Note
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9. Nested logit model¶
Bayesian estimation of a nested logit model.
Michel Bierlaire, EPFL Mon Nov 03 2025, 20:02:56
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 import 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 lognested
from biogeme.nests import NestsForNestedLogit, OneNestForNestedLogit
logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b09_nested')
Example b09_nested
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, 0, 0)
b_cost = Beta('b_cost', 0, None, 0, 0)
nest_parameter = Beta('nest_parameter', 1, 1, 3, 0)
Definition of the utility functions.
v_train = asc_train + b_time * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
v_swissmetro = 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
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}
Definition of nests. Only the non-trivial nests must be defined. A trivial nest is a nest containing exactly one alternative. In this example, we create a nest for the existing modes, that is train (1) and car (3).
existing = OneNestForNestedLogit(
nest_param=nest_parameter, list_of_alternatives=[1, 3], name='existing'
)
nests = NestsForNestedLogit(choice_set=list(v), tuple_of_nests=(existing,))
The following elements do not appear in any nest and are assumed each to be alone in a separate nest: {2}. If it is not the intention, check the assignment of alternatives to nests.
Definition of the model. This is the contribution of each observation to the log likelihood function. The choice model is a nested logit, with availability conditions.
log_probability = lognested(v, av, nests, CHOICE)
Create the Biogeme object.
the_biogeme = BIOGEME(
database,
log_probability,
)
the_biogeme.model_name = 'b09_nested'
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 0:01:00.219654
Posterior predictive log-likelihood (sum of log mean p) -5234.05
Expected log-likelihood E[log L(Y|θ)] -5239.42
Best-draw log-likelihood (posterior upper bound) -5236.95
LOO (Leave-One-Out Cross-Validation) -5244.90
LOO Standard Error 62.49
Effective number of parameters (p_LOO) 10.85
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 -0.512124 ... 3462.981777 4175.576651
1 asc_car -0.166689 ... 3342.661567 3943.674385
2 b_time -0.900190 ... 3291.896591 4101.519864
3 b_cost -0.857870 ... 4318.675427 4783.129155
4 nest_parameter 2.056541 ... 3774.351367 4538.800107
[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 nest_parameter [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 nest_parameter [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]
We calculate the correlation between the error terms of the alternatives.
corr = nests.correlation(
parameters=summary_results.get_beta_values(),
alternatives_names={1: 'Train', 2: 'Swissmetro', 3: 'Car'},
)
print(corr)
Train Swissmetro Car
Train 1.000000 0.0 0.763558
Swissmetro 0.000000 1.0 0.000000
Car 0.763558 0.0 1.000000
Total running time of the script: (0 minutes 1.140 seconds)