# 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 Probability in MVFilter  
An MVFilter fuses the noise detection results of multiple SVFilters, while an SVFilter fuses the classification results of multiple classifiers. Therefore, the errors made by each classifier are the basis to infer the errors made by an MVFilter.  
Let P(E1\({}_{i}\)) and P(E2\({}_{i}\)) be the probability that classifier i makes an E1 and E2 error, respectively. To clarify the analysis, it is assumed that all the various classifiers in an SVFilter have the same probability of making an error. Therefore, we assume that \(P(E1_{i})=P(E1)\) and \(P(E2_{i})=P(E2)\). The most commonly used SVFilters include the majority filter (MF) and consensus filter (CF). The analysis here is based on MF, while a similar analysis can be conducted for CF.  
MF makes an E1 (or E2) error when more than half of these classifiers make an E1 (or E2) error. If the number of classifiers in MF is \(y\), then we have:  
\[P(E1_{MF})=\sum_{j>y/2}^{j=y}\left(P(E1)^{j}(1-P(E1))^{y-j}\right)\left(\begin{array} []{c}y\\ j\end{array}\right)\]  
\[P(E2_{MF})=\sum_{j>y/2}^{j=y}\left(P(E2)^{j}(1-P(E2))^{y-j}\right)\left( \begin{array}{c}y\\ j\end{array}\right)\]  
Suppose an MVFilter consists of \(t\) majority filters (MMF). Let \(P(E1_{MF_{i}})\) and \(P(E2_{MF_{i}})\) denote the probability that each MF makes an E1 and E2 error, respectively. To simplify the analysis, it is assumed that \(P(E1_{MF_{i}})=P(E1_{MF})\) and \(P(E2_{MF_{i}})=P(E2_{MF})\). The decision rule of an MVFilter is \"if at least m of the t SVFilters think data is mislabeled, then these data are identified as mislabeled\". This m value, called the decision point, will influence the probabilities of making an error for an MVFilter. Let MMF represent an MVFilter consisting of multiple majority filters, then the following relationships can be found:  
\[P(E1_{MMF})=\]  
\[\sum_{j\geq m}^{j=t}\left(P(E1_{MF})^{j}(1-P(E1_{MF}))^{t-j}\right)\left( \begin{array}{c}t\\ j\end{array}\right)\]  
\[P(E2_{MMF})=\]  
\[\sum_{j>t-m}^{j=t}\left(P(E2_{MF})^{j}(1-P(E2_{MF}))^{t-j}\right)\left( \begin{array}{c}t\\ j\end{array}\right)\]  
The decision point value m can be any number between one and \(t\). Among all possible values, the representative decision points include \(m=1\), \(m=t/2\), \(m=t\). When m = 1, data will be identified as mislabeled if at least one SVFilter thinks these data are mislabeled. When \(m=t\), data will be identified as mislabeled only if all the \(t\) SVFilters think these data are mislabeled. Conceptually, the noise detection rule is too loose if the decision point is one, while the rule is too strict if the decision point is \(t\). In this sense, \(m=t/2\) is usually moderate. For these three representative decision points, we have the following relationships:  
\[P(E1_{MMF}|m=1)=1-(1-P(E1_{MF}))^{t}\] \[P(E2_{MMF}|m=1)=P(E2_{MF})^{t}\] \[P(E1_{MMF}|m=t/2)=\] \[\sum_{j>t/2}^{j=t}\left(P(E2_{MF})^{j}(1-P(E2_{MF}))^{t-j}\right) \left(\begin{array}{c}t\\ j\end{array}\right)\] \[P(E1_{MMF}|m=t)=P(E1_{MF})^{t}\]  
\[P(E2_{MMF}|m=t)=1-(1-P(E2_{MF}))^{t}\]  
For the above relationships, normally we have:  
\[P(E1_{MMF}|m=t)<P(E1_{MMF}|m=t/2)<\] \[P(E1_{MMF}|m=1)\] \[P(E2_{MMF}|m=1)<P(E2_{MMF}|m=t/2)<\] \[P(E2_{MMF}|m=t)\]  
As \(P(E1_{MMF})\) and \(P(E2_{MMF})\) are conflicting, the optimal decision point should make a trade-off between these two probabilities. Therefore, if the probability of making errors is the only concern of MVFilter, the optimal decision point (ODP) is \(ODP=\underset{m=1:j}{\arg\min}(P(E1_{MMF})+P(E2_{MMF}))\).