# Enhanced Label Noise Filtering with Multiple Voting
## 3 Analysis of Decision Point, Error Probability, and Cost for MVFilter
### Relationship between the Decision Point and Error Cost  
In Section 3.1, for an MVFilter, the relationships between the optimal decision point and probabilities of making errors are analyzed. In this section, the costs of misrecognitions are considered. We will further analyze the relationships between the decision point and expected costs.  
Misrecognition/error costs allow us to specify the relative importance of different kinds of errors. In fact, many applications have unequal misrecognition costs. In our previous work [41] while studying the behaviors of the supervised feature selection algorithm, we noticed a trade-off of a smaller and bigger number of noise-free data preferences among various algorithms. As a consequence of this trade-off, different costs should be determined for different errors. A smaller number of noise-free data yields a higher type 1 error cost compared to the type 2 error cost.  
The various misrecognition costs are defined by a cost matrix. The cost matrix reflects the domain specific costs representing the cost sensitive model in the critical medical domain. Therefore, associative costs for a different type of error are finalized by the domain expert keeping the clinical context and consequences in mind.  
As shown in Table 1, cost matrix C usually has the following structure, wherein the cost matrix rows correspond to predicted results, while columns correspond to actual results, i.e., row/column = predict/actual.  
For correctly classified mislabeled (or noise-free) data, the cost is zero, and hence, it is normally assumed that \(C_{00}=C_{11}=0\) in the above matrix. With this assumption, the expected cost of an MVFilter is:  
\[ExpectedCost_{MVFilter}=\]  
\[P(E1_{MVFilter})C_{01}+P(E2_{MVFilter})C_{10}\]  
As Section 3.1 shows, \(P(E1_{MVFilter})\) and \(P(E2_{MVFilter})\) are correlated with the decision point value. Therefore, \(ExpectedCost_{MVFilter}\) is determined by both the decision point value and the cost matrix. If the cost matrix is fixed, then \(ExpectedCost_{MVFilter}\) is only influenced by the decision point value. Therefore, the cost concerned optimal decision point should be:  
\[ODP=\underset{m=1:j}{\arg\min}(P(E1_{MVFilter})C_{01}+P(E2_{MVFilter})C_{10})\]  
In this equation, if \(C_{01}\gg C_{10}\), \(P(E1_{MVFilter})\) will be the dominant factor to determine ODP value. The ODP is the decision point that can minimize \(P(E1_{MVFilter})\). From the analysis in Section 3.1, we know that it is highly probable that \(ODP=t\). On the other hand, if \(C_{10}\gg C_{01}\), \(P(E2_{MVFilter})\) will be the dominant factor to determine the OPD value. In this case, it is likely that \(ODP=1\).  
It should be noted that the ODP can be determined from the above analysis only in some extreme cases (for example, when \(C_{01}\gg C_{10}\) or \(C_{10}\gg C_{01}\)). However, for the other cases, directly calculating the ODP is extremely difficult. In addition, the above equation of the ODP is obtained by making several assumptions. Therefore, it is not very useful to calculate the ODP value through mathematically inferring since the calculated ODP is influenced by the assumptions.