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7. Latent class model¶
Bayesian estimation of a discrete mixture of logit (or latent class model).
Michel Bierlaire, EPFL Mon Nov 03 2025, 13:36:51
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,
)
from biogeme.bayesian_estimation import (
BayesianResults,
BayesianResultsSummary,
get_pandas_estimated_parameters,
)
from biogeme.biogeme import BIOGEME
from biogeme.expressions import Beta, log
from biogeme.models import logit
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_time = Beta('b_time', 0, None, None, 0)
b_cost = Beta('b_cost', 0, None, None, 0)
Class membership probability.
prob_class1 = Beta('prob_class1', 0.5, 0, 1, 0)
prob_class2 = 1 - prob_class1
Definition of the utility functions for latent_old class 1, where the time coefficient is zero.
v_train_class_1 = asc_train + b_cost * TRAIN_COST_SCALED
v_swissmetro_class_1 = asc_sm + b_cost * SM_COST_SCALED
v_car_class_1 = asc_car + b_cost * CAR_CO_SCALED
Associate utility functions with the numbering of alternatives.
v_class_1 = {1: v_train_class_1, 2: v_swissmetro_class_1, 3: v_car_class_1}
Definition of the utility functions for latent_old class 2, whete the time coefficient is estimated.
v_train_class_2 = asc_train + b_time * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
v_swissmetro_class_2 = asc_sm + b_time * SM_TT_SCALED + b_cost * SM_COST_SCALED
v_car_class_2 = asc_car + b_time * CAR_TT_SCALED + b_cost * CAR_CO_SCALED
Associate utility functions with the numbering of alternatives.
v_class_2 = {1: v_train_class_2, 2: v_swissmetro_class_2, 3: v_car_class_2}
Associate the availability conditions with the alternatives.
av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP}
The choice model is a discrete mixture of logit, with availability conditions
choice_probability_class_1 = logit(v_class_1, av, CHOICE)
choice_probability_class_2 = logit(v_class_2, av, CHOICE)
prob = (
prob_class1 * choice_probability_class_1 + prob_class2 * choice_probability_class_2
)
log_probability = log(prob)
Create the Biogeme object
the_biogeme = BIOGEME(database, log_probability)
the_biogeme.model_name = 'b07_discrete_mixture'
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 0:01:17.790031
Posterior predictive log-likelihood (sum of log mean p) -5208.05
Expected log-likelihood E[log L(Y|θ)] -5210.99
Best-draw log-likelihood (posterior upper bound) -5208.53
LOO (Leave-One-Out Cross-Validation) -5213.94
LOO Standard Error 53.21
Effective number of parameters (p_LOO) 5.89
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 -0.399529 ... 4062.552923 5060.974518
1 b_cost -1.266278 ... 5583.692324 5172.725233
2 asc_car 0.123778 ... 4449.233525 4867.994695
3 b_time -2.813642 ... 3765.422393 4294.823690
4 prob_class1 0.252521 ... 4497.808463 4639.971249
[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 log_like [chain, draw, obs] [4, 2000, 6768]
16 posterior prob_class1 [chain, draw] [4, 2000]
17 prior asc_car [chain, draw] [1, 2000]
18 prior asc_train [chain, draw] [1, 2000]
19 prior b_cost [chain, draw] [1, 2000]
20 prior b_time [chain, draw] [1, 2000]
21 prior log_like [chain, draw, obs] [1, 2000, 6768]
22 prior prob_class1 [chain, draw] [1, 2000]
23 sample_stats acceptance_rate [chain, draw] [4, 2000]
24 sample_stats diverging [chain, draw] [4, 2000]
25 sample_stats energy [chain, draw] [4, 2000]
26 sample_stats lp [chain, draw] [4, 2000]
27 sample_stats n_steps [chain, draw] [4, 2000]
28 sample_stats step_size [chain, draw] [4, 2000]
29 sample_stats tree_depth [chain, draw] [4, 2000]
Total running time of the script: (0 minutes 1.134 seconds)