## 3 Methodology
### Confidence Truth Serum  
#### 3.3.1 Algorithm Overview  
Confidence Truth Serum (CTS) introduces the concept of Bayesian Truth Serum into the ensemble strategy. It uses base classifiers to represent individual decisions within a group and establishes a predictive mechanism that allows the truth to reside in the hands of a knowledgeable minority. This mechanism prevents the wisdom of a single individual within the group from being disregarded due to majority voting, thus recovering the correct answers held by the minority and providing a comprehensive understanding of the uncertainty surrounding the event. Unlike the approach in reference [6], this ensemble strategy considers both the classification results and the confidence level of the base classifiers.  
Specifically, for each sample \(\mathbf{x}_{i}\), the basic classifier \(f_{j}\) classifies independently, yielding the class output result \(O_{i}^{j}\) and the prediction probability \(C_{i}^{j}\), that is,  
\[\mathbf{x}_{i}\overset{f_{j}}{\rightarrow}\left(O_{i}^{j}=f_{j}(\mathbf{x}_{i}),C_{i} ^{j}=P_{j}(O_{i}^{j}|\mathbf{x}_{i})\right) \tag{5}\]  
Based on comprehensive strategy \(\mathbf{\mathcal{T}}\), the final classification result is provided for the sample: \(O_{i}\).  
\[O_{i}=\mathbf{\mathcal{T}}\left((O_{i}^{1},C_{i}^{1}),(O_{i}^{2},C_{i}^{2}), \cdots,(O_{i}^{N},C_{i}^{N})\right) \tag{6}\]  
\(C_{i}^{j}\), often referred to as classification confidence in many machine learning models, primarily serves to quantify each classifier's certainty in a given prediction, thereby optimizing the decision-making process in ensemble learning. The calculation method for confidence is effectively and reasonably defined by us based on the fundamental principles of the classifiers used and the relevant information provided during their prediction processes. By clarifying the calculation methods for the confidence levels of each base learner on the data, we ensure that each classifier has a confidence value for each data point, which is then applied to subsequent computational processes.  
Based on the above definitions, we can obtain the output results of all base classifiers on the training set.  
\[\mathbf{C}=\left(\begin{array}{cccc}P_{1}\big{(}O_{1}^{1}|\mathbf{x}_{1}\big{)}&P_{2} \big{(}O_{1}^{2}|\mathbf{x}_{1}\big{)}&\cdots&P_{N}\big{(}O_{1}^{N}|\mathbf{x}_{1}\big{)} \\ P_{1}\big{(}O_{2}^{1}|\mathbf{x}_{2}\big{)}&P_{2}\big{(}O_{2}^{2}|\mathbf{x}_{2}\big{)} &\cdots&P_{N}\big{(}O_{2}^{N}|\mathbf{x}_{2}\big{)}\\ \vdots&\vdots&\ddots&\vdots\\ P_{1}\big{(}O_{M}^{1}|\mathbf{x_{M}}\big{)}&P_{2}\big{(}O_{M}^{2}|\mathbf{x_{M}}\big{)} &\cdots&P_{N}\big{(}O_{M}^{N}|\mathbf{x_{M}}\big{)}\end{array}\right) \tag{7}\]  
\[\mathbf{O}=\left(\begin{array}{cccc}f_{1}(\mathbf{x}_{1})&f_{2}(\mathbf{x}_{1})&\cdots&f _{N}(\mathbf{x}_{1})\\ f_{1}(\mathbf{x}_{2})&f_{2}(\mathbf{x}_{2})&\cdots&f_{N}(\mathbf{x}_{2})\\ \vdots&\vdots&\ddots&\vdots\\ f_{1}(\mathbf{x_{M}})&f_{2}(\mathbf{x_{M}})&\cdots&f_{N}(\mathbf{x_{M}})\end{array}\right) \tag{8}\]  
Similarly, for a test sample \(\mathbf{x_{p}}\), the classification results of all base classifiers can be obtained:  
\[\mathbf{O}\big{(}\mathbf{x_{p}}\big{)}=\big{(}f_{1}(\mathbf{x_{p}}),f_{2}(\mathbf{x_{p}}), \ldots f_{N}(\mathbf{x_{p}})\big{)}=\left(O_{p}^{1},O_{p}^{2},\ldots O_{p}^{N}\right) \tag{9}\]  
\[\mathbf{C}\big{(}\mathbf{x_{p}}\big{)}=\left(P_{1}\Big{(}O_{1}^{1}|\mathbf{x_{p}}\Big{)},P_ {1}\Big{(}O_{1}^{1}|\mathbf{x_{p}}\Big{)},\ldots P_{1}\Big{(}O_{1}^{1}|\mathbf{x_{p}} \Big{)}\right)=\left(C_{p}^{1},C_{p}^{2},\ldots C_{p}^{N}\right) \tag{10}\]