Note
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1a. Estimation of a logit model (Bayesian)¶
This example illustrates the Bayesian estimation of a logit model using Biogeme and the Swissmetro stated-preference dataset.
The choice situation involves three transportation alternatives:
train,
car,
Swissmetro.
The script is organized into sections separated by # %% markers.
These markers are important because the examples are automatically
converted into Jupyter notebooks for the documentation. Each # %%
marker defines a new notebook cell.
Tested with Biogeme 3.3.3.
Michel Bierlaire, EPFL Tue Jun 09 2026, 15:30:00
from pathlib import Path
from IPython.core.display_functions import display
Import the processed Swissmetro dataset and all variables used in the specification.
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
from biogeme.models import loglogit
Configure the logger. DEBUG provides detailed information about the execution of the example.
logger = blog.get_screen_logger(level=blog.DEBUG)
logger.info('Example b01a_logit.py')
[INFO] 2026-06-16 21:37:48,822 Example b01a_logit.py <plot_b01a_logit.py:59>
Alternative-specific constants. By default, Biogeme assigns a normal prior distribution centered on the starting value. Bounds, when specified, are also used to truncate the prior.
asc_car = Beta('asc_car', 0, None, None, 0)
asc_train = Beta('asc_train', 0, None, None, 0)
The Swissmetro constant is normalized to zero for identification purposes. Setting the last argument of Beta to 1 fixes the parameter at its default value and removes it from the estimation.
asc_sm = Beta('asc_sm', 0, None, None, 1)
Coefficients associated with travel time and travel cost. The upper bound is set to zero to enforce a non-positive marginal utility.
b_time = Beta('b_time', 0, None, 0, 0)
b_cost = Beta('b_cost', 0, None, 0, 0)
Utility functions for the three alternatives.
v_train = asc_train + b_time * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
v_sm = asc_sm + b_time * SM_TT_SCALED + b_cost * SM_COST_SCALED
v_car = asc_car + b_time * CAR_TT_SCALED + b_cost * CAR_CO_SCALED
Mapping between alternative identifiers and utility functions.
v = {1: v_train, 2: v_sm, 3: v_car}
Availability conditions associated with each alternative.
av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP}
Log of the choice probability for the logit model. This expression defines the contribution of one observation to the log likelihood.
log_probability = loglogit(v, av, CHOICE)
Create the Biogeme object.
the_biogeme = BIOGEME(database, log_probability)
the_biogeme.model_name = 'b01a_logit'
[DEBUG] 2026-06-16 21:37:48,823 READ FILE biogeme.toml : automatic <parameters.py:184>
[INFO] 2026-06-16 21:37:48,823 Default values of the Biogeme parameters are used. <parameters.py:222>
[WARNING] 2026-06-16 21:37:48,825 File biogeme.toml has been created <parameters.py:259>
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 0:00:12.070779
Posterior predictive log-likelihood (sum of log mean p) -5329.75
Expected log-likelihood E[log L(Y|θ)] -5333.30
Best-draw log-likelihood (posterior upper bound) -5331.26
LOO (Leave-One-Out Cross-Validation) -5336.84
LOO Standard Error 59.64
Effective number of parameters (p_LOO) 7.09
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 -0.700990 -0.700994 ... 1.000527 3409.726462 4290.972112
1 asc_car -0.154262 -0.153463 ... 1.000833 3478.259900 4665.015623
2 b_time -1.279176 -1.279355 ... 1.000398 3611.838832 4177.952522
3 b_cost -1.084311 -1.084529 ... 1.000116 5397.458136 5039.059388
[4 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 log_like [chain, draw, obs] [4, 2000, 6768]
16 prior asc_car [chain, draw] [1, 2000]
17 prior asc_train [chain, draw] [1, 2000]
18 prior b_cost [chain, draw] [1, 2000]
19 prior b_time [chain, draw] [1, 2000]
20 prior log_like [chain, draw, obs] [1, 2000, 6768]
21 sample_stats acceptance_rate [chain, draw] [4, 2000]
22 sample_stats diverging [chain, draw] [4, 2000]
23 sample_stats energy [chain, draw] [4, 2000]
24 sample_stats lp [chain, draw] [4, 2000]
25 sample_stats n_steps [chain, draw] [4, 2000]
26 sample_stats step_size [chain, draw] [4, 2000]
27 sample_stats tree_depth [chain, draw] [4, 2000]
Total running time of the script: (0 minutes 1.539 seconds)