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Gaussian MIMIC model: maximum likelihood estimation¶
This example estimates a Gaussian MIMIC model with one latent variable and no choice component. The observed Likert indicators are treated as continuous responses and modeled with Gaussian measurement equations.
The example introduces the latent-variable-only setting used later in the hybrid-choice tutorial. The latent-variable structure itself is imported from a separate semantic specification file. This script adds the Gaussian measurement configuration, the normalization constraints required for identification, and the maximum-likelihood estimation setup.
The script performs the following steps:
load the latent-variable and indicator specifications,
define a Gaussian measurement configuration for all indicators,
define the normalization constraints used for identification,
resolve the semantic specification into an estimable model,
build the Biogeme expressions from the resolved model,
integrate over the latent variable using Monte Carlo draws,
construct the corresponding log-likelihood,
estimate the model, or reload previously saved estimation results,
display the estimated parameters as pandas and LaTeX tables.
Michel Bierlaire Mon Jun 15 2026, 09:54:37
from __future__ import annotations
from likert_spec import likert_indicators, likert_types
from number_of_draws import NUMBER_OF_DRAWS
from one_latent_variable_spec import DEFAULT_SIGMA_START, latent_variables
from optima import read_data
import biogeme.biogeme_logging as blog
from biogeme.biogeme import BIOGEME
from biogeme.expressions import log
from biogeme.latent_variables import (
BuildContext,
EstimationMode,
Fixing,
IndicatorMeasurementSpec,
MeasurementConfiguration,
MeasurementIntercept,
MeasurementLoading,
MeasurementModel,
NormalizationPlan,
PositiveParameterSpec,
PositivityMode,
build_biogeme_model,
resolve_model,
)
from biogeme.results_processing import (
get_latex_estimated_parameters,
get_latex_general_statistics,
get_pandas_estimated_parameters,
)
logger = blog.get_screen_logger(level=blog.INFO)
Gaussian measurement configuration for all Likert indicators.
measurement_configuration = MeasurementConfiguration(
specifications=[
IndicatorMeasurementSpec(
indicator_name=indicator.name,
measurement_model=MeasurementModel.GAUSSIAN,
measurement_sigma=PositiveParameterSpec(start=DEFAULT_SIGMA_START),
)
for indicator in likert_indicators
]
)
Load the Optima data.¶
database = read_data()
Define the build context for maximum likelihood estimation.¶
The build context specifies how the semantic latent-variable specification is translated into estimable Biogeme expressions. We start from the default maximum-likelihood context and explicitly select the log-exp parameterization for positive parameters.
default_context = BuildContext.default(EstimationMode.MAXIMUM_LIKELIHOOD)
context = BuildContext(
estimation_mode=default_context.estimation_mode,
draw_type=default_context.draw_type,
positivity_mode=PositivityMode.LOG_EXP,
naming=default_context.naming,
ordinal_eps=default_context.ordinal_eps,
ordinal_enforce_order=default_context.ordinal_enforce_order,
)
Identification constraints for the latent variable¶
A latent variable is not directly observed, so its location and scale are not identified unless we impose normalization constraints.
We use a reference-indicator strategy based on indicator Envir01:
The intercept of the reference indicator is fixed to 0. This anchors the location of the latent-variable scale.
The loading of the reference indicator on the latent variable is fixed to -1. This anchors the scale of the latent variable and fixes its orientation. The negative sign means that higher values of the latent variable imply lower expected values for
Envir01.
With these two constraints in place, the remaining intercepts and loadings are interpreted relative to the reference indicator.
normalization_plan = NormalizationPlan()
normalization_plan.add(
Fixing(MeasurementIntercept('Envir01'), 0.0, note='reference indicator: location')
)
normalization_plan.add(
Fixing(
MeasurementLoading('car_centric_attitude', 'Envir01'),
-1.0,
note='reference indicator: scale and orientation',
)
)
Resolve the semantic specification.¶
The resolver combines the latent-variable specification, the indicator definitions, the Gaussian measurement configuration, and the normalization plan into an internal resolved model.
resolved_model = resolve_model(
latent_variables=latent_variables,
likert_indicators=likert_indicators,
likert_types=likert_types,
measurement_configuration=measurement_configuration,
context=context,
normalization_plan=normalization_plan,
)
Build the Biogeme expressions.¶
The builder translates the resolved model into expressions that can be used by Biogeme for maximum-likelihood estimation.
built_model = build_biogeme_model(resolved_model)
Log-likelihood¶
The builder provides the Monte-Carlo integrated measurement likelihood. Taking its logarithm yields the log-likelihood used for maximum-likelihood estimation.
integrated_likelihood = built_model.integrated_likelihood
log_likelihood = log(integrated_likelihood)
Estimate the model with Biogeme.¶
Existing results are reloaded from the YAML file when available.
biogeme = BIOGEME(
database,
log_likelihood,
number_of_draws=NUMBER_OF_DRAWS,
calculating_second_derivatives='never',
numerically_safe=False,
max_iterations=5_000,
group_of_parameters=built_model.parameter_groups,
)
biogeme.model_name = 'plot_h02_lv_mimic_gauss'
yaml_file_name = f'saved_results/{biogeme.model_name}.yaml'
results = biogeme.estimate_or_load(yaml_file_name=yaml_file_name)
Biogeme parameters read from biogeme.toml.
Estimation results are read from saved_results/plot_h02_lv_mimic_gauss.yaml. No estimation is performed.
Display a compact summary and the estimated parameters.
print(results.short_summary())
pandas_results = get_pandas_estimated_parameters(
estimation_results=results,
group_of_parameters=built_model.parameter_groups,
)
for group_name, pandas_table in pandas_results.items():
print(group_name if group_name else 'Estimated parameters')
print(pandas_table)
general_statistics = get_latex_general_statistics(estimation_results=results)
print(general_statistics)
estimated_parameters = get_latex_estimated_parameters(
estimation_results=results,
group_of_parameters=built_model.parameter_groups,
)
for group_name, latex_table in estimated_parameters.items():
print(group_name if group_name else 'Estimated parameters')
print(latex_table)
Results for model plot_h02_lv_mimic_gauss
Nbr of parameters: 32
Sample size: 889
Excluded data: 0
Final log likelihood: -13029.98
Akaike Information Criterion: 26123.95
Bayesian Information Criterion: 26277.24
Structural equation
Name ... BHHH p-value
0 struct_car_centric_attitude_intercept ... 0.000000
1 struct_car_centric_attitude_top_manager ... 0.523173
2 struct_car_centric_attitude_car_oriented_parents ... 0.020318
3 struct_car_centric_attitude_high_education ... 0.000033
4 struct_car_centric_attitude_low_education ... 0.325572
5 struct_car_centric_attitude_used_to_go_to_scho... ... 0.858527
6 struct_car_centric_attitude_sigma_log ... 0.970816
[7 rows x 5 columns]
Measurement equation: Envir01
Name Value ... BHHH t-stat. BHHH p-value
7 measurement_Envir01_sigma_log 0.009691 ... 0.215424 0.829437
[1 rows x 5 columns]
Measurement equation: Envir02
Name ... BHHH p-value
8 measurement_intercept_Envir02 ... 0.000000
9 measurement_coefficient_car_centric_attitude_E... ... 0.000000
10 measurement_Envir02_sigma_log ... 0.363014
[3 rows x 5 columns]
Measurement equation: Envir06
Name ... BHHH p-value
11 measurement_intercept_Envir06 ... 0.0
12 measurement_coefficient_car_centric_attitude_E... ... 0.0
13 measurement_Envir06_sigma_log ... 0.0
[3 rows x 5 columns]
Measurement equation: Mobil03
Name ... BHHH p-value
14 measurement_intercept_Mobil03 ... 0.000000
15 measurement_coefficient_car_centric_attitude_M... ... 0.006289
16 measurement_Mobil03_sigma_log ... 0.000000
[3 rows x 5 columns]
Measurement equation: Mobil05
Name ... BHHH p-value
17 measurement_intercept_Mobil05 ... 0.000000e+00
18 measurement_coefficient_car_centric_attitude_M... ... 2.253014e-08
19 measurement_Mobil05_sigma_log ... 0.000000e+00
[3 rows x 5 columns]
Measurement equation: Mobil08
Name ... BHHH p-value
20 measurement_intercept_Mobil08 ... 0.000000e+00
21 measurement_coefficient_car_centric_attitude_M... ... 9.726642e-10
22 measurement_Mobil08_sigma_log ... 3.364021e-08
[3 rows x 5 columns]
Measurement equation: Mobil09
Name ... BHHH p-value
23 measurement_intercept_Mobil09 ... 0.000000e+00
24 measurement_coefficient_car_centric_attitude_M... ... 1.798117e-12
25 measurement_Mobil09_sigma_log ... 3.289191e-03
[3 rows x 5 columns]
Measurement equation: Mobil10
Name ... BHHH p-value
26 measurement_intercept_Mobil10 ... 0.000000e+00
27 measurement_coefficient_car_centric_attitude_M... ... 2.731662e-09
28 measurement_Mobil10_sigma_log ... 0.000000e+00
[3 rows x 5 columns]
Measurement equation: LifSty07
Name ... BHHH p-value
29 measurement_intercept_LifSty07 ... 0.000000
30 measurement_coefficient_car_centric_attitude_L... ... 0.076143
31 measurement_LifSty07_sigma_log ... 0.000000
[3 rows x 5 columns]
%% General statistics
\section{General statistics}
\begin{tabular}{ll}
Number of estimated parameters & 32 \\
Sample size & 889 \\
Excluded observations & 0 \\
Init log likelihood & -26577.53 \\
Final log likelihood & -13029.98 \\
Likelihood ratio test for the init. model & 27095.1 \\
Rho-square for the init. model & 0.51 \\
Rho-square-bar for the init. model & 0.509 \\
Akaike Information Criterion & 26123.95 \\
Bayesian Information Criterion & 26277.24 \\
Final gradient norm & 2.5847E-01 \\
Number of draws & 50000 \\
Draws generation time & 0:00:13.248030 \\
Types of draws & struct\_car\_centric\_attitude\_draws: NORMAL\_MLHS\_ANTI \\
Bootstrapping time & None \\
Algorithm & \verb$BFGS with trust region for simple bound constraints$ \\
Cause of termination & \verb$Relative gradient = 5.4e-06 <= 6.1e-06$ \\
Number of function evaluations & \verb$807$ \\
Number of gradient evaluations & \verb$459$ \\
Number of hessian evaluations & \verb$0$ \\
Number of iterations & \verb$348$ \\
Optimization time & \verb$0:25:49.527588$ \\
Proportion of Hessian calculation & \verb$0/229 = 0.0%$ \\
Relative gradient & \verb$5.356e-06$ \\
\end{tabular}
Structural equation
\begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l}
& & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\
Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\
number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\
\hline
0 & struct\_car\_centric\_attitude\_intercept & -2&64 & 0&115 & -23&0 & 0&00 \\
1 & struct\_car\_centric\_attitude\_top\_manager & 0&0914 & 0&143 & 0&638 & 0&523 \\
2 & struct\_car\_centric\_attitude\_car\_oriented\_parents & 0&237 & 0&102 & 2&32 & 0&0203 \\
3 & struct\_car\_centric\_attitude\_high\_education & -0&541 & 0&130 & -4&15 & 3&27e-05 \\
4 & struct\_car\_centric\_attitude\_low\_education & 0&123 & 0&125 & 0&983 & 0&326 \\
5 & struct\_car\_centric\_attitude\_used\_to\_go\_to\_school\_by\_car & 0&0998 & 0&560 & 0&178 & 0&859 \\
6 & struct\_car\_centric\_attitude\_sigma\_log & -0&00271 & 0&0740 & -0&0366 & 0&971 \\
\end{tabular}
Measurement equation: Envir01
\begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l}
& & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\
Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\
number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\
\hline
7 & measurement\_Envir01\_sigma\_log & 0&00969 & 0&0450 & 0&215 & 0&829 \\
\end{tabular}
Measurement equation: Envir02
\begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l}
& & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\
Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\
number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\
\hline
8 & measurement\_intercept\_Envir02 & 1&76 & 0&160 & 11&0 & 0&00 \\
9 & measurement\_coefficient\_car\_centric\_attitude\_Envir02 & -0&597 & 0&0574 & -10&4 & 0&00 \\
10 & measurement\_Envir02\_sigma\_log & 0&0247 & 0&0271 & 0&910 & 0&363 \\
\end{tabular}
Measurement equation: Envir06
\begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l}
& & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\
Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\
number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\
\hline
11 & measurement\_intercept\_Envir06 & 3&30 & 0&101 & 32&8 & 0&00 \\
12 & measurement\_coefficient\_car\_centric\_attitude\_Envir06 & -0&376 & 0&0385 & -9&77 & 0&00 \\
13 & measurement\_Envir06\_sigma\_log & -0&303 & 0&0183 & -16&5 & 0&00 \\
\end{tabular}
Measurement equation: Mobil03
\begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l}
& & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\
Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\
number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\
\hline
14 & measurement\_intercept\_Mobil03 & 2&86 & 0&156 & 18&3 & 0&00 \\
15 & measurement\_coefficient\_car\_centric\_attitude\_Mobil03 & -0&150 & 0&0550 & -2&73 & 0&00629 \\
16 & measurement\_Mobil03\_sigma\_log & 0&289 & 0&0286 & 10&1 & 0&00 \\
\end{tabular}
Measurement equation: Mobil05
\begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l}
& & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\
Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\
number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\
\hline
17 & measurement\_intercept\_Mobil05 & 2&42 & 0&172 & 14&0 & 0&00 \\
18 & measurement\_coefficient\_car\_centric\_attitude\_Mobil05 & -0&339 & 0&0607 & -5&59 & 2&25e-08 \\
19 & measurement\_Mobil05\_sigma\_log & 0&320 & 0&0288 & 11&1 & 0&00 \\
\end{tabular}
Measurement equation: Mobil08
\begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l}
& & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\
Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\
number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\
\hline
20 & measurement\_intercept\_Mobil08 & 3&22 & 0&150 & 21&4 & 0&00 \\
21 & measurement\_coefficient\_car\_centric\_attitude\_Mobil08 & 0&336 & 0&0550 & 6&11 & 9&73e-10 \\
22 & measurement\_Mobil08\_sigma\_log & 0&141 & 0&0255 & 5&52 & 3&36e-08 \\
\end{tabular}
Measurement equation: Mobil09
\begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l}
& & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\
Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\
number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\
\hline
23 & measurement\_intercept\_Mobil09 & 2&95 & 0&133 & 22&1 & 0&00 \\
24 & measurement\_coefficient\_car\_centric\_attitude\_Mobil09 & -0&351 & 0&0498 & -7&05 & 1&80e-12 \\
25 & measurement\_Mobil09\_sigma\_log & 0&0575 & 0&0196 & 2&94 & 0&00329 \\
\end{tabular}
Measurement equation: Mobil10
\begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l}
& & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\
Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\
number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\
\hline
26 & measurement\_intercept\_Mobil10 & 5&30 & 0&266 & 19&9 & 0&00 \\
27 & measurement\_coefficient\_car\_centric\_attitude\_Mobil10 & 0&533 & 0&0896 & 5&95 & 2&73e-09 \\
28 & measurement\_Mobil10\_sigma\_log & 0&594 & 0&0428 & 13&9 & 0&00 \\
\end{tabular}
Measurement equation: LifSty07
\begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l}
& & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\
Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\
number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\
\hline
29 & measurement\_intercept\_LifSty07 & 2&49 & 0&141 & 17&7 & 0&00 \\
30 & measurement\_coefficient\_car\_centric\_attitude\_LifSty07 & 0&0866 & 0&0488 & 1&77 & 0&0761 \\
31 & measurement\_LifSty07\_sigma\_log & 0&217 & 0&0247 & 8&76 & 0&00 \\
\end{tabular}
Total running time of the script: (0 minutes 1.083 seconds)