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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')
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,))
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, 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'
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 0:01:00.323492
Posterior predictive log-likelihood (sum of log mean p) -5234.08
Expected log-likelihood E[log L(Y|θ)] -5239.43
Best-draw log-likelihood (posterior upper bound) -5236.94
LOO (Leave-One-Out Cross-Validation) -5244.88
LOO Standard Error 62.49
Effective number of parameters (p_LOO) 10.80
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 -1.056517 ... 2983.359140 3567.481626
1 asc_car -0.347840 ... 3081.463560 3409.419691
2 b_time -1.850810 ... 3923.110568 4445.632886
3 b_cost -1.767597 ... 3905.582600 4542.012141
4 scale_parameter 0.484927 ... 3625.135474 3963.390778
[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 scale_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 scale_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]
Total running time of the script: (0 minutes 1.146 seconds)