## 3 Methodology
### Confidence Truth Serum
#### 3.3.2 Ensemble Strategy \(\mathbf{\mathcal{T}}\)  
1. According to the explanation of the Bayesian Truth Serum algorithm in Section 3.2, we need to calculate the prior probability \(P\) and posterior probability \(Q\) for each answer option (i.e., the classification result for each sample). Prior probability and posterior probability are concepts rooted in statistics. The prior refers to the estimated likelihood of an event occurring based on prior knowledge or historical information, before any data are observed. It reflects an initial belief about the occurrence of an event in the absence of new evidence. The posterior probability, on the other hand, is the updated probability of the event after observing the data, representing the belief after taking new evidence into account and making a prediction. Therefore, in this paper, the prior probability \(P\) is defined as the result of each base classifier in the ensemble \(\mathbb{F}\) predicting the classification outcomes of other base classifiers, also referred to as peer prediction information. The posterior probability \(Q\) is calculated by aggregating the category prediction probabilities (classification confidence) of all base classifiers for an unknown event. Taking a supervised binary classification task as an example, the calculation process is as follows:  
Based on the matrix \(\mathbf{O}\), for the \(k\)-th base classifier, calculate the consistency ratio of the other \(N-1\) classifiers agreeing with its classification result according to Equation (11), denoted as \(y_{i}^{k}\) (representing the belief score of the \(k\)-th classifier for the \(i\)-th sample):  
\[y_{i}^{k}=\frac{\sum_{j\
eq k,\ O_{i}^{j}=O_{i}^{k}}C_{i}^{j}}{\sum_{j\
eq k}C_ {i}^{j}} \tag{11}\]  
The result \(y_{i}^{k}\) is used as the new label for each training sample, and \(C_{i}^{k}\) is added as an additional one-dimensional feature to the sample, forming the feature set \(\mathbf{\mathbf{\gamma}}_{i}^{\prime}=\left\{X_{1},X_{2},X_{3}.....X_{n},C_{i}^{k}\right\}\). This forms a new training dataset \(\mathbf{D_{k}}=\left\{\mathbf{\mathbf{\gamma}}_{i},y_{i}^{k}\right\}\), which is then divided into \(\mathbf{D_{k}^{+}}\) and \(\mathbf{D_{k}^{-}}\) according to the original label of the samples, \"1\" or \"0\". Here, confidence is added as a new feature because traditional regression model training usually utilizes feature vectors by optimizing an objective function, such as using the least squares error to fit the model \(\hat{y}=w^{T}+b\). However, in ensemble learning, the classification confidence \(C_{i}^{k}\) of each base learner for a sample is also an important source of information. Adding classification confidence as a new one-dimensional feature increases the dimensionality of the training samples, allowing the regression model to utilize sample information more comprehensively, leading to a more optimal parameter model and, thus, improving prediction accuracy. To avoid the phenomenon where directly applying Equation (11) results in the prior and posterior probabilities always being identical, it is necessary to approximate the prior by training a regression model (see reference [6]); therefore, we, respectively, train two regression models for each individual classifier based on the two training sets \(\mathbf{D_{k}^{+}}\) and \(\mathbf{D_{k}^{-}}\). We repeat the above process for each base classifier. For each test sample \((\mathbf{x}_{i},l_{i})\), we calculate the prior probability of label 1 using the corresponding regression model based on the actual prediction results of each classifier. \[\overline{p}_{k}(x_{i})= \begin{cases}p_{k}^{+}(x_{i})&,\quad if\ O_{i}^{k}=1 \\ 1-p_{k}^{-}(x_{i})&,\quad if\ O_{i}^{k}=0\end{cases}