib import Path
import pymc as pm
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,
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 b26triangular_panel_mixture.py')
Example b26triangular_panel_mixture.py
The scale parameters must stay away from zero. We define a small but positive lower bound
POSITIVE_LOWER_BOUND = 1.0e-5
Define a random parameter with a triangular distribution. The triangular distribution is not directly available from Biogeme. It has to be generated by a function provided by the user, based on PyMC available distributions.
See the PyMC documentation: https://www.pymc.io/projects/docs/en/stable/api/distributions.html
Mean of the distribution.
b_time = Beta('b_time', 0, None, None, 0)
Scale of the distribution. It is advised not to use 0 as starting value for the following parameter.
b_time_s = Beta('b_time_s', 1, POSITIVE_LOWER_BOUND, None, 0)
Distribution of the draws
TriangularFactory = partial(
pm.Triangular,
lower=-1.0,
c=0.0,
upper=1.0,
)
Associate the function with a name
DISTRIBUTIONS = {'TRIANGULAR': TriangularFactory}
Define a random parameter with a triangular distribution, designed to be used for Monte-Carlo simulation.
b_time_rnd = DistributedParameter(
'b_time_rnd',
b_time
+ b_time_s
* Draws('b_time_rnd_err_term', 'TRIANGULAR', dict_of_distributions=DISTRIBUTIONS),
)
Parameters to be estimated.
b_cost = Beta('b_cost', 0, None, None, 0)
The constants are distributed across individuals, to address serial correlation. In a panel setting, the corresponding draws are generated at the individual level. Wrapping them in DistributedParameter ensures they are expanded consistently when combined with observation-level variables.
asc_car = Beta('asc_car', 0, None, None, 0)
asc_car_s = Beta('asc_car_s', 1, None, None, 0)
asc_car_rnd = DistributedParameter(
'asc_car_rnd',
asc_car
+ asc_car_s
* Draws('asc_car_eps', 'TRIANGULAR', dict_of_distributions=DISTRIBUTIONS),
)
asc_train = Beta('asc_train', 0, None, None, 0)
asc_train_s = Beta('asc_train_s', 1, None, None, 0)
asc_train_rnd = DistributedParameter(
'asc_train_rnd',
asc_train
+ asc_train_s
* Draws('asc_train_eps', 'TRIANGULAR', dict_of_distributions=DISTRIBUTIONS),
)
asc_sm = Beta('asc_sm', 0, None, None, 1)
asc_sm_s = Beta('asc_sm_s', 1, None, None, 0)
asc_sm_rnd = DistributedParameter(
'asc_sm_rnd',
asc_sm
+ asc_sm_s * Draws('asc_sm_eps', 'TRIANGULAR', dict_of_distributions=DISTRIBUTIONS),
)
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 to the random parameters, the likelihood of one observation is given by the logit model (called the kernel).
conditional_log_probability = loglogit(v, av, CHOICE)
Create the Biogeme object.
the_biogeme = BIOGEME(
database,
conditional_log_probability,
)
the_biogeme.model_name = 'b26triangular_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 2000
Total number of draws 8000
Acceptance rate target 0.9
Run time 0:13:41.519060
Posterior predictive log-likelihood (sum of log mean p) -2182.64
Expected log-likelihood E[log L(Y|θ)] -2363.93
Best-draw log-likelihood (posterior upper bound) -2259.81
LOO (Leave-One-Out Cross-Validation) -3020.46
LOO Standard Error 79.89
Effective number of parameters (p_LOO) 837.82
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 asc_train -0.402460 ... 3071.528780 3744.116801
1 asc_train_s 2.873625 ... 7.245802 10.896058
2 b_time -6.022726 ... 3546.323885 4916.398275
3 b_cost -3.308140 ... 3832.302584 4600.112667
4 asc_sm_s 1.848717 ... 7.273402 11.831726
5 asc_car 0.373026 ... 3252.382355 4829.100317
6 asc_car_s 4.635733 ... 7.198686 10.602251
7 b_time_s 8.784831 ... 2137.646087 3394.346458
[8 rows x 12 columns]
Report the variables stored in the Bayesian estimation results.
display(summary_results.report_stored_variables())
group ... shape
0 constant_data ... [6768]
1 constant_data ... [6768]
2 constant_data ... [6768]
3 constant_data ... [6768]
4 constant_data ... [6768]
5 constant_data ... [6768]
6 constant_data ... [6768]
7 constant_data ... [6768]
8 constant_data ... [6768]
9 constant_data ... [6768]
10 log_likelihood ... [4, 2000, 752]
11 posterior ... [4, 2000]
12 posterior ... [4, 2000, 752]
13 posterior ... [4, 2000, 6768]
14 posterior ... [4, 2000, 752]
15 posterior ... [4, 2000]
16 posterior ... [4, 2000, 752]
17 posterior ... [4, 2000, 6768]
18 posterior ... [4, 2000, 752]
19 posterior ... [4, 2000]
20 posterior ... [4, 2000]
21 posterior ... [4, 2000, 752]
22 posterior ... [4, 2000, 6768]
23 posterior ... [4, 2000, 752]
24 posterior ... [4, 2000]
25 posterior ... [4, 2000]
26 posterior ... [4, 2000]
27 posterior ... [4, 2000, 6768]
28 posterior ... [4, 2000, 752]
29 posterior ... [4, 2000, 752]
30 posterior ... [4, 2000]
31 posterior ... [4, 2000, 752]
32 prior ... [1, 2000]
33 prior ... [1, 2000, 752]
34 prior ... [1, 2000, 6768]
35 prior ... [1, 2000, 752]
36 prior ... [1, 2000]
37 prior ... [1, 2000, 752]
38 prior ... [1, 2000, 6768]
39 prior ... [1, 2000, 752]
40 prior ... [1, 2000]
41 prior ... [1, 2000]
42 prior ... [1, 2000, 752]
43 prior ... [1, 2000, 6768]
44 prior ... [1, 2000, 752]
45 prior ... [1, 2000]
46 prior ... [1, 2000]
47 prior ... [1, 2000]
48 prior ... [1, 2000, 6768]
49 prior ... [1, 2000, 752]
50 prior ... [1, 2000, 752]
51 prior ... [1, 2000]
52 prior ... [1, 2000, 752]
53 sample_stats ... [4, 2000]
54 sample_stats ... [4, 2000]
55 sample_stats ... [4, 2000]
56 sample_stats ... [4, 2000]
57 sample_stats ... [4, 2000]
58 sample_stats ... [4, 2000]
59 sample_stats ... [4, 2000]
[60 rows x 4 columns]
Total running time of the script: (0 minutes 1.139 seconds)