.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/bayesian_swissmetro/plot_b05_normal_mixture.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_bayesian_swissmetro_plot_b05_normal_mixture.py: .. _plot_b05_normal_mixture: 5. Mixture of logit models: normal distribution =============================================== Example of a normal mixture of logit models, using Bayesian inference. Michel Bierlaire, EPFL Thu Nov 20 2025, 11:26:01 .. GENERATED FROM PYTHON SOURCE LINES 12-20 .. code-block:: Python import biogeme.biogeme_logging as blog from IPython.core.display_functions import display from biogeme.bayesian_estimation import BayesianResults, get_pandas_estimated_parameters from biogeme.biogeme import BIOGEME from biogeme.expressions import Beta, DistributedParameter, Draws from biogeme.models import loglogit .. GENERATED FROM PYTHON SOURCE LINES 21-22 See the data processing script: :ref:`swissmetro_data`. .. GENERATED FROM PYTHON SOURCE LINES 22-39 .. code-block:: Python 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, ) logger = blog.get_screen_logger(level=blog.INFO) logger.info('Example b05_normal_mixtures.py') .. rst-class:: sphx-glr-script-out .. code-block:: none Example b05_normal_mixtures.py .. GENERATED FROM PYTHON SOURCE LINES 40-41 The scale parameters must stay away from zero. We define a small but positive lower bound .. GENERATED FROM PYTHON SOURCE LINES 41-43 .. code-block:: Python POSITIVE_LOWER_BOUND = 1.0e-5 .. GENERATED FROM PYTHON SOURCE LINES 44-45 Parameters to be estimated .. GENERATED FROM PYTHON SOURCE LINES 45-50 .. code-block:: Python 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_cost = Beta('b_cost', 0, None, 0, 0) .. GENERATED FROM PYTHON SOURCE LINES 51-53 Define a random parameter, normally distributed, designed to be used for Monte-Carlo simulation. .. GENERATED FROM PYTHON SOURCE LINES 53-55 .. code-block:: Python b_time = Beta('b_time', 0, None, 0, 0) .. GENERATED FROM PYTHON SOURCE LINES 56-57 It is advised *not* to use 0 as starting value for the following parameter. .. GENERATED FROM PYTHON SOURCE LINES 57-60 .. code-block:: Python b_time_s = Beta('b_time_s', 10, POSITIVE_LOWER_BOUND, None, 0) b_time_eps = Draws('b_time_eps', 'NORMAL') .. GENERATED FROM PYTHON SOURCE LINES 61-63 The purpose of the `DistributedParameter` operator is to explicitly store the simulated individual-level parameters in the output file. .. GENERATED FROM PYTHON SOURCE LINES 63-65 .. code-block:: Python b_time_rnd = DistributedParameter('b_time_rnd', b_time + b_time_s * b_time_eps) .. GENERATED FROM PYTHON SOURCE LINES 66-67 Definition of the utility functions. .. GENERATED FROM PYTHON SOURCE LINES 67-71 .. code-block:: Python v_train = asc_train + b_time_rnd * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED v_swissmetro = asc_sm + b_time_rnd * SM_TT_SCALED + b_cost * SM_COST_SCALED v_car = asc_car + b_time_rnd * CAR_TT_SCALED + b_cost * CAR_CO_SCALED .. GENERATED FROM PYTHON SOURCE LINES 72-73 Associate utility functions with the numbering of alternatives. .. GENERATED FROM PYTHON SOURCE LINES 73-75 .. code-block:: Python v = {1: v_train, 2: v_swissmetro, 3: v_car} .. GENERATED FROM PYTHON SOURCE LINES 76-77 Associate the availability conditions with the alternatives. .. GENERATED FROM PYTHON SOURCE LINES 77-79 .. code-block:: Python av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP} .. GENERATED FROM PYTHON SOURCE LINES 80-86 When performing maximum likelihood estimation, in order to obtain the loglikelihood, we would first calculate the kernel conditional on b_time_rnd, and then integrate over b_time_rnd using Monte-Carlo. However, when performing Bayesian estimation, the random parameters will be explicitly simulated. Therefore, what the algorithm needs is the *conditional* log likelihood, which is simply a (log) logit here. This is one of the most important advantage of this estimation method: it does not require to calculate the complicated integrals. .. GENERATED FROM PYTHON SOURCE LINES 86-88 .. code-block:: Python conditional_log_likelihood = loglogit(v, av, CHOICE) .. GENERATED FROM PYTHON SOURCE LINES 89-90 These notes will be included as such in the report file. .. GENERATED FROM PYTHON SOURCE LINES 90-92 .. code-block:: Python USER_NOTES = 'Example of Bayesian estimation of a mixture of logit models with three alternatives' .. GENERATED FROM PYTHON SOURCE LINES 93-94 Create the Biogeme object. .. GENERATED FROM PYTHON SOURCE LINES 94-101 .. code-block:: Python the_biogeme = BIOGEME( database, conditional_log_likelihood, user_notes=USER_NOTES, ) the_biogeme.model_name = 'b05_normal_mixture' .. rst-class:: sphx-glr-script-out .. code-block:: none Biogeme parameters read from biogeme.toml. .. GENERATED FROM PYTHON SOURCE LINES 102-103 Estimate the parameters. .. GENERATED FROM PYTHON SOURCE LINES 103-110 .. code-block:: Python try: bayesian_results = BayesianResults.from_netcdf( filename=f'saved_results/{the_biogeme.model_name}.nc' ) except FileNotFoundError: bayesian_results = the_biogeme.bayesian_estimation() .. rst-class:: sphx-glr-script-out .. code-block:: none Loaded NetCDF file size: 1.8 GB load finished in 9243 ms (9.24 s) .. GENERATED FROM PYTHON SOURCE LINES 111-112 Get the results in a pandas table .. GENERATED FROM PYTHON SOURCE LINES 112-116 .. code-block:: Python pandas_results = get_pandas_estimated_parameters( estimation_results=bayesian_results, ) display(pandas_results) .. rst-class:: sphx-glr-script-out .. code-block:: none Diagnostics computation took 69.8 seconds (cached). Name Value (mean) Value (median) ... R hat ESS (bulk) ESS (tail) 0 asc_train -0.400545 -0.399224 ... 1.000274 4764.255073 5518.165472 1 asc_car 0.139390 0.139290 ... 1.001596 2727.944492 5227.745062 2 b_time -2.272711 -2.269103 ... 1.004406 1225.734728 2834.903303 3 b_time_s 1.675588 1.672388 ... 1.006639 868.962343 1806.335511 4 b_cost -1.288904 -1.287596 ... 1.000337 4855.099327 5827.038372 [5 rows x 12 columns] .. rst-class:: sphx-glr-timing **Total running time of the script:** (1 minutes 19.148 seconds) .. _sphx_glr_download_auto_examples_bayesian_swissmetro_plot_b05_normal_mixture.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_b05_normal_mixture.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_b05_normal_mixture.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_b05_normal_mixture.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_