Post-hoc tuning the cut-off point of decision function

The diabetes dataset

To illustrate the tuning of the decision threshold, we will use the diabetes dataset. This dataset is available on OpenML: https://www.openml.org/d/37. We use the fetch_openml function to fetch this dataset.

from sklearn.datasets import fetch_openml

diabetes = fetch_openml(data_id=37, as_frame=True, parser="pandas")
data, target = diabetes.data, diabetes.target

We look at the target to understand the type of problem we are dealing with.

target.value_counts()

class
tested_negative    500
tested_positive    268
Name: count, dtype: int64

We can see that we are dealing with a binary classification problem. Since the labels are not encoded as 0 and 1, we make it explicit that we consider the class labeled “tested_negative” as the negative class (which is also the most frequent) and the class labeled “tested_positive” the positive as the positive class:

neg_label, pos_label = target.value_counts().index

We can also observe that this binary problem is slightly imbalanced where we have around twice more samples from the negative class than from the positive class. When it comes to evaluation, we should consider this aspect to interpret the results.

Our vanilla classifier

We define a basic predictive model composed of a scaler followed by a logistic regression classifier.

from sklearn.linear_model import LogisticRegression
from sklearn.pipeline import make_pipeline
from sklearn.preprocessing import StandardScaler

model = make_pipeline(StandardScaler(), LogisticRegression())
model

Pipeline(steps=[('standardscaler', StandardScaler()),
                ('logisticregression', LogisticRegression())])

We evaluate our model using cross-validation. We use the accuracy and the balanced accuracy to report the performance of our model. The balanced accuracy is a metric that is less sensitive to class imbalance and will allow us to put the accuracy score in perspective.

Cross-validation allows us to study the variance of the decision threshold across different splits of the data. However, the dataset is rather small and it would be detrimental to use more than 5 folds to evaluate the dispersion. Therefore, we use a RepeatedStratifiedKFold where we apply several repetitions of 5-fold cross-validation.

import pandas as pd

from sklearn.model_selection import RepeatedStratifiedKFold, cross_validate

scoring = ["accuracy", "balanced_accuracy"]
cv_scores = [
    "train_accuracy",
    "test_accuracy",
    "train_balanced_accuracy",
    "test_balanced_accuracy",
]
cv = RepeatedStratifiedKFold(n_splits=5, n_repeats=10, random_state=42)
cv_results_vanilla_model = pd.DataFrame(
    cross_validate(
        model,
        data,
        target,
        scoring=scoring,
        cv=cv,
        return_train_score=True,
        return_estimator=True,
    )
)
cv_results_vanilla_model[cv_scores].aggregate(["mean", "std"]).T

Our predictive model succeeds to grasp the relationship between the data and the target. The training and testing scores are close to each other, meaning that our predictive model is not overfitting. We can also observe that the balanced accuracy is lower than the accuracy, due to the class imbalance previously mentioned.

For this classifier, we let the decision threshold, used convert the probability of the positive class into a class prediction, to its default value: 0.5. However, this threshold might not be optimal. If our interest is to maximize the balanced accuracy, we should select another threshold that would maximize this metric.

The TunedThresholdClassifierCV meta-estimator allows to tune the decision threshold of a classifier given a metric of interest.

Tuning the decision threshold

We create a TunedThresholdClassifierCV and configure it to maximize the balanced accuracy. We evaluate the model using the same cross-validation strategy as previously.

from sklearn.model_selection import TunedThresholdClassifierCV

tuned_model = TunedThresholdClassifierCV(estimator=model, scoring="balanced_accuracy")
cv_results_tuned_model = pd.DataFrame(
    cross_validate(
        tuned_model,
        data,
        target,
        scoring=scoring,
        cv=cv,
        return_train_score=True,
        return_estimator=True,
    )
)
cv_results_tuned_model[cv_scores].aggregate(["mean", "std"]).T

In comparison with the vanilla model, we observe that the balanced accuracy score increased. Of course, it comes at the cost of a lower accuracy score. It means that our model is now more sensitive to the positive class but makes more mistakes on the negative class.

However, it is important to note that this tuned predictive model is internally the same model as the vanilla model: they have the same fitted coefficients.

import matplotlib.pyplot as plt

vanilla_model_coef = pd.DataFrame(
    [est[-1].coef_.ravel() for est in cv_results_vanilla_model["estimator"]],
    columns=diabetes.feature_names,
)
tuned_model_coef = pd.DataFrame(
    [est.estimator_[-1].coef_.ravel() for est in cv_results_tuned_model["estimator"]],
    columns=diabetes.feature_names,
)

fig, ax = plt.subplots(ncols=2, figsize=(12, 4), sharex=True, sharey=True)
vanilla_model_coef.boxplot(ax=ax[0])
ax[0].set_ylabel("Coefficient value")
ax[0].set_title("Vanilla model")
tuned_model_coef.boxplot(ax=ax[1])
ax[1].set_title("Tuned model")
_ = fig.suptitle("Coefficients of the predictive models")

Only the decision threshold of each model was changed during the cross-validation.

decision_threshold = pd.Series(
    [est.best_threshold_ for est in cv_results_tuned_model["estimator"]],
)
ax = decision_threshold.plot.kde()
ax.axvline(
    decision_threshold.mean(),
    color="k",
    linestyle="--",
    label=f"Mean decision threshold: {decision_threshold.mean():.2f}",
)
ax.set_xlabel("Decision threshold")
ax.legend(loc="upper right")
_ = ax.set_title(
    "Distribution of the decision threshold \nacross different cross-validation folds"
)

In average, a decision threshold around 0.32 maximizes the balanced accuracy, which is different from the default decision threshold of 0.5. Thus tuning the decision threshold is particularly important when the output of the predictive model is used to make decisions. Besides, the metric used to tune the decision threshold should be chosen carefully. Here, we used the balanced accuracy but it might not be the most appropriate metric for the problem at hand. The choice of the “right” metric is usually problem-dependent and might require some domain knowledge. Refer to the example entitled, Post-tuning the decision threshold for cost-sensitive learning, for more details.

Total running time of the script: (0 minutes 41.564 seconds)

Download Jupyter notebook: plot_tuned_decision_threshold.ipynb

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