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17. Mixture with lognormal distribution¶
Bayesian estimation of a mixture of logit models. The mixing distribution is distributed as a log normal.
Michel Bierlaire, EPFL Sat Nov 15 2025, 18:20:02
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, exp
from biogeme.models import loglogit
logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b17_lognormal_mixture.py')
Example b17_lognormal_mixture.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.
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, normally distributed, designed to be used for Monte-Carlo simulation.
b_time_param = Beta('b_time', 0, None, None, 0)
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, 2, 0)
Define a random parameter, log normally distributed, designed to be used for Monte-Carlo simulation.
b_time_eps = Draws('b_time_eps', 'NORMAL')
b_time_rnd = DistributedParameter(
'b_time_rnd', -exp(b_time_param + 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}
Conditional to b_time_rnd, we have a 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 = 'b17_lognormal_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 0:03:23.066740
Posterior predictive log-likelihood (sum of log mean p) -4063.57
Expected log-likelihood E[log L(Y|θ)] -4413.86
Best-draw log-likelihood (posterior upper bound) -4117.38
LOO (Leave-One-Out Cross-Validation) -5087.16
LOO Standard Error 50.36
Effective number of parameters (p_LOO) 1023.59
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 -0.347316 -0.347581 ... 1.000644 2543.525278 3947.035274
1 b_time 0.574652 0.574737 ... 1.000738 1824.241399 2507.492539
2 b_cost -1.386627 -1.385219 ... 1.001746 2734.046323 3961.219536
3 asc_car 0.175382 0.173789 ... 1.000502 1897.821773 3050.235129
4 b_time_s 1.256082 1.251646 ... 1.001949 834.754534 1633.067158
[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 1.132 seconds)