Note
Go to the end to download the full example code.
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
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
See the data processing script: Data preparation for Swissmetro.
from swissmetro_data import CHOICE, TRAIN_COST_SCALED, TRAIN_TT_SCALED, database
We define a small but positive lower bound
POSITIVE_LOWER_BOUND = 1.0e-5
logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b18b_ordinal_probit.py')
Example b18b_ordinal_probit.py
Parameters to be estimated
b_time = Beta('b_time', 0, None, None, 0)
b_cost = Beta('b_cost', 0, None, None, 0)
Threshold parameters for the ordered probit.
\(\tau_1 \leq 0\).
tau1 = Beta('tau1', -1, None, 0, 0)
\(\delta_2 \geq 0\).
delta2 = Beta('delta2', 2, POSITIVE_LOWER_BOUND, None, 0)
\(\tau_2 = \tau_1 + \delta_2\)
tau2 = tau1 + delta2
Utility
utility = b_time * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
Associate each discrete indicator with an interval.
log_probability = OrderedLogProbit(
eta=utility,
cutpoints=[tau1, tau2],
y=CHOICE,
categories=[1, 2, 3],
neutral_labels=[],
)
Create the Biogeme object.
the_biogeme = BIOGEME(database, log_probability)
the_biogeme.model_name = 'b18b_ordinal_probit'
Biogeme parameters read from biogeme.toml.
Estimate the parameters.
try:
results = BayesianResults.from_netcdf(
filename=f'saved_results/{the_biogeme.model_name}.nc'
)
except FileNotFoundError:
results = the_biogeme.bayesian_estimation()
*** 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
print(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 0:01:20.937926
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
pandas_results = get_pandas_estimated_parameters(estimation_results=results)
display(pandas_results)
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]
Total running time of the script: (2 minutes 17.846 seconds)