# Enhanced Label Noise Filtering with Multiple Voting
## 4 Novel Approach to Determine the Optimal Decision Point  
In this section, we present our approach that can select the optimal decision point for an MVFilter by considering both cost information and the dataset itself. Given a noisy training dataset, we define that the ODP is the value that can minimize the expected cost of an MVFilter. As pointed out in Section 3, mathematically inferring the ODP is difficult. Therefore, instead of directly inferring, we try to estimate the ODP implicitly. For a noisy training dataset E, if we already know which data in E are mislabeled, it is trivial to decide the ODP. We just need to explore all the possible decision points. The OPD will be the point that minimizes the overall costs of misrecognitions. Of course, the mislabeled data distribution in E is unknown since our mission is to identify mislabeled data from E. However, if there exists another noisy dataset E' similar to E and with a known mislabeled data distribution, then we could implicitly estimate ODP from E' instead of E since their ODPs should be similar. This actually is the key idea of our approach. Given a noisy dataset E to handle, we will generate another dataset E'. The new generated E' requires: (1) E' and E are from the same/similar data distribution, and (2) the mislabeled data distributions in E and E' are similar. If such an E' could be generated, we can easily get the ODP based on E' since the mislabeled data distribution in E' is known. In many real applications, in addition to the noisy dataset E, usually another validation dataset E\({}_{nf}\) is available. E\({}_{nf}\) contains only noise-free data and coming from the same data distribution as E. As there are no mislabeled data in E\({}_{nf}\), the artificial erroneous labels are put into E\({}_{nf}\). Here, we assume that the prior knowledge of the noise ratio in E is available, which is used to determine the erroneous \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Actual Mislabeled** & **Actual Noise-Free** \\ \hline Predict mislabeled and eliminate & C(0,0) = C\({}_{00}\) & C(0,1) = C\({}_{01}\) \\ \hline Predict noise-free and retain & C(1,0) = C\({}_{10}\) & C(0,0) = C\({}_{11}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cost matrix of the mislabeled data filter. labels put into E\({}_{nf}\). Through the above procedures, E\({}_{nf}\) can be converted to E'. The optimal decision point from E' can be used to estimate the actual decision point in E.  
As the actual mislabeled data distribution in E is not available, we put erroneous labels in E\({}_{nf}\) in a random manner based on the prior noise ratio information. Although the mislabeled data in E are also stochastic, the mislabeled data distribution in E and E' can have a great difference. In this case, the ODP value obtained from E' is actually not optimal for E. To solve this problem, the ODP is estimated several times. This method uses the _numIter_ parameter to control the specified number of iterations. Each time E' changes since random erroneous labels are put into E\({}_{nf}\). For each time, all the possible decision points (from one to t) will be explored, and accordingly, the cost of misrecognition is recorded. The average cost of each decision point value is obtained by taking the mean value of this decision point multiple times the misrecognition costs. Finally, the decision point having the least average cost is selected as the optimal decision point. The details of our algorithm are shown in Algorithm 3.