.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/bayesian_swissmetro/plot_b18b_ordinal_probit.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_bayesian_swissmetro_plot_b18b_ordinal_probit.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_bayesian_swissmetro_plot_b18b_ordinal_probit.py:


18. Ordinal probit model
========================

Bayesian estimation of an ordinal probit model.  This is just to illustrate the
syntax, as the data are not ordered.  But the example assume, for the
sake of it, that the alternatives are ordered as 1->2->3

Michel Bierlaire, EPFL
Mon Nov 17 2025, 16:44:27

.. GENERATED FROM PYTHON SOURCE LINES 13-20

.. code-block:: Python


    from IPython.core.display_functions import display

    import biogeme.biogeme_logging as blog
    from biogeme.bayesian_estimation import BayesianResults, get_pandas_estimated_parameters
    from biogeme.biogeme import BIOGEME
    from biogeme.expressions import Beta, OrderedLogProbit







.. GENERATED FROM PYTHON SOURCE LINES 21-22

See the data processing script: :ref:`swissmetro_data`.

.. GENERATED FROM PYTHON SOURCE LINES 22-24

.. code-block:: Python

    from swissmetro_data import CHOICE, TRAIN_COST_SCALED, TRAIN_TT_SCALED, database








.. GENERATED FROM PYTHON SOURCE LINES 25-26

We define a small but positive lower bound

.. GENERATED FROM PYTHON SOURCE LINES 26-31

.. code-block:: Python

    POSITIVE_LOWER_BOUND = 1.0e-5

    logger = blog.get_screen_logger(level=blog.INFO)
    logger.info('Example b18b_ordinal_probit.py')





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Example b18b_ordinal_probit.py 




.. GENERATED FROM PYTHON SOURCE LINES 32-33

Parameters to be estimated

.. GENERATED FROM PYTHON SOURCE LINES 33-36

.. code-block:: Python

    b_time = Beta('b_time', 0, None, None, 0)
    b_cost = Beta('b_cost', 0, None, None, 0)








.. GENERATED FROM PYTHON SOURCE LINES 37-38

Threshold parameters for the ordered probit.

.. GENERATED FROM PYTHON SOURCE LINES 40-41

:math:`\tau_1 \leq 0`.

.. GENERATED FROM PYTHON SOURCE LINES 41-43

.. code-block:: Python

    tau1 = Beta('tau1', -1, None, 0, 0)








.. GENERATED FROM PYTHON SOURCE LINES 44-45

:math:`\delta_2 \geq 0`.

.. GENERATED FROM PYTHON SOURCE LINES 45-47

.. code-block:: Python

    delta2 = Beta('delta2', 2, POSITIVE_LOWER_BOUND, None, 0)








.. GENERATED FROM PYTHON SOURCE LINES 48-49

:math:`\tau_2 = \tau_1 + \delta_2`

.. GENERATED FROM PYTHON SOURCE LINES 49-51

.. code-block:: Python

    tau2 = tau1 + delta2








.. GENERATED FROM PYTHON SOURCE LINES 52-53

Utility

.. GENERATED FROM PYTHON SOURCE LINES 53-55

.. code-block:: Python

    utility = b_time * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED








.. GENERATED FROM PYTHON SOURCE LINES 56-57

Associate each discrete indicator with an interval.

.. GENERATED FROM PYTHON SOURCE LINES 57-65

.. code-block:: Python

    log_probability = OrderedLogProbit(
        eta=utility,
        cutpoints=[tau1, tau2],
        y=CHOICE,
        categories=[1, 2, 3],
        neutral_labels=[],
    )








.. GENERATED FROM PYTHON SOURCE LINES 66-67

Create the Biogeme object.

.. GENERATED FROM PYTHON SOURCE LINES 67-70

.. code-block:: Python

    the_biogeme = BIOGEME(database, log_probability)
    the_biogeme.model_name = 'b18b_ordinal_probit'





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Biogeme parameters read from biogeme.toml. 




.. GENERATED FROM PYTHON SOURCE LINES 71-72

Estimate the parameters.

.. GENERATED FROM PYTHON SOURCE LINES 72-79

.. code-block:: Python

    try:
        results = BayesianResults.from_netcdf(
            filename=f'saved_results/{the_biogeme.model_name}.nc'
        )
    except FileNotFoundError:
        results = the_biogeme.bayesian_estimation()





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    *** Initial values of the parameters are obtained from the file __b18b_ordinal_probit.iter 
    Cannot read file __b18b_ordinal_probit.iter. Statement is ignored. 
    Starting values for the algorithm: {} 
    /Users/bierlair/MyFiles/github/biogeme/src/biogeme/biogeme.py:832: UserWarning: Note: JAX currently sees 1 CPU device. To parallelize across CPU devices, set XLA_FLAGS as above and restart Python/Jupyter.
    macOS / Linux (bash/zsh):
      export XLA_FLAGS="--xla_force_host_platform_device_count=<number_of_cores>"

    Jupyter (new cell, before `import jax`):
      %env XLA_FLAGS="--xla_force_host_platform_device_count=<number_of_cores>"

      warning_cpu_devices()
    Detected CPU devices: 1 | System logical cores: 12
    Current XLA_FLAGS: (none set)
    Platform: Darwin 24.6.0 | Python: 3.13.1
 
    Auto sampling: JAX available (devices=1, platforms=cpu) → numpyro/vectorized 
    /Users/bierlair/MyFiles/github/biogeme/src/biogeme/biogeme.py:859: UserWarning: The effect of Potentials on other parameters is ignored during prior predictive sampling. This is likely to lead to invalid or biased predictive samples.
      pm.sample_prior_predictive(

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    posterior_predictive_loglike finished in 295 ms
    expected_log_likelihood finished in 11 ms
    best_draw_log_likelihood finished in 10 ms
    waic_res finished in 629 ms
    waic finished in 630 ms
    loo_res finished in 7949 ms (7.95 s)
    loo finished in 7949 ms (7.95 s)
    Diagnostics computation took 24.6 seconds (cached). 
    File b18b_ordinal_probit.html has been generated. 
    Save simulation results on b18b_ordinal_probit.nc 
    Saved Bayesian results (posterior + metadata) to b18b_ordinal_probit.nc 




.. GENERATED FROM PYTHON SOURCE LINES 80-82

.. code-block:: Python

    print(results.short_summary())





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    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                                                 0:01:20.937926
    Posterior predictive log-likelihood (sum of log mean p)  -5788.53
    Expected log-likelihood E[log L(Y|θ)]                    -5791.09
    Best-draw log-likelihood (posterior upper bound)         -5789.06
    WAIC (Widely Applicable Information Criterion)           -5793.68
    WAIC Standard Error                                      50.22
    Effective number of parameters (p_WAIC)                  5.14
    LOO (Leave-One-Out Cross-Validation)                     -5793.68
    LOO Standard Error                                       50.22
    Effective number of parameters (p_LOO)                   5.15




.. GENERATED FROM PYTHON SOURCE LINES 83-85

.. code-block:: Python

    pandas_results = get_pandas_estimated_parameters(estimation_results=results)
    display(pandas_results)




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

         Name  Value (mean)  Value (median)  ...     R hat   ESS (bulk)   ESS (tail)
    0  b_time      0.018185        0.018066  ...  1.000283  3567.792410  3915.862286
    1  b_cost      0.687145        0.686680  ...  1.000589  3889.787673  3888.711159
    2    tau1     -0.604802       -0.604485  ...  1.000811  3684.885412  4010.346058
    3  delta2      1.755283        1.755430  ...  1.000272  4091.844948  3768.913913

    [4 rows x 12 columns]





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (2 minutes 17.846 seconds)


.. _sphx_glr_download_auto_examples_bayesian_swissmetro_plot_b18b_ordinal_probit.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_b18b_ordinal_probit.ipynb <plot_b18b_ordinal_probit.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_b18b_ordinal_probit.py <plot_b18b_ordinal_probit.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_b18b_ordinal_probit.zip <plot_b18b_ordinal_probit.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_