POPSRegression#

class popsregression.POPSRegression(*, max_iter=300, tol=0.001, alpha_1=1e-06, alpha_2=1e-06, lambda_1=1e-06, lambda_2=1e-06, alpha_init=None, lambda_init=None, compute_score=False, fit_intercept=False, copy_X=True, verbose=False, mode_threshold=1e-08, resample_density=1.0, resampling_method='uniform', percentile_clipping=0.0, leverage_percentile=50.0, posterior='hypercube')#

Bayesian regression for low-noise data with misspecification uncertainty.

Fits a linear model using BayesianRidge, then estimates weight uncertainties accounting for model misspecification using the POPS (Pointwise Optimal Parameter Sets) algorithm [1]. Unlike standard Bayesian regression, the aleatoric noise precision alpha_ is not used for predictions, as it should be negligible in the low-noise regime.

Standard Bayesian regression can only estimate epistemic and aleatoric uncertainties. In the low-noise limit, weight uncertainties (sigma_ in BayesianRidge) are significantly underestimated as they only account for epistemic uncertainties that decay with increasing data. POPS corrects this by estimating misspecification uncertainty from pointwise optimal parameter sets.

Parameters:

max_iterint, default=300

Maximum number of iterations for the BayesianRidge convergence loop.

tolfloat, default=1e-3

Convergence threshold. Stop the algorithm if the coefficient vector has converged.

alpha_1float, default=1e-6

Shape parameter for the Gamma distribution prior over alpha_.

alpha_2float, default=1e-6

Inverse scale (rate) parameter for the Gamma distribution prior over alpha_.

lambda_1float, default=1e-6

Shape parameter for the Gamma distribution prior over lambda_.

lambda_2float, default=1e-6

Inverse scale (rate) parameter for the Gamma distribution prior over lambda_.

alpha_initfloat, default=None

Initial value for alpha_ (precision of the noise). If None, alpha_init is 1 / Var(y).

lambda_initfloat, default=None

Initial value for lambda_ (precision of the weights). If None, lambda_init is 1.

compute_scorebool, default=False

If True, compute the log marginal likelihood at each step.

fit_interceptbool, default=False

Whether to fit an intercept. If True, a constant column is appended to X (rather than centering) so that the intercept participates in the POPS posterior estimation.

copy_Xbool, default=True

If True, X will be copied; else, it may be overwritten.

verbosebool, default=False

Verbose mode when fitting the model.

mode_thresholdfloat, default=1e-8

Eigenvalue threshold (relative to max) for determining the effective dimensionality of the POPS posterior. Eigenvalues below mode_threshold * max_eigenvalue are discarded.

resample_densityfloat, default=1.0

Number of resampled points per training point. The actual number of samples is max(100, int(resample_density * n_samples)).

resampling_method{‘uniform’, ‘sobol’, ‘latin’, ‘halton’}, default=’uniform’

Quasi-random sampling method for generating points within the POPS hypercube posterior.

percentile_clippingfloat, default=0.0

Percentile to clip from each end when determining hypercube bounds. The hypercube spans the [percentile_clipping, 100 - percentile_clipping] range. Should be between 0 and 50.

leverage_percentilefloat, default=50.0

Only training points with leverage scores above this percentile are used for POPS posterior estimation. Higher values accelerate fitting by focusing on high-leverage points.

posterior{‘hypercube’, ‘ensemble’}, default=’hypercube’

Form of the POPS parameter posterior:

'hypercube': fit an axis-aligned box in PCA space (default).
'ensemble': use raw pointwise corrections directly.

Attributes:

coef_ndarray of shape (n_features,): Coefficients of the regression model (posterior mean).
intercept_float: Independent term in the decision function. Set to 0.0 if fit_intercept=False.
alpha_float: Estimated precision of the noise. Not used for prediction.
lambda_float: Estimated precision of the weights.
sigma_ndarray of shape (n_features, n_features): Estimated epistemic variance-covariance matrix of the weights.
misspecification_sigma_ndarray of shape (n_features, n_features): Estimated misspecification variance-covariance matrix from POPS.
posterior_samples_ndarray of shape (n_features, n_posterior_samples): Samples from the POPS posterior, representing plausible weight perturbations.
scores_ndarray of shape (n_iter_,): Value of the log marginal likelihood at each iteration. Only available if compute_score=True.
n_iter_int: The actual number of iterations to reach convergence.
n_features_in_int: Number of features seen during fit.
feature_names_in_ndarray of shape (n_features_in_,): Names of features seen during fit. Defined only when X has feature names that are all strings.

See also

sklearn.linear_model.BayesianRidge: Bayesian ridge regression without misspecification correction.
sklearn.linear_model.ARDRegression: Bayesian ARD regression.

References

[1]

Swinburne, T.D. and Perez, D. (2025). “Parameter uncertainties for imperfect surrogate models in the low-noise regime.” Machine Learning: Science and Technology, 6, 015008. :doi:`10.1088/2632-2153/ad9fce`

Examples

>>> import numpy as np
>>> from popsregression import POPSRegression
>>> rng = np.random.RandomState(0)
>>> X = np.array([[1, 1], [1, 2], [2, 2], [2, 3]])
>>> y = np.dot(X, np.array([1, 2])) + 0.01 * rng.randn(4)
>>> reg = POPSRegression()
>>> reg.fit(X, y)
POPSRegression()
>>> reg.predict(np.array([[3, 5]]))
array([...])

Methods

`fit`(X, y[, sample_weight])	Fit the POPS regression model.
`get_metadata_routing`()	Get metadata routing of this object.
`get_params`([deep])	Get parameters for this estimator.
`predict`(X[, return_std, return_bounds, ...])	Predict using the POPS regression model.
`score`(X, y[, sample_weight])	Return coefficient of determination on test data.
`set_fit_request`(*[, sample_weight])	Configure whether metadata should be requested to be passed to the `fit` method.
`set_params`(**params)	Set the parameters of this estimator.
`set_predict_request`(*[, return_bounds, ...])	Configure whether metadata should be requested to be passed to the `predict` method.
`set_score_request`(*[, sample_weight])	Configure whether metadata should be requested to be passed to the `score` method.

fit(X, y, sample_weight=None)#

Fit the POPS regression model.

Parameters:

Xarray-like of shape (n_samples, n_features): Training data.
yarray-like of shape (n_samples,): Target values.
sample_weightarray-like of shape (n_samples,), default=None: Individual weights for each sample.

Returns:

selfobject: Returns the instance itself.

get_metadata_routing()#

Get metadata routing of this object.

Please check User Guide on how the routing mechanism works.

Returns:

routingMetadataRequest: A MetadataRequest encapsulating routing information.

get_params(deep=True)#

Get parameters for this estimator.

Parameters:

deepbool, default=True: If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:

paramsdict: Parameter names mapped to their values.

predict(X, return_std=False, return_bounds=False, return_epistemic_std=False)#

Predict using the POPS regression model.

In addition to the standard return_std from BayesianRidge, this method can return prediction bounds (min/max over the posterior) and epistemic-only uncertainty.

Parameters:

Xarray-like of shape (n_samples, n_features): Samples to predict for.
return_stdbool, default=False: If True, return the combined (misspecification + epistemic) standard deviation.
return_boundsbool, default=False: If True, return the max and min predictions over the POPS posterior samples.
return_epistemic_stdbool, default=False: If True, return the epistemic-only standard deviation (from sigma_, excluding misspecification).

Returns:

y_meanndarray of shape (n_samples,): Predicted mean values.
y_stdndarray of shape (n_samples,): Combined standard deviation. Only returned if return_std=True.
y_maxndarray of shape (n_samples,): Upper bound from posterior samples. Only returned if return_bounds=True.
y_minndarray of shape (n_samples,): Lower bound from posterior samples. Only returned if return_bounds=True.
y_epistemic_stdndarray of shape (n_samples,): Epistemic-only standard deviation. Only returned if return_epistemic_std=True.

score(X, y, sample_weight=None)#

Return coefficient of determination on test data.

The coefficient of determination, $R^2$, is defined as $(1 - \frac{u}{v})$, where $u$ is the residual sum of squares ((y_true - y_pred)** 2).sum() and $v$ is the total sum of squares ((y_true - y_true.mean()) ** 2).sum(). The best possible score is 1.0 and it can be negative (because the model can be arbitrarily worse). A constant model that always predicts the expected value of y, disregarding the input features, would get a $R^2$ score of 0.0.

Parameters:

Xarray-like of shape (n_samples, n_features): Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead with shape (n_samples, n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.
yarray-like of shape (n_samples,) or (n_samples, n_outputs): True values for X.
sample_weightarray-like of shape (n_samples,), default=None: Sample weights.

Returns:

scorefloat: $R^2$ of self.predict(X) w.r.t. y.

Notes

The $R^2$ score used when calling score on a regressor uses multioutput='uniform_average' from version 0.23 to keep consistent with default value of r2_score(). This influences the score method of all the multioutput regressors (except for MultiOutputRegressor).

set_fit_request(*, sample_weight: bool | None | str = '$UNCHANGED$') → POPSRegression#

Configure whether metadata should be requested to be passed to the fit method.

Note that this method is only relevant when this estimator is used as a sub-estimator within a meta-estimator and metadata routing is enabled with enable_metadata_routing=True (see sklearn.set_config()). Please check the User Guide on how the routing mechanism works.

The options for each parameter are:

True: metadata is requested, and passed to fit if provided. The request is ignored if metadata is not provided.
False: metadata is not requested and the meta-estimator will not pass it to fit.
None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
str: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.

Added in version 1.3.

Parameters:

sample_weightstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED: Metadata routing for sample_weight parameter in fit.

Returns:

selfobject: The updated object.

set_params(**params)#

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Parameters:

**paramsdict: Estimator parameters.

Returns:

selfestimator instance: Estimator instance.

Configure whether metadata should be requested to be passed to the predict method.

Note that this method is only relevant when this estimator is used as a sub-estimator within a meta-estimator and metadata routing is enabled with enable_metadata_routing=True (see sklearn.set_config()). Please check the User Guide on how the routing mechanism works.

The options for each parameter are:

True: metadata is requested, and passed to predict if provided. The request is ignored if metadata is not provided.
False: metadata is not requested and the meta-estimator will not pass it to predict.
None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
str: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.

Added in version 1.3.

Parameters:

return_boundsstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED: Metadata routing for return_bounds parameter in predict.
return_epistemic_stdstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED: Metadata routing for return_epistemic_std parameter in predict.
return_stdstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED: Metadata routing for return_std parameter in predict.

Returns:

selfobject: The updated object.

set_score_request(*, sample_weight: bool | None | str = '$UNCHANGED$') → POPSRegression#

Configure whether metadata should be requested to be passed to the score method.

Note that this method is only relevant when this estimator is used as a sub-estimator within a meta-estimator and metadata routing is enabled with enable_metadata_routing=True (see sklearn.set_config()). Please check the User Guide on how the routing mechanism works.

The options for each parameter are:

True: metadata is requested, and passed to score if provided. The request is ignored if metadata is not provided.
False: metadata is not requested and the meta-estimator will not pass it to score.
None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
str: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.

Added in version 1.3.

Parameters:

sample_weightstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED: Metadata routing for sample_weight parameter in score.

Returns:

selfobject: The updated object.

Examples using `popsregression.POPSRegression`#

POPS vs BayesianRidge: Uncertainty for Low-Noise Surrogates

POPSRegression#

Examples using popsregression.POPSRegression#

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Examples using `popsregression.POPSRegression`#