# Shapley Values as a Strategy for Ensemble Weights Estimation
## 2 Related Work
Even though the authors apply similar strategies, our approach differs from [19] in that we use a less localized assessment of detection performance. In our approach, binary cross-entropy values of models are used for Shapley value calculations as opposed to using predictions and ground-truth values directly. The goal of the authors in [19] is to improve the ensemble performance at the level of individual prediction points, while our approach evaluates the contribution of ensemble members by using their performance measures as a basis for Shapley value calculations. The method proposed in [19] is also utilizing Shapley approximation methods, thus differing from our proposed approach of calculating Shapley values exactly. The authors of this study did not consider the possibility to use Shapley values produced by their method as an ensemble weight selection strategy. To test if this could be a viable solution, we compared our proposition with the strategy proposed by [19]. Shapley values [18] were also applied in the ranking-based ensemble selection approach [32] that was based on the inducted subgraph game from combinatorics. This approach evaluated the contribution of every available classifier by incorporating ensemble characteristics such as individual accuracies and group variety into a Shapley value. The researchers validated the effectiveness of the approach by comparing it to five other techniques using 26 UCI benchmark datasets using AdaBoost [4] with a decision stump as their base learner. The results of this new ranking approach showed improvement over the original ensemble and outperformed other tested approaches in more than half cases. This approach evaluated the contribution of every available classifier by incorporating ensemble characteristics such as individual accuracies and group variety into a Shapley value. The difference in approach in [32] from ours is that we choose not to measure ensemble diversity explicitly. Ensemble heterogeneity and the possibility to apply Shapley values to evaluate classifiers have also been explored by [33]. The authors proposed a heterogeneous ensemble model based on the generalized Shapley value and the Choquet integral. The proposed method employs a fuzzy measure that uses model accuracy and diversity to construct a machine learning classifier. The predictions for such a classifier are aggregated using the Shapley-Choquet integral. The authors prove that the proposed classifier can outperform four existing classifiers, as well as five homogeneous model ensembles and three heterogeneous model ensembles using four banking-related datasets. Similarly, as in our approach, authors applied their model in the banking-related domain for credit risk modeling and compared ranks of classifiers using the Friedman test. This method differs from our strategy due to the application of fuzzy measures and advocating the use of heterogeneous ensembles. Recently, Shapley model ranking techniques have been applied in the field of federated learning [31]. The authors propose a method to approximate federated Shapley values with increased efficiency. The presented research also suggests that Shapley values could be used in a wide range of data evaluation tasks. The comparison of related research and their application areas are presented in Table 1.  
Our proposal differs from existing approaches by two main aspects: (a) weighting strategies utilize binary cross-entropy in combination with Shapely value calculations, as opposed to relying on individual predictions on a case-based level; and (b) Shapley value is calculated by utilizing empty coalition as a benchmark value corresponding to expected binary cross-entropy of random guessing with respect to class imbalance.