# Bayesian Feature Fusion Using Factor Graph in Reduced Normal Form
## 3 Results
### Inference
#### 3.2.2 Missing Values' Management  
One of the most important characteristics of the Bayesian approach is its capability to treat missing values. In Figure 6 the effect of the absence of some detections on the classification performance is shown. All detections are correctly injected in the network except for \(k\) of them that are completely missing, and for which uniform distributions are injected in the network.  
For increasing number of missing variables, we compute all the possible missing variables' combinations and average the obtained metrics: classification accuracy, Jensen-Shannon Divergence and Conditional Entropy.  
In Figure 6d we show also the number of completely uncertain classification that is always zero except for high number of missing variables. With all variable missing, we have a completely uncertain classification for all presented examples.  
Figure 4: Upper Row: Injected Variable and model’s answer. Middle Row: Injected Variable with Error on Variable \(X_{TF}\) and related answer. Lower Row: Injected Variable with the completely missing variable \(X_{TF}\) and related answer.  
Figure 5: Induced distribution on the observed variable from the Class.  
The graph demonstrates that, in the average, also with three completely missing detections (e.g., \(X_{FF}\), \(X_{FB}\), \(X_{TB}\) or \(X_{FF}\), \(X_{TB}\) and \(X_{BB}\), etc.) the classification accuracy decreases less than 10% and also other metrics confirm the robustness of this model to the missing values. Please note that the Conditional Entropy describes an increasing in the uncertainty, in other words the classification becomes less sharp.  
To emphasize the capability of the model to treat missing values, following the same procedure described in Section 3.1, we performed a 5-fold Stratified Cross Validation but, now, including the records with missing values in the Training Set and in the Test Set at each split (also in this case all duplicated records and records also present in the Training Set are removed from the Test Set). In Table 3 the confusion matrix is shown together with per class precision, recall and F1-Score. The overall classification accuracy is 76.2% and the macro-average F1-Score is 0.7474.  
In the same configuration, if we do not take in account missing values in Training Set, we obtain a decrease in the accuracy classification (62.6%) and of the macro-average F1-Score (0.6506). This could suggest of including missing values also in the Training Set to increase the accuracy in presence of the missing values. Unfortunately, we can't conclude this because the dimension of the effective Test Set for two simulations are different since, in the first case, several records in the Training Set are present also in the Test Set and hence are removed from it.  
Moreover, we trained the model using all 114 records without missing values and performed a classification task only on the unique 80 records that contain missing values for the \"back\" variables (\(X_{BB}\), \(X_{FB}\), \(X_{TB}\)). The classification accuracy for these records is 58.8% and the macro-average F1-Score is 0.6274. These simulations confirm the high flexibility and robustness of the model to manage missing values.  
Figure 6: (**a**) Classification accuracy, (**b**) Jensen-Shannon divergence on Class Variable, (**c**) Conditional Entropy of Class Variable and (**d**) the number of completely uncertain classifications varying the number of missing variables.