# Analysis of Score-Level Fusion Rules for Deepfake Detection
## 3 Ensemble Methods for Deepfake Detection
### Fusion Methods Based on Weighted Average  
To extend the preliminary analysis presented in [10], we evaluated whether fusion methods based on weighted averages could be more effective. In fact, the baselines of deepfake detection methods are often known, and certain methods are more efficient than others. Weighting these more and, thus, assigning them greater relative importance in the fusion process could allow exploiting the complementarity without losing precision on correctly classified samples. In general, fusion methods based on the weighted average  
Figure 2: Taxonomy of the analyzed score-level fusion methods.  
use weights assigned to each detector for linearly combining the scores assigned to each sample by such classification systems.  
\[\textit{avg\_weight\_score}=\frac{\sum\limits_{i=1}^{N}P_{i}(\text{ deepfake}|x)w_{i}}{\sum\limits_{i=1}^{N}w_{i}} \tag{6}\]  
There are various methods of assigning such weights (e.g., [38; 39; 40; 41]). In this study, we assigned the weights according to three different methods, namely accuracy-based, correlation-based and based on mutual information.  
In the first approach, we computed the accuracy of the single detector on the training set and then used such values as weights in the implementation of the previous equation. Specifically, the accuracy was computed using a threshold equal to 0.5 on the scores for classifying each sample as deepfake, whether its score was greater or equal to such a threshold, or as real, otherwise.  
Similarly, through the correlation-based method, we assigned the weights to the fusion module as the Pearson correlation coefficients resulting from the correlation analysis between the scores predicted by the single detection systems and the actual classes (i.e., the real samples were labeled as 0, and the manipulated ones as 1). Thus, unlike the previous method, the correlation-based method allows assigning the weights according to the reliability of the single deepfake detector estimated at the score level rather than at the decision level.  
In particular, the Pearson correlation coefficient (PCC) is a measure of linear correlation between two variables, represented by the scores and the actual classes in this case. Hence, the more the scores obtained from a specific deepfake detector are correlated with the target, the greater the weight it will have in the fusion. Through this method, when a correlation coefficient is obtained, the corresponding weight was considered as 0 (i.e., the detection system was not considered).  
The fusion method based on mutual information employs a similar approach by assigning the weights according to the mutual information (MI) computed from the set of scores obtained by the training set and the actual classes. Specifically, the mutual information between two variables is a non-negative value, obtained through an estimation method based on entropy, which measures the dependency between the variables [42; 43; 44] and between the predicted scores and the actual labels in this case. Thus, unlike the previous method, the MI-based method is able to capture even non-linear dependency between scores and targets, thus, assigning the weights accordingly. Such a value is equal to zero if and only if two random variables are independent, while the higher the values, the higher the dependency between variables.