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6. Mixture of logit models: uniform distribution¶
Example of a uniform mixture of logit models, using Bayesian inference.
Michel Bierlaire, EPFL Thu Nov 20 2025, 11:30:45
from pathlib import Path
from IPython.core.display_functions import display
See the data processing script: Data preparation for Swissmetro.
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, DistributedParameter, Draws
from biogeme.models import loglogit
logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b06_unif_mixture.py')
Example b06_unif_mixture.py
Parameters to be estimated.
asc_car = Beta('asc_car', 0, None, None, 0)
asc_train = Beta('asc_train', 0, None, None, 0)
asc_sm = Beta('asc_sm', 0, None, None, 1)
b_cost = Beta('b_cost', 0, None, None, 0)
Define a random parameter, uniformly distributed, designed to be used for Monte-Carlo simulation.
b_time = Beta('b_time', 0, None, None, 0)
b_time_s = Beta('b_time_s', 1, None, None, 0)
b_time_eps = Draws('b_time_eps', 'UNIFORMSYM')
b_time_rnd = DistributedParameter('b_time_rnd', b_time + b_time_s * b_time_eps)
Definition of the utility functions.
v_train = asc_train + b_time_rnd * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
v_swissmetro = asc_sm + b_time_rnd * SM_TT_SCALED + b_cost * SM_COST_SCALED
v_car = asc_car + 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}
In order to obtain the loglikelihood, we would first calculate the kernel conditional on b_time_rnd, and the integrate over b_time_rnd using Monte-Carlo. However, when performing Bayesian estimation, the random parameters will be explicitly simulated. Therefore, what the algorithm needs is the conditional log likelihood, which is simply a (log) logit here.
conditional_log_likelihood = loglogit(v, av, CHOICE)
Create the Biogeme object.
the_biogeme = BIOGEME(database, conditional_log_likelihood)
the_biogeme.model_name = 'b06_unif_mixture'
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 0:05:04.297235
Posterior predictive log-likelihood (sum of log mean p) -4189.64
Expected log-likelihood E[log L(Y|θ)] -4537.88
Best-draw log-likelihood (posterior upper bound) -4289.51
LOO (Leave-One-Out Cross-Validation) -5207.58
LOO Standard Error 52.55
Effective number of parameters (p_LOO) 1017.94
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 -0.385251 -0.386907 ... 0.999860 4004.538764 4928.859229
1 b_time -2.330903 -2.327276 ... 1.000641 1535.637884 2226.834647
2 b_time_s -0.003793 0.010845 ... 1.735559 6.094277 34.786804
3 b_cost -1.280172 -1.279601 ... 0.999993 6958.057685 5332.780312
4 asc_car 0.146294 0.145507 ... 1.000952 2843.418549 3778.927111
[5 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 b_time_eps [chain, draw, obs] [4, 2000, 6768]
16 posterior b_time_rnd [chain, draw, obs] [4, 2000, 6768]
17 posterior b_time_s [chain, draw] [4, 2000]
18 posterior log_like [chain, draw, obs] [4, 2000, 6768]
19 prior asc_car [chain, draw] [1, 2000]
20 prior asc_train [chain, draw] [1, 2000]
21 prior b_cost [chain, draw] [1, 2000]
22 prior b_time [chain, draw] [1, 2000]
23 prior b_time_eps [chain, draw, obs] [1, 2000, 6768]
24 prior b_time_rnd [chain, draw, obs] [1, 2000, 6768]
25 prior b_time_s [chain, draw] [1, 2000]
26 prior log_like [chain, draw, obs] [1, 2000, 6768]
27 sample_stats acceptance_rate [chain, draw] [4, 2000]
28 sample_stats diverging [chain, draw] [4, 2000]
29 sample_stats energy [chain, draw] [4, 2000]
30 sample_stats lp [chain, draw] [4, 2000]
31 sample_stats n_steps [chain, draw] [4, 2000]
32 sample_stats step_size [chain, draw] [4, 2000]
33 sample_stats tree_depth [chain, draw] [4, 2000]
Total running time of the script: (0 minutes 1.140 seconds)