The performance of algorithms, methods, and models tends to depend heavily on the distribution of cases on which they are applied, this distribution being specific to the applicative domain. After performing an evaluation in several domains, it is highly informative to compute a (weighted) mean performance and, as shown in this paper, to scrutinize what happens during this averaging. To achieve this goal, we adopt a probabilistic framework and consider a performance as a probability measure (e.g., a normalized confusion matrix for a classification task). It appears that the corresponding weighted mean is known to be the summarization, and that only some remarkable scores assign to the summarized performance a value equal to a weighted arithmetic mean of the values assigned to the domain-specific performances. These scores include the family of ranking scores, a continuum parameterized by user preferences, and that the weights to consider in the arithmetic mean depend on the user preferences. Based on this, we rigorously define four domains, named easiest, most difficult, preponderant, and bottleneck domains, as functions of user preferences. After establishing the theory in a general setting, regardless of the task, we develop new visual tools for two-class classification.

In the computer vision and machine learning communities, as well as in many other research domains, rigorous evaluation of any new method, including classifiers, is essential. One key component of the evaluation process is the ability to compare and rank methods. However, ranking classifiers and accurately comparing their performances, especially when taking application-specific preferences into account, remains challenging. For instance, commonly used evaluation tools like Receiver Operating Characteristic (ROC) and Precision/Recall (PR) spaces display performances based on two scores. Hence, they are inherently limited in their ability to compare classifiers across a broader range of scores and lack the capability to establish a clear ranking among classifiers. In this paper, we present a novel versatile tool, named the Tile, that organizes an infinity of ranking scores in a single 2D map for two-class classifiers, including common evaluation scores such as the accuracy, the true positive rate, the positive predictive value, Jaccard's coefficient, and all F-beta scores. Furthermore, we study the properties of the underlying ranking scores, such as the influence of the priors or the correspondences with the ROC space, and depict how to characterize any other score by comparing them to the Tile. Overall, we demonstrate that the Tile is a powerful tool that effectively captures all the rankings in a single visualization and allows interpreting them.

In binary classification tasks, accurate representation of probabilistic predictions is essential for various real-world applications such as predicting payment defaults or assessing medical risks. The model must then be well-calibrated to ensure alignment between predicted probabilities and actual outcomes. However, when score heterogeneity deviates from the underlying data probability distribution, traditional calibration metrics lose reliability, failing to align score distribution with actual probabilities. In this study, we highlight approaches that prioritize optimizing the alignment between predicted scores and true probability distributions over minimizing traditional performance or calibration metrics. When employing tree-based models such as Random Forest and XGBoost, our analysis emphasizes the flexibility these models offer in tuning hyperparameters to minimize the Kullback-Leibler (KL) divergence between predicted and true distributions. Through extensive empirical analysis across 10 UCI datasets and simulations, we demonstrate that optimizing tree-based models based on KL divergence yields superior alignment between predicted scores and actual probabilities without significant performance loss. In real-world scenarios, the reference probability is determined a priori as a Beta distribution estimated through maximum likelihood. Conversely, minimizing traditional calibration metrics may lead to suboptimal results, characterized by notable performance declines and inferior KL values. Our findings reveal limitations in traditional calibration metrics, which could undermine the reliability of predictive models for critical decision-making.