10. Nested logit model normalized from bottom

Bayesian estimation of a nested logit model where the normalization is done at the

bottom level.

Michel Bierlaire, EPFL Mon Nov 03 2025, 20:07:02

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
from biogeme.models import lognested_mev_mu
from biogeme.nests import NestsForNestedLogit, OneNestForNestedLogit

logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b10nested_bottom.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_time = Beta('b_time', 0, None, 0, 0)
b_cost = Beta('b_cost', 0, None, 0, 0)

This is the scale parameter of the choice model. It is usually normalized to one. In this example, we normalize the nest parameter instead, and estimate the scale parameter for the model.

scale_parameter = Beta('scale_parameter', 0.5, POSITIVE_LOWER_BOUND, 1.0, 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. The nest parameter is normalized to 1.

nest_parameter = 1.0
existing = OneNestForNestedLogit(
    nest_param=nest_parameter, list_of_alternatives=[1, 3], name='existing'
)

nests = NestsForNestedLogit(choice_set=list(v), tuple_of_nests=(existing,))

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, where the scale parameter mu is explicitly involved.

log_probability = lognested_mev_mu(v, av, nests, CHOICE, scale_parameter)

Create the Biogeme object.

the_biogeme = BIOGEME(database, log_probability)
the_biogeme.model_name = 'b10_nested_bottom'

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())

Present the parameter estimates in a pandas table.

pandas_results = get_pandas_estimated_parameters(
    estimation_results=summary_results,
)
display(pandas_results)

Report the variables stored in the Bayesian estimation results.

display(summary_results.report_stored_variables())