1.16. Probability calibration

When performing classification you often want not only to predict the class label, but also obtain a probability of the respective label. This probability gives you some kind of confidence on the prediction. Some models can give you poor estimates of the class probabilities and some even do not support probability prediction (e.g., some instances of SgDClassifier). The calibration module allows you to better calibrate the probabilities of a given model, or to add support for probability prediction.

Well calibrated classifiers are probabilistic classifiers for which the output of the predict_proba method can be directly interpreted as a confidence level. For instance, a well calibrated (binary) classifier should classify the samples such that among the samples to which it gave a predict_proba value close to, say, 0.8, approximately 80% actually belong to the positive class.

Before we show how to re-calibrate a classifier, we first need a way to detect how good a classifier is calibrated.

Note

Strictly proper scoring rules for probabilistic predictions like sklearn.metrics.brier_score_loss and sklearn.metrics.log_loss assess calibration (reliability) and discriminative power (resolution) of a model, as well as the randomness of the data (uncertainty) at the same time. This follows from the well-known Brier score decomposition of Murphy [1]. As it is not clear which term dominates, the score is of limited use for assessing calibration alone (unless one computes each term of the decomposition). A lower Brier loss, for instance, does not necessarily mean a better calibrated model, it could also mean a worse calibrated model with much more discriminatory power, e.g. using many more features.

1.16.1. Calibration curves

Calibration curves, also referred to as reliability diagrams (Wilks 1995 [2]), compare how well the probabilistic predictions of a binary classifier are calibrated. It plots the frequency of the positive label (to be more precise, an estimation of the conditional event probability \(P(Y=1|\text{predict_proba})\)) on the y-axis against the predicted probability predict_proba of a model on the x-axis. The tricky part is to get values for the y-axis. In scikit-learn, this is accomplished by binning the predictions such that the x-axis represents the average predicted probability in each bin. The y-axis is then the fraction of positives given the predictions of that bin, i.e. the proportion of samples whose class is the positive class (in each bin).

The top calibration curve plot is created with CalibrationDisplay.from_estimator, which uses calibration_curve to calculate the per bin average predicted probabilities and fraction of positives. CalibrationDisplay.from_estimator takes as input a fitted classifier, which is used to calculate the predicted probabilities. The classifier thus must have predict_proba method. For the few classifiers that do not have a predict_proba method, it is possible to use CalibratedClassifierCV to calibrate the classifier outputs to probabilities.

The bottom histogram gives some insight into the behavior of each classifier by showing the number of samples in each predicted probability bin.

LogisticRegression is more likely to return well calibrated predictions by itself as it has a canonical link function for its loss, i.e. the logit-link for the Log loss. In the unpenalized case, this leads to the so-called balance property, see [8] and Logistic regression. In the plot above, data is generated according to a linear mechanism, which is consistent with the LogisticRegression model (the model is ‘well specified’), and the value of the regularization parameter C is tuned to be appropriate (neither too strong nor too low). As a consequence, this model returns accurate predictions from its predict_proba method. In contrast to that, the other shown models return biased probabilities; with different biases per model.

gaussianNB (Naive Bayes) tends to push probabilities to 0 or 1 (note the counts in the histograms). This is mainly because it makes the assumption that features are conditionally independent given the class, which is not the case in this dataset which contains 2 redundant features.

RandomForestClassifier shows the opposite behavior: the histograms show peaks at probabilities approximately 0.2 and 0.9, while probabilities close to 0 or 1 are very rare. An explanation for this is given by Niculescu-Mizil and Caruana [3]: “Methods such as bagging and random forests that average predictions from a base set of models can have difficulty making predictions near 0 and 1 because variance in the underlying base models will bias predictions that should be near zero or one away from these values. Because predictions are restricted to the interval [0,1], errors caused by variance tend to be one-sided near zero and one. For example, if a model should predict p = 0 for a case, the only way bagging can achieve this is if all bagged trees predict zero. If we add noise to the trees that bagging is averaging over, this noise will cause some trees to predict values larger than 0 for this case, thus moving the average prediction of the bagged ensemble away from 0. We observe this effect most strongly with random forests because the base-level trees trained with random forests have relatively high variance due to feature subsetting.” As a result, the calibration curve shows a characteristic sigmoid shape, indicating that the classifier could trust its “intuition” more and return probabilities closer to 0 or 1 typically.

LinearSVC (SVC) shows an even more sigmoid curve than the random forest, which is typical for maximum-margin methods (compare Niculescu-Mizil and Caruana [3]), which focus on difficult to classify samples that are close to the decision boundary (the support vectors).

1.16.2. Calibrating a classifier

Calibrating a classifier consists of fitting a regressor (called a calibrator) that maps the output of the classifier (as given by decision_function or predict_proba) to a calibrated probability in [0, 1]. Denoting the output of the classifier for a given sample by \(f_i\), the calibrator tries to predict the conditional event probability \(P(y_i = 1 | f_i)\).

Ideally, the calibrator is fit on a dataset independent of the training data used to fit the classifier in the first place. This is because performance of the classifier on its training data would be better than for novel data. Using the classifier output of training data to fit the calibrator would thus result in a biased calibrator that maps to probabilities closer to 0 and 1 than it should.

1.16.3. Usage

The CalibratedClassifierCV class is used to calibrate a classifier.

CalibratedClassifierCV uses a cross-validation approach to ensure unbiased data is always used to fit the calibrator. The data is split into k (train_set, test_set) couples (as determined by cv). When ensemble=True (default), the following procedure is repeated independently for each cross-validation split:

1. a clone of base_estimator is trained on the train subset

2. the trained base_estimator makes predictions on the test subset

3. the predictions are used to fit a calibrator (either a sigmoid or isotonic regressor) (when the data is multiclass, a calibrator is fit for every class)

This results in an ensemble of k (classifier, calibrator) couples where each calibrator maps the output of its corresponding classifier into [0, 1]. Each couple is exposed in the calibrated_classifiers_ attribute, where each entry is a calibrated classifier with a predict_proba method that outputs calibrated probabilities. The output of predict_proba for the main CalibratedClassifierCV instance corresponds to the average of the predicted probabilities of the k estimators in the calibrated_classifiers_ list. The output of predict is the class that has the highest probability.

It is important to choose cv carefully when using ensemble=True. All classes should be present in both train and test subsets for every split. When a class is absent in the train subset, the predicted probability for that class will default to 0 for the (classifier, calibrator) couple of that split. This skews the predict_proba as it averages across all couples. When a class is absent in the test subset, the calibrator for that class (within the (classifier, calibrator) couple of that split) is fit on data with no positive class. This results in ineffective calibration.

When ensemble=False, cross-validation is used to obtain ‘unbiased’ predictions for all the data, via cross_val_predict. These unbiased predictions are then used to train the calibrator. The attribute calibrated_classifiers_ consists of only one (classifier, calibrator) couple where the classifier is the base_estimator trained on all the data. In this case the output of predict_proba for CalibratedClassifierCV is the predicted probabilities obtained from the single (classifier, calibrator) couple.

The main advantage of ensemble=True is to benefit from the traditional ensembling effect (similar to Bagging meta-estimator). The resulting ensemble should both be well calibrated and slightly more accurate than with ensemble=False. The main advantage of using ensemble=False is computational: it reduces the overall fit time by training only a single base classifier and calibrator pair, decreases the final model size and increases prediction speed.

Alternatively an already fitted classifier can be calibrated by setting cv="prefit". In this case, the data is not split and all of it is used to fit the regressor. It is up to the user to make sure that the data used for fitting the classifier is disjoint from the data used for fitting the regressor.

CalibratedClassifierCV supports the use of two regression techniques for calibration via the method parameter: "sigmoid" and "isotonic".

1.16.3.1. Sigmoid

The sigmoid regressor, method="sigmoid" is based on Platt’s logistic model [4]:

\[p(y_i = 1 | f_i) = \frac{1}{1 + \exp(A f_i + B)} \,,\]

where \(y_i\) is the true label of sample \(i\) and \(f_i\) is the output of the un-calibrated classifier for sample \(i\). \(A\) and \(B\) are real numbers to be determined when fitting the regressor via maximum likelihood.

The sigmoid method assumes the calibration curve can be corrected by applying a sigmoid function to the raw predictions. This assumption has been empirically justified in the case of Support Vector Machines with common kernel functions on various benchmark datasets in section 2.1 of Platt 1999 [4] but does not necessarily hold in general. Additionally, the logistic model works best if the calibration error is symmetrical, meaning the classifier output for each binary class is normally distributed with the same variance [7]. This can be a problem for highly imbalanced classification problems, where outputs do not have equal variance.

In general this method is most effective for small sample sizes or when the un-calibrated model is under-confident and has similar calibration errors for both high and low outputs.

1.16.3.2. Isotonic

The method="isotonic" fits a non-parametric isotonic regressor, which outputs a step-wise non-decreasing function, see sklearn.isotonic. It minimizes:

\[\sum_{i=1}^{n} (y_i - \hat{f}_i)^2\]

subject to \(\hat{f}_i \geq \hat{f}_j\) whenever \(f_i \geq f_j\). \(y_i\) is the true label of sample \(i\) and \(\hat{f}_i\) is the output of the calibrated classifier for sample \(i\) (i.e., the calibrated probability). This method is more general when compared to ‘sigmoid’ as the only restriction is that the mapping function is monotonically increasing. It is thus more powerful as it can correct any monotonic distortion of the un-calibrated model. However, it is more prone to overfitting, especially on small datasets [6].

Overall, ‘isotonic’ will perform as well as or better than ‘sigmoid’ when there is enough data (greater than ~ 1000 samples) to avoid overfitting [3].

Note

Impact on ranking metrics like AUC

It is generally expected that calibration does not affect ranking metrics such as ROC-AUC. However, these metrics might differ after calibration when using method="isotonic" since isotonic regression introduces ties in the predicted probabilities. This can be seen as within the uncertainty of the model predictions. In case, you strictly want to keep the ranking and thus AUC scores, use method="sigmoid" which is a strictly monotonic transformation and thus keeps the ranking.

1.16.3.3. Multiclass support

Both isotonic and sigmoid regressors only support 1-dimensional data (e.g., binary classification output) but are extended for multiclass classification if the base_estimator supports multiclass predictions. For multiclass predictions, CalibratedClassifierCV calibrates for each class separately in a OneVsRestClassifier fashion [5]. When predicting probabilities, the calibrated probabilities for each class are predicted separately. As those probabilities do not necessarily sum to one, a postprocessing is performed to normalize them.

Examples

• Probability Calibration curves

• Probability Calibration for 3-class classification

• Probability calibration of classifiers

• Comparison of Calibration of Classifiers

References

[1]

Allan H. Murphy (1973). “A New Vector Partition of the Probability Score” Journal of Applied Meteorology and Climatology

[2]

On the combination of forecast probabilities for consecutive precipitation periods. Wea. Forecasting, 5, 640–650., Wilks, D. S., 1990a

[3] (1,2,3)

Predicting good Probabilities with Supervised Learning, A. Niculescu-Mizil & R. Caruana, ICML 2005

[4] (1,2)

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

[5]

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

[6]

Predicting accurate probabilities with a ranking loss. Menon AK, Jiang XJ, Vembu S, Elkan C, Ohno-Machado L. Proc Int Conf Mach Learn. 2012;2012:703-710

[7]

Beyond sigmoids: How to obtain well-calibrated probabilities from binary classifiers with beta calibration Kull, M., Silva Filho, T. M., & Flach, P. (2017).

[8]

Mario V. Wüthrich, Michael Merz (2023). “Statistical Foundations of Actuarial Learning and its Applications” Springer Actuarial