sklearn/calibration.py

"""Calibration of predicted probabilities."""

# Author: Alexandre Gramfort <alexandre.gramfort@telecom-paristech.fr>
#         Balazs Kegl <balazs.kegl@gmail.com>
#         Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>
#         Mathieu Blondel <mathieu@mblondel.org>
#
# License: BSD 3 clause

import warnings
from inspect import signature
from functools import partial

from math import log
import numpy as np
from joblib import Parallel

from scipy.special import expit
from scipy.special import xlogy
from scipy.optimize import fmin_bfgs

from .base import (
    BaseEstimator,
    ClassifierMixin,
    RegressorMixin,
    clone,
    MetaEstimatorMixin,
    is_classifier,
)
from .preprocessing import label_binarize, LabelEncoder
from .utils import (
    column_or_1d,
    indexable,
    check_matplotlib_support,
)

from .utils.multiclass import check_classification_targets
from .utils.fixes import delayed
from .utils.validation import (
    _check_fit_params,
    _check_sample_weight,
    _num_samples,
    check_consistent_length,
    check_is_fitted,
)
from .utils import _safe_indexing
from .isotonic import IsotonicRegression
from .svm import LinearSVC
from .model_selection import check_cv, cross_val_predict
from .metrics._base import _check_pos_label_consistency
from .metrics._plot.base import _get_response


class CalibratedClassifierCV(ClassifierMixin, MetaEstimatorMixin, BaseEstimator):
    """Probability calibration with isotonic regression or logistic regression.

    This class uses cross-validation to both estimate the parameters of a
    classifier and subsequently calibrate a classifier. With default
    `ensemble=True`, for each cv split it
    fits a copy of the base estimator to the training subset, and calibrates it
    using the testing subset. For prediction, predicted probabilities are
    averaged across these individual calibrated classifiers. When
    `ensemble=False`, cross-validation is used to obtain unbiased predictions,
    via :func:`~sklearn.model_selection.cross_val_predict`, which are then
    used for calibration. For prediction, the base estimator, trained using all
    the data, is used. This is the method implemented when `probabilities=True`
    for :mod:`sklearn.svm` estimators.

    Already fitted classifiers can be calibrated via the parameter
    `cv="prefit"`. In this case, no cross-validation is used and all provided
    data is used for calibration. The user has to take care manually that data
    for model fitting and calibration are disjoint.

    The calibration is based on the :term:`decision_function` method of the
    `base_estimator` if it exists, else on :term:`predict_proba`.

    Read more in the :ref:`User Guide <calibration>`.

    Parameters
    ----------
    base_estimator : estimator instance, default=None
        The classifier whose output need to be calibrated to provide more
        accurate `predict_proba` outputs. The default classifier is
        a :class:`~sklearn.svm.LinearSVC`.

    method : {'sigmoid', 'isotonic'}, default='sigmoid'
        The method to use for calibration. Can be 'sigmoid' which
        corresponds to Platt's method (i.e. a logistic regression model) or
        'isotonic' which is a non-parametric approach. It is not advised to
        use isotonic calibration with too few calibration samples
        ``(<<1000)`` since it tends to overfit.

    cv : int, cross-validation generator, iterable or "prefit", \
            default=None
        Determines the cross-validation splitting strategy.
        Possible inputs for cv are:

        - None, to use the default 5-fold cross-validation,
        - integer, to specify the number of folds.
        - :term:`CV splitter`,
        - An iterable yielding (train, test) splits as arrays of indices.

        For integer/None inputs, if ``y`` is binary or multiclass,
        :class:`~sklearn.model_selection.StratifiedKFold` is used. If ``y`` is
        neither binary nor multiclass, :class:`~sklearn.model_selection.KFold`
        is used.

        Refer to the :ref:`User Guide <cross_validation>` for the various
        cross-validation strategies that can be used here.

        If "prefit" is passed, it is assumed that `base_estimator` has been
        fitted already and all data is used for calibration.

        .. versionchanged:: 0.22
            ``cv`` default value if None changed from 3-fold to 5-fold.

    n_jobs : int, default=None
        Number of jobs to run in parallel.
        ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
        ``-1`` means using all processors.

        Base estimator clones are fitted in parallel across cross-validation
        iterations. Therefore parallelism happens only when `cv != "prefit"`.

        See :term:`Glossary <n_jobs>` for more details.

        .. versionadded:: 0.24

    ensemble : bool, default=True
        Determines how the calibrator is fitted when `cv` is not `'prefit'`.
        Ignored if `cv='prefit'`.

        If `True`, the `base_estimator` is fitted using training data, and
        calibrated using testing data, for each `cv` fold. The final estimator
        is an ensemble of `n_cv` fitted classifier and calibrator pairs, where
        `n_cv` is the number of cross-validation folds. The output is the
        average predicted probabilities of all pairs.

        If `False`, `cv` is used to compute unbiased predictions, via
        :func:`~sklearn.model_selection.cross_val_predict`, which are then
        used for calibration. At prediction time, the classifier used is the
        `base_estimator` trained on all the data.
        Note that this method is also internally implemented  in
        :mod:`sklearn.svm` estimators with the `probabilities=True` parameter.

        .. versionadded:: 0.24

    Attributes
    ----------
    classes_ : ndarray of shape (n_classes,)
        The class labels.

    n_features_in_ : int
        Number of features seen during :term:`fit`. Only defined if the
        underlying base_estimator exposes such an attribute when fit.

        .. versionadded:: 0.24

    feature_names_in_ : ndarray of shape (`n_features_in_`,)
        Names of features seen during :term:`fit`. Only defined if the
        underlying base_estimator exposes such an attribute when fit.

        .. versionadded:: 1.0

    calibrated_classifiers_ : list (len() equal to cv or 1 if `cv="prefit"` \
            or `ensemble=False`)
        The list of classifier and calibrator pairs.

        - When `cv="prefit"`, the fitted `base_estimator` and fitted
          calibrator.
        - When `cv` is not "prefit" and `ensemble=True`, `n_cv` fitted
          `base_estimator` and calibrator pairs. `n_cv` is the number of
          cross-validation folds.
        - When `cv` is not "prefit" and `ensemble=False`, the `base_estimator`,
          fitted on all the data, and fitted calibrator.

        .. versionchanged:: 0.24
            Single calibrated classifier case when `ensemble=False`.

    See Also
    --------
    calibration_curve : Compute true and predicted probabilities
        for a calibration curve.

    References
    ----------
    .. [1] Obtaining calibrated probability estimates from decision trees
           and naive Bayesian classifiers, B. Zadrozny & C. Elkan, ICML 2001

    .. [2] Transforming Classifier Scores into Accurate Multiclass
           Probability Estimates, B. Zadrozny & C. Elkan, (KDD 2002)

    .. [3] Probabilistic Outputs for Support Vector Machines and Comparisons to
           Regularized Likelihood Methods, J. Platt, (1999)

    .. [4] Predicting Good Probabilities with Supervised Learning,
           A. Niculescu-Mizil & R. Caruana, ICML 2005

    Examples
    --------
    >>> from sklearn.datasets import make_classification
    >>> from sklearn.naive_bayes import GaussianNB
    >>> from sklearn.calibration import CalibratedClassifierCV
    >>> X, y = make_classification(n_samples=100, n_features=2,
    ...                            n_redundant=0, random_state=42)
    >>> base_clf = GaussianNB()
    >>> calibrated_clf = CalibratedClassifierCV(base_estimator=base_clf, cv=3)
    >>> calibrated_clf.fit(X, y)
    CalibratedClassifierCV(base_estimator=GaussianNB(), cv=3)
    >>> len(calibrated_clf.calibrated_classifiers_)
    3
    >>> calibrated_clf.predict_proba(X)[:5, :]
    array([[0.110..., 0.889...],
           [0.072..., 0.927...],
           [0.928..., 0.071...],
           [0.928..., 0.071...],
           [0.071..., 0.928...]])
    >>> from sklearn.model_selection import train_test_split
    >>> X, y = make_classification(n_samples=100, n_features=2,
    ...                            n_redundant=0, random_state=42)
    >>> X_train, X_calib, y_train, y_calib = train_test_split(
    ...        X, y, random_state=42
    ... )
    >>> base_clf = GaussianNB()
    >>> base_clf.fit(X_train, y_train)
    GaussianNB()
    >>> calibrated_clf = CalibratedClassifierCV(
    ...     base_estimator=base_clf,
    ...     cv="prefit"
    ... )
    >>> calibrated_clf.fit(X_calib, y_calib)
    CalibratedClassifierCV(base_estimator=GaussianNB(), cv='prefit')
    >>> len(calibrated_clf.calibrated_classifiers_)
    1
    >>> calibrated_clf.predict_proba([[-0.5, 0.5]])
    array([[0.936..., 0.063...]])
    """

    def __init__(
        self,
        base_estimator=None,
        *,
        method="sigmoid",
        cv=None,
        n_jobs=None,
        ensemble=True,
    ):
        self.base_estimator = base_estimator
        self.method = method
        self.cv = cv
        self.n_jobs = n_jobs
        self.ensemble = ensemble

    def fit(self, X, y, sample_weight=None, **fit_params):
        """Fit the calibrated model.

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features)
            Training data.

        y : array-like of shape (n_samples,)
            Target values.

        sample_weight : array-like of shape (n_samples,), default=None
            Sample weights. If None, then samples are equally weighted.

        **fit_params : dict
            Parameters to pass to the `fit` method of the underlying
            classifier.

        Returns
        -------
        self : object
            Returns an instance of self.
        """
        check_classification_targets(y)
        X, y = indexable(X, y)
        if sample_weight is not None:
            sample_weight = _check_sample_weight(sample_weight, X)

        for sample_aligned_params in fit_params.values():
            check_consistent_length(y, sample_aligned_params)

        if self.base_estimator is None:
            # we want all classifiers that don't expose a random_state
            # to be deterministic (and we don't want to expose this one).
            base_estimator = LinearSVC(random_state=0)
        else:
            base_estimator = self.base_estimator

        self.calibrated_classifiers_ = []
        if self.cv == "prefit":
            # `classes_` should be consistent with that of base_estimator
            check_is_fitted(self.base_estimator, attributes=["classes_"])
            self.classes_ = self.base_estimator.classes_

            pred_method, method_name = _get_prediction_method(base_estimator)
            n_classes = len(self.classes_)
            predictions = _compute_predictions(pred_method, method_name, X, n_classes)

            calibrated_classifier = _fit_calibrator(
                base_estimator,
                predictions,
                y,
                self.classes_,
                self.method,
                sample_weight,
            )
            self.calibrated_classifiers_.append(calibrated_classifier)
        else:
            # Set `classes_` using all `y`
            label_encoder_ = LabelEncoder().fit(y)
            self.classes_ = label_encoder_.classes_
            n_classes = len(self.classes_)

            # sample_weight checks
            fit_parameters = signature(base_estimator.fit).parameters
            supports_sw = "sample_weight" in fit_parameters
            if sample_weight is not None and not supports_sw:
                estimator_name = type(base_estimator).__name__
                warnings.warn(
                    f"Since {estimator_name} does not appear to accept sample_weight, "
                    "sample weights will only be used for the calibration itself. This "
                    "can be caused by a limitation of the current scikit-learn API. "
                    "See the following issue for more details: "
                    "https://github.com/scikit-learn/scikit-learn/issues/21134. Be "
                    "warned that the result of the calibration is likely to be "
                    "incorrect."
                )

            # Check that each cross-validation fold can have at least one
            # example per class
            if isinstance(self.cv, int):
                n_folds = self.cv
            elif hasattr(self.cv, "n_splits"):
                n_folds = self.cv.n_splits
            else:
                n_folds = None
            if n_folds and np.any(
                [np.sum(y == class_) < n_folds for class_ in self.classes_]
            ):
                raise ValueError(
                    f"Requesting {n_folds}-fold "
                    "cross-validation but provided less than "
                    f"{n_folds} examples for at least one class."
                )
            cv = check_cv(self.cv, y, classifier=True)

            if self.ensemble:
                parallel = Parallel(n_jobs=self.n_jobs)
                self.calibrated_classifiers_ = parallel(
                    delayed(_fit_classifier_calibrator_pair)(
                        clone(base_estimator),
                        X,
                        y,
                        train=train,
                        test=test,
                        method=self.method,
                        classes=self.classes_,
                        supports_sw=supports_sw,
                        sample_weight=sample_weight,
                        **fit_params,
                    )
                    for train, test in cv.split(X, y)
                )
            else:
                this_estimator = clone(base_estimator)
                _, method_name = _get_prediction_method(this_estimator)
                fit_params = (
                    {"sample_weight": sample_weight}
                    if sample_weight is not None and supports_sw
                    else None
                )
                pred_method = partial(
                    cross_val_predict,
                    estimator=this_estimator,
                    X=X,
                    y=y,
                    cv=cv,
                    method=method_name,
                    n_jobs=self.n_jobs,
                    fit_params=fit_params,
                )
                predictions = _compute_predictions(
                    pred_method, method_name, X, n_classes
                )

                if sample_weight is not None and supports_sw:
                    this_estimator.fit(X, y, sample_weight=sample_weight)
                else:
                    this_estimator.fit(X, y)
                # Note: Here we don't pass on fit_params because the supported
                # calibrators don't support fit_params anyway
                calibrated_classifier = _fit_calibrator(
                    this_estimator,
                    predictions,
                    y,
                    self.classes_,
                    self.method,
                    sample_weight,
                )
                self.calibrated_classifiers_.append(calibrated_classifier)

        first_clf = self.calibrated_classifiers_[0].base_estimator
        if hasattr(first_clf, "n_features_in_"):
            self.n_features_in_ = first_clf.n_features_in_
        if hasattr(first_clf, "feature_names_in_"):
            self.feature_names_in_ = first_clf.feature_names_in_
        return self

    def predict_proba(self, X):
        """Calibrated probabilities of classification.

        This function returns calibrated probabilities of classification
        according to each class on an array of test vectors X.

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features)
            The samples, as accepted by `base_estimator.predict_proba`.

        Returns
        -------
        C : ndarray of shape (n_samples, n_classes)
            The predicted probas.
        """
        check_is_fitted(self)
        # Compute the arithmetic mean of the predictions of the calibrated
        # classifiers
        mean_proba = np.zeros((_num_samples(X), len(self.classes_)))
        for calibrated_classifier in self.calibrated_classifiers_:
            proba = calibrated_classifier.predict_proba(X)
            mean_proba += proba

        mean_proba /= len(self.calibrated_classifiers_)

        return mean_proba

    def predict(self, X):
        """Predict the target of new samples.

        The predicted class is the class that has the highest probability,
        and can thus be different from the prediction of the uncalibrated classifier.

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features)
            The samples, as accepted by `base_estimator.predict`.

        Returns
        -------
        C : ndarray of shape (n_samples,)
            The predicted class.
        """
        check_is_fitted(self)
        return self.classes_[np.argmax(self.predict_proba(X), axis=1)]

    def _more_tags(self):
        return {
            "_xfail_checks": {
                "check_sample_weights_invariance": (
                    "Due to the cross-validation and sample ordering, removing a sample"
                    " is not strictly equal to putting is weight to zero. Specific unit"
                    " tests are added for CalibratedClassifierCV specifically."
                ),
            }
        }


def _fit_classifier_calibrator_pair(
    estimator,
    X,
    y,
    train,
    test,
    supports_sw,
    method,
    classes,
    sample_weight=None,
    **fit_params,
):
    """Fit a classifier/calibration pair on a given train/test split.

    Fit the classifier on the train set, compute its predictions on the test
    set and use the predictions as input to fit the calibrator along with the
    test labels.

    Parameters
    ----------
    estimator : estimator instance
        Cloned base estimator.

    X : array-like, shape (n_samples, n_features)
        Sample data.

    y : array-like, shape (n_samples,)
        Targets.

    train : ndarray, shape (n_train_indices,)
        Indices of the training subset.

    test : ndarray, shape (n_test_indices,)
        Indices of the testing subset.

    supports_sw : bool
        Whether or not the `estimator` supports sample weights.

    method : {'sigmoid', 'isotonic'}
        Method to use for calibration.

    classes : ndarray, shape (n_classes,)
        The target classes.

    sample_weight : array-like, default=None
        Sample weights for `X`.

    **fit_params : dict
        Parameters to pass to the `fit` method of the underlying
        classifier.

    Returns
    -------
    calibrated_classifier : _CalibratedClassifier instance
    """
    fit_params_train = _check_fit_params(X, fit_params, train)
    X_train, y_train = _safe_indexing(X, train), _safe_indexing(y, train)
    X_test, y_test = _safe_indexing(X, test), _safe_indexing(y, test)

    if sample_weight is not None and supports_sw:
        sw_train = _safe_indexing(sample_weight, train)
        estimator.fit(X_train, y_train, sample_weight=sw_train, **fit_params_train)
    else:
        estimator.fit(X_train, y_train, **fit_params_train)

    n_classes = len(classes)
    pred_method, method_name = _get_prediction_method(estimator)
    predictions = _compute_predictions(pred_method, method_name, X_test, n_classes)

    sw_test = None if sample_weight is None else _safe_indexing(sample_weight, test)
    calibrated_classifier = _fit_calibrator(
        estimator, predictions, y_test, classes, method, sample_weight=sw_test
    )
    return calibrated_classifier


def _get_prediction_method(clf):
    """Return prediction method.

    `decision_function` method of `clf` returned, if it
    exists, otherwise `predict_proba` method returned.

    Parameters
    ----------
    clf : Estimator instance
        Fitted classifier to obtain the prediction method from.

    Returns
    -------
    prediction_method : callable
        The prediction method.
    method_name : str
        The name of the prediction method.
    """
    if hasattr(clf, "decision_function"):
        method = getattr(clf, "decision_function")
        return method, "decision_function"
    elif hasattr(clf, "predict_proba"):
        method = getattr(clf, "predict_proba")
        return method, "predict_proba"
    else:
        raise RuntimeError(
            "'base_estimator' has no 'decision_function' or 'predict_proba' method."
        )


def _compute_predictions(pred_method, method_name, X, n_classes):
    """Return predictions for `X` and reshape binary outputs to shape
    (n_samples, 1).

    Parameters
    ----------
    pred_method : callable
        Prediction method.

    method_name: str
        Name of the prediction method

    X : array-like or None
        Data used to obtain predictions.

    n_classes : int
        Number of classes present.

    Returns
    -------
    predictions : array-like, shape (X.shape[0], len(clf.classes_))
        The predictions. Note if there are 2 classes, array is of shape
        (X.shape[0], 1).
    """
    predictions = pred_method(X=X)

    if method_name == "decision_function":
        if predictions.ndim == 1:
            predictions = predictions[:, np.newaxis]
    elif method_name == "predict_proba":
        if n_classes == 2:
            predictions = predictions[:, 1:]
    else:  # pragma: no cover
        # this branch should be unreachable.
        raise ValueError(f"Invalid prediction method: {method_name}")
    return predictions


def _fit_calibrator(clf, predictions, y, classes, method, sample_weight=None):
    """Fit calibrator(s) and return a `_CalibratedClassifier`
    instance.

    `n_classes` (i.e. `len(clf.classes_)`) calibrators are fitted.
    However, if `n_classes` equals 2, one calibrator is fitted.

    Parameters
    ----------
    clf : estimator instance
        Fitted classifier.

    predictions : array-like, shape (n_samples, n_classes) or (n_samples, 1) \
                    when binary.
        Raw predictions returned by the un-calibrated base classifier.

    y : array-like, shape (n_samples,)
        The targets.

    classes : ndarray, shape (n_classes,)
        All the prediction classes.

    method : {'sigmoid', 'isotonic'}
        The method to use for calibration.

    sample_weight : ndarray, shape (n_samples,), default=None
        Sample weights. If None, then samples are equally weighted.

    Returns
    -------
    pipeline : _CalibratedClassifier instance
    """
    Y = label_binarize(y, classes=classes)
    label_encoder = LabelEncoder().fit(classes)
    pos_class_indices = label_encoder.transform(clf.classes_)
    calibrators = []
    for class_idx, this_pred in zip(pos_class_indices, predictions.T):
        if method == "isotonic":
            calibrator = IsotonicRegression(out_of_bounds="clip")
        elif method == "sigmoid":
            calibrator = _SigmoidCalibration()
        else:
            raise ValueError(
                f"'method' should be one of: 'sigmoid' or 'isotonic'. Got {method}."
            )
        calibrator.fit(this_pred, Y[:, class_idx], sample_weight)
        calibrators.append(calibrator)

    pipeline = _CalibratedClassifier(clf, calibrators, method=method, classes=classes)
    return pipeline


class _CalibratedClassifier:
    """Pipeline-like chaining a fitted classifier and its fitted calibrators.

    Parameters
    ----------
    base_estimator : estimator instance
        Fitted classifier.

    calibrators : list of fitted estimator instances
        List of fitted calibrators (either 'IsotonicRegression' or
        '_SigmoidCalibration'). The number of calibrators equals the number of
        classes. However, if there are 2 classes, the list contains only one
        fitted calibrator.

    classes : array-like of shape (n_classes,)
        All the prediction classes.

    method : {'sigmoid', 'isotonic'}, default='sigmoid'
        The method to use for calibration. Can be 'sigmoid' which
        corresponds to Platt's method or 'isotonic' which is a
        non-parametric approach based on isotonic regression.
    """

    def __init__(self, base_estimator, calibrators, *, classes, method="sigmoid"):
        self.base_estimator = base_estimator
        self.calibrators = calibrators
        self.classes = classes
        self.method = method

    def predict_proba(self, X):
        """Calculate calibrated probabilities.

        Calculates classification calibrated probabilities
        for each class, in a one-vs-all manner, for `X`.

        Parameters
        ----------
        X : ndarray of shape (n_samples, n_features)
            The sample data.

        Returns
        -------
        proba : array, shape (n_samples, n_classes)
            The predicted probabilities. Can be exact zeros.
        """
        n_classes = len(self.classes)
        pred_method, method_name = _get_prediction_method(self.base_estimator)
        predictions = _compute_predictions(pred_method, method_name, X, n_classes)

        label_encoder = LabelEncoder().fit(self.classes)
        pos_class_indices = label_encoder.transform(self.base_estimator.classes_)

        proba = np.zeros((_num_samples(X), n_classes))
        for class_idx, this_pred, calibrator in zip(
            pos_class_indices, predictions.T, self.calibrators
        ):
            if n_classes == 2:
                # When binary, `predictions` consists only of predictions for
                # clf.classes_[1] but `pos_class_indices` = 0
                class_idx += 1
            proba[:, class_idx] = calibrator.predict(this_pred)

        # Normalize the probabilities
        if n_classes == 2:
            proba[:, 0] = 1.0 - proba[:, 1]
        else:
            denominator = np.sum(proba, axis=1)[:, np.newaxis]
            # In the edge case where for each class calibrator returns a null
            # probability for a given sample, use the uniform distribution
            # instead.
            uniform_proba = np.full_like(proba, 1 / n_classes)
            proba = np.divide(
                proba, denominator, out=uniform_proba, where=denominator != 0
            )

        # Deal with cases where the predicted probability minimally exceeds 1.0
        proba[(1.0 < proba) & (proba <= 1.0 + 1e-5)] = 1.0

        return proba


def _sigmoid_calibration(predictions, y, sample_weight=None):
    """Probability Calibration with sigmoid method (Platt 2000)

    Parameters
    ----------
    predictions : ndarray of shape (n_samples,)
        The decision function or predict proba for the samples.

    y : ndarray of shape (n_samples,)
        The targets.

    sample_weight : array-like of shape (n_samples,), default=None
        Sample weights. If None, then samples are equally weighted.

    Returns
    -------
    a : float
        The slope.

    b : float
        The intercept.

    References
    ----------
    Platt, "Probabilistic Outputs for Support Vector Machines"
    """
    predictions = column_or_1d(predictions)
    y = column_or_1d(y)

    F = predictions  # F follows Platt's notations

    # Bayesian priors (see Platt end of section 2.2):
    # It corresponds to the number of samples, taking into account the
    # `sample_weight`.
    mask_negative_samples = y <= 0
    if sample_weight is not None:
        prior0 = (sample_weight[mask_negative_samples]).sum()
        prior1 = (sample_weight[~mask_negative_samples]).sum()
    else:
        prior0 = float(np.sum(mask_negative_samples))
        prior1 = y.shape[0] - prior0
    T = np.zeros_like(y, dtype=np.float64)
    T[y > 0] = (prior1 + 1.0) / (prior1 + 2.0)
    T[y <= 0] = 1.0 / (prior0 + 2.0)
    T1 = 1.0 - T

    def objective(AB):
        # From Platt (beginning of Section 2.2)
        P = expit(-(AB[0] * F + AB[1]))
        loss = -(xlogy(T, P) + xlogy(T1, 1.0 - P))
        if sample_weight is not None:
            return (sample_weight * loss).sum()
        else:
            return loss.sum()

    def grad(AB):
        # gradient of the objective function
        P = expit(-(AB[0] * F + AB[1]))
        TEP_minus_T1P = T - P
        if sample_weight is not None:
            TEP_minus_T1P *= sample_weight
        dA = np.dot(TEP_minus_T1P, F)
        dB = np.sum(TEP_minus_T1P)
        return np.array([dA, dB])

    AB0 = np.array([0.0, log((prior0 + 1.0) / (prior1 + 1.0))])
    AB_ = fmin_bfgs(objective, AB0, fprime=grad, disp=False)
    return AB_[0], AB_[1]


class _SigmoidCalibration(RegressorMixin, BaseEstimator):
    """Sigmoid regression model.

    Attributes
    ----------
    a_ : float
        The slope.

    b_ : float
        The intercept.
    """

    def fit(self, X, y, sample_weight=None):
        """Fit the model using X, y as training data.

        Parameters
        ----------
        X : array-like of shape (n_samples,)
            Training data.

        y : array-like of shape (n_samples,)
            Training target.

        sample_weight : array-like of shape (n_samples,), default=None
            Sample weights. If None, then samples are equally weighted.

        Returns
        -------
        self : object
            Returns an instance of self.
        """
        X = column_or_1d(X)
        y = column_or_1d(y)
        X, y = indexable(X, y)

        self.a_, self.b_ = _sigmoid_calibration(X, y, sample_weight)
        return self

    def predict(self, T):
        """Predict new data by linear interpolation.

        Parameters
        ----------
        T : array-like of shape (n_samples,)
            Data to predict from.

        Returns
        -------
        T_ : ndarray of shape (n_samples,)
            The predicted data.
        """
        T = column_or_1d(T)
        return expit(-(self.a_ * T + self.b_))


def calibration_curve(
    y_true, y_prob, *, pos_label=None, normalize=False, n_bins=5, strategy="uniform"
):
    """Compute true and predicted probabilities for a calibration curve.

    The method assumes the inputs come from a binary classifier, and
    discretize the [0, 1] interval into bins.

    Calibration curves may also be referred to as reliability diagrams.

    Read more in the :ref:`User Guide <calibration>`.

    Parameters
    ----------
    y_true : array-like of shape (n_samples,)
        True targets.

    y_prob : array-like of shape (n_samples,)
        Probabilities of the positive class.

    pos_label : int or str, default=None
        The label of the positive class.

        .. versionadded:: 1.1

    normalize : bool, default=False
        Whether y_prob needs to be normalized into the [0, 1] interval, i.e.
        is not a proper probability. If True, the smallest value in y_prob
        is linearly mapped onto 0 and the largest one onto 1.

    n_bins : int, default=5
        Number of bins to discretize the [0, 1] interval. A bigger number
        requires more data. Bins with no samples (i.e. without
        corresponding values in `y_prob`) will not be returned, thus the
        returned arrays may have less than `n_bins` values.

    strategy : {'uniform', 'quantile'}, default='uniform'
        Strategy used to define the widths of the bins.

        uniform
            The bins have identical widths.
        quantile
            The bins have the same number of samples and depend on `y_prob`.

    Returns
    -------
    prob_true : ndarray of shape (n_bins,) or smaller
        The proportion of samples whose class is the positive class, in each
        bin (fraction of positives).

    prob_pred : ndarray of shape (n_bins,) or smaller
        The mean predicted probability in each bin.

    References
    ----------
    Alexandru Niculescu-Mizil and Rich Caruana (2005) Predicting Good
    Probabilities With Supervised Learning, in Proceedings of the 22nd
    International Conference on Machine Learning (ICML).
    See section 4 (Qualitative Analysis of Predictions).

    Examples
    --------
    >>> import numpy as np
    >>> from sklearn.calibration import calibration_curve
    >>> y_true = np.array([0, 0, 0, 0, 1, 1, 1, 1, 1])
    >>> y_pred = np.array([0.1, 0.2, 0.3, 0.4, 0.65, 0.7, 0.8, 0.9,  1.])
    >>> prob_true, prob_pred = calibration_curve(y_true, y_pred, n_bins=3)
    >>> prob_true
    array([0. , 0.5, 1. ])
    >>> prob_pred
    array([0.2  , 0.525, 0.85 ])
    """
    y_true = column_or_1d(y_true)
    y_prob = column_or_1d(y_prob)
    check_consistent_length(y_true, y_prob)
    pos_label = _check_pos_label_consistency(pos_label, y_true)

    if normalize:  # Normalize predicted values into interval [0, 1]
        y_prob = (y_prob - y_prob.min()) / (y_prob.max() - y_prob.min())
    elif y_prob.min() < 0 or y_prob.max() > 1:
        raise ValueError(
            "y_prob has values outside [0, 1] and normalize is set to False."
        )

    labels = np.unique(y_true)
    if len(labels) > 2:
        raise ValueError(
            f"Only binary classification is supported. Provided labels {labels}."
        )
    y_true = y_true == pos_label

    if strategy == "quantile":  # Determine bin edges by distribution of data
        quantiles = np.linspace(0, 1, n_bins + 1)
        bins = np.percentile(y_prob, quantiles * 100)
    elif strategy == "uniform":
        bins = np.linspace(0.0, 1.0, n_bins + 1)
    else:
        raise ValueError(
            "Invalid entry to 'strategy' input. Strategy "
            "must be either 'quantile' or 'uniform'."
        )

    binids = np.searchsorted(bins[1:-1], y_prob)

    bin_sums = np.bincount(binids, weights=y_prob, minlength=len(bins))
    bin_true = np.bincount(binids, weights=y_true, minlength=len(bins))
    bin_total = np.bincount(binids, minlength=len(bins))

    nonzero = bin_total != 0
    prob_true = bin_true[nonzero] / bin_total[nonzero]
    prob_pred = bin_sums[nonzero] / bin_total[nonzero]

    return prob_true, prob_pred


class CalibrationDisplay:
    """Calibration curve (also known as reliability diagram) visualization.

    It is recommended to use
    :func:`~sklearn.calibration.CalibrationDisplay.from_estimator` or
    :func:`~sklearn.calibration.CalibrationDisplay.from_predictions`
    to create a `CalibrationDisplay`. All parameters are stored as attributes.

    Read more about calibration in the :ref:`User Guide <calibration>` and
    more about the scikit-learn visualization API in :ref:`visualizations`.

    .. versionadded:: 1.0

    Parameters
    ----------
    prob_true : ndarray of shape (n_bins,)
        The proportion of samples whose class is the positive class (fraction
        of positives), in each bin.

    prob_pred : ndarray of shape (n_bins,)
        The mean predicted probability in each bin.

    y_prob : ndarray of shape (n_samples,)
        Probability estimates for the positive class, for each sample.

    estimator_name : str, default=None
        Name of estimator. If None, the estimator name is not shown.

    pos_label : str or int, default=None
        The positive class when computing the calibration curve.
        By default, `estimators.classes_[1]` is considered as the
        positive class.

        .. versionadded:: 1.1

    Attributes
    ----------
    line_ : matplotlib Artist
        Calibration curve.

    ax_ : matplotlib Axes
        Axes with calibration curve.

    figure_ : matplotlib Figure
        Figure containing the curve.

    See Also
    --------
    calibration_curve : Compute true and predicted probabilities for a
        calibration curve.
    CalibrationDisplay.from_predictions : Plot calibration curve using true
        and predicted labels.
    CalibrationDisplay.from_estimator : Plot calibration curve using an
        estimator and data.

    Examples
    --------
    >>> from sklearn.datasets import make_classification
    >>> from sklearn.model_selection import train_test_split
    >>> from sklearn.linear_model import LogisticRegression
    >>> from sklearn.calibration import calibration_curve, CalibrationDisplay
    >>> X, y = make_classification(random_state=0)
    >>> X_train, X_test, y_train, y_test = train_test_split(
    ...     X, y, random_state=0)
    >>> clf = LogisticRegression(random_state=0)
    >>> clf.fit(X_train, y_train)
    LogisticRegression(random_state=0)
    >>> y_prob = clf.predict_proba(X_test)[:, 1]
    >>> prob_true, prob_pred = calibration_curve(y_test, y_prob, n_bins=10)
    >>> disp = CalibrationDisplay(prob_true, prob_pred, y_prob)
    >>> disp.plot()
    <...>
    """

    def __init__(
        self, prob_true, prob_pred, y_prob, *, estimator_name=None, pos_label=None
    ):
        self.prob_true = prob_true
        self.prob_pred = prob_pred
        self.y_prob = y_prob
        self.estimator_name = estimator_name
        self.pos_label = pos_label

    def plot(self, *, ax=None, name=None, ref_line=True, **kwargs):
        """Plot visualization.

        Extra keyword arguments will be passed to
        :func:`matplotlib.pyplot.plot`.

        Parameters
        ----------
        ax : Matplotlib Axes, default=None
            Axes object to plot on. If `None`, a new figure and axes is
            created.

        name : str, default=None
            Name for labeling curve. If `None`, use `estimator_name` if
            not `None`, otherwise no labeling is shown.

        ref_line : bool, default=True
            If `True`, plots a reference line representing a perfectly
            calibrated classifier.

        **kwargs : dict
            Keyword arguments to be passed to :func:`matplotlib.pyplot.plot`.

        Returns
        -------
        display : :class:`~sklearn.calibration.CalibrationDisplay`
            Object that stores computed values.
        """
        check_matplotlib_support("CalibrationDisplay.plot")
        import matplotlib.pyplot as plt

        if ax is None:
            fig, ax = plt.subplots()

        name = self.estimator_name if name is None else name
        info_pos_label = (
            f"(Positive class: {self.pos_label})" if self.pos_label is not None else ""
        )

        line_kwargs = {}
        if name is not None:
            line_kwargs["label"] = name
        line_kwargs.update(**kwargs)

        ref_line_label = "Perfectly calibrated"
        existing_ref_line = ref_line_label in ax.get_legend_handles_labels()[1]
        if ref_line and not existing_ref_line:
            ax.plot([0, 1], [0, 1], "k:", label=ref_line_label)
        self.line_ = ax.plot(self.prob_pred, self.prob_true, "s-", **line_kwargs)[0]

        # We always have to show the legend for at least the reference line
        ax.legend(loc="lower right")

        xlabel = f"Mean predicted probability {info_pos_label}"
        ylabel = f"Fraction of positives {info_pos_label}"
        ax.set(xlabel=xlabel, ylabel=ylabel)

        self.ax_ = ax
        self.figure_ = ax.figure
        return self

    @classmethod
    def from_estimator(
        cls,
        estimator,
        X,
        y,
        *,
        n_bins=5,
        strategy="uniform",
        pos_label=None,
        name=None,
        ref_line=True,
        ax=None,
        **kwargs,
    ):
        """Plot calibration curve using a binary classifier and data.

        A calibration curve, also known as a reliability diagram, uses inputs
        from a binary classifier and plots the average predicted probability
        for each bin against the fraction of positive classes, on the
        y-axis.

        Extra keyword arguments will be passed to
        :func:`matplotlib.pyplot.plot`.

        Read more about calibration in the :ref:`User Guide <calibration>` and
        more about the scikit-learn visualization API in :ref:`visualizations`.

        .. versionadded:: 1.0

        Parameters
        ----------
        estimator : estimator instance
            Fitted classifier or a fitted :class:`~sklearn.pipeline.Pipeline`
            in which the last estimator is a classifier. The classifier must
            have a :term:`predict_proba` method.

        X : {array-like, sparse matrix} of shape (n_samples, n_features)
            Input values.

        y : array-like of shape (n_samples,)
            Binary target values.

        n_bins : int, default=5
            Number of bins to discretize the [0, 1] interval into when
            calculating the calibration curve. A bigger number requires more
            data.

        strategy : {'uniform', 'quantile'}, default='uniform'
            Strategy used to define the widths of the bins.

            - `'uniform'`: The bins have identical widths.
            - `'quantile'`: The bins have the same number of samples and depend
              on predicted probabilities.

        pos_label : str or int, default=None
            The positive class when computing the calibration curve.
            By default, `estimators.classes_[1]` is considered as the
            positive class.

            .. versionadded:: 1.1

        name : str, default=None
            Name for labeling curve. If `None`, the name of the estimator is
            used.

        ref_line : bool, default=True
            If `True`, plots a reference line representing a perfectly
            calibrated classifier.

        ax : matplotlib axes, default=None
            Axes object to plot on. If `None`, a new figure and axes is
            created.

        **kwargs : dict
            Keyword arguments to be passed to :func:`matplotlib.pyplot.plot`.

        Returns
        -------
        display : :class:`~sklearn.calibration.CalibrationDisplay`.
            Object that stores computed values.

        See Also
        --------
        CalibrationDisplay.from_predictions : Plot calibration curve using true
            and predicted labels.

        Examples
        --------
        >>> import matplotlib.pyplot as plt
        >>> from sklearn.datasets import make_classification
        >>> from sklearn.model_selection import train_test_split
        >>> from sklearn.linear_model import LogisticRegression
        >>> from sklearn.calibration import CalibrationDisplay
        >>> X, y = make_classification(random_state=0)
        >>> X_train, X_test, y_train, y_test = train_test_split(
        ...     X, y, random_state=0)
        >>> clf = LogisticRegression(random_state=0)
        >>> clf.fit(X_train, y_train)
        LogisticRegression(random_state=0)
        >>> disp = CalibrationDisplay.from_estimator(clf, X_test, y_test)
        >>> plt.show()
        """
        method_name = f"{cls.__name__}.from_estimator"
        check_matplotlib_support(method_name)

        if not is_classifier(estimator):
            raise ValueError("'estimator' should be a fitted classifier.")

        y_prob, pos_label = _get_response(
            X, estimator, response_method="predict_proba", pos_label=pos_label
        )

        name = name if name is not None else estimator.__class__.__name__
        return cls.from_predictions(
            y,
            y_prob,
            n_bins=n_bins,
            strategy=strategy,
            pos_label=pos_label,
            name=name,
            ref_line=ref_line,
            ax=ax,
            **kwargs,
        )

    @classmethod
    def from_predictions(
        cls,
        y_true,
        y_prob,
        *,
        n_bins=5,
        strategy="uniform",
        pos_label=None,
        name=None,
        ref_line=True,
        ax=None,
        **kwargs,
    ):
        """Plot calibration curve using true labels and predicted probabilities.

        Calibration curve, also known as reliability diagram, uses inputs
        from a binary classifier and plots the average predicted probability
        for each bin against the fraction of positive classes, on the
        y-axis.

        Extra keyword arguments will be passed to
        :func:`matplotlib.pyplot.plot`.

        Read more about calibration in the :ref:`User Guide <calibration>` and
        more about the scikit-learn visualization API in :ref:`visualizations`.

        .. versionadded:: 1.0

        Parameters
        ----------
        y_true : array-like of shape (n_samples,)
            True labels.

        y_prob : array-like of shape (n_samples,)
            The predicted probabilities of the positive class.

        n_bins : int, default=5
            Number of bins to discretize the [0, 1] interval into when
            calculating the calibration curve. A bigger number requires more
            data.

        strategy : {'uniform', 'quantile'}, default='uniform'
            Strategy used to define the widths of the bins.

            - `'uniform'`: The bins have identical widths.
            - `'quantile'`: The bins have the same number of samples and depend
              on predicted probabilities.

        pos_label : str or int, default=None
            The positive class when computing the calibration curve.
            By default, `estimators.classes_[1]` is considered as the
            positive class.

            .. versionadded:: 1.1

        name : str, default=None
            Name for labeling curve.

        ref_line : bool, default=True
            If `True`, plots a reference line representing a perfectly
            calibrated classifier.

        ax : matplotlib axes, default=None
            Axes object to plot on. If `None`, a new figure and axes is
            created.

        **kwargs : dict
            Keyword arguments to be passed to :func:`matplotlib.pyplot.plot`.

        Returns
        -------
        display : :class:`~sklearn.calibration.CalibrationDisplay`.
            Object that stores computed values.

        See Also
        --------
        CalibrationDisplay.from_estimator : Plot calibration curve using an
            estimator and data.

        Examples
        --------
        >>> import matplotlib.pyplot as plt
        >>> from sklearn.datasets import make_classification
        >>> from sklearn.model_selection import train_test_split
        >>> from sklearn.linear_model import LogisticRegression
        >>> from sklearn.calibration import CalibrationDisplay
        >>> X, y = make_classification(random_state=0)
        >>> X_train, X_test, y_train, y_test = train_test_split(
        ...     X, y, random_state=0)
        >>> clf = LogisticRegression(random_state=0)
        >>> clf.fit(X_train, y_train)
        LogisticRegression(random_state=0)
        >>> y_prob = clf.predict_proba(X_test)[:, 1]
        >>> disp = CalibrationDisplay.from_predictions(y_test, y_prob)
        >>> plt.show()
        """
        method_name = f"{cls.__name__}.from_estimator"
        check_matplotlib_support(method_name)

        prob_true, prob_pred = calibration_curve(
            y_true, y_prob, n_bins=n_bins, strategy=strategy, pos_label=pos_label
        )
        name = "Classifier" if name is None else name
        pos_label = _check_pos_label_consistency(pos_label, y_true)

        disp = cls(
            prob_true=prob_true,
            prob_pred=prob_pred,
            y_prob=y_prob,
            estimator_name=name,
            pos_label=pos_label,
        )
        return disp.plot(ax=ax, ref_line=ref_line, **kwargs)