## 3 Methodology
### Confidence Truth Serum with Single Regression (CTS-SR)  
As previously mentioned, in a binary classification problem, the CTS algorithm requires the construction of a total of \(2N\) regression models for \(N\) base classifiers. This incurs significant memory overhead, especially in the context of ensemble learning where the number of classifiers is large. Additionally, each model is trained only on samples with specific labels, so when the training sample set is small, the regression model has limited data for training. To reduce memory overhead and make full use of the training samples to fit the regression model, this paper proposes a variant of the CTS algorithm, called CTS-SR, which only requires the construction of a single regression model based on all data.  
Specifically, in this algorithm, we add the classifier's classification result, named \(f_{k}(\mathbf{x}_{i})\), for each sample as a one-dimensional feature, in addition to the confidence named \(\mathbf{C}_{i}^{k}\) as a new feature, that is, \(\mathbf{x}_{i}=\left\{\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3}.....\mathbf{x}_{n},\mathbf{C}_{i}^{ k},\mathbf{f}_{k}(\mathbf{x}_{i})\right\}\). Still, we let \(y_{i}^{k}\), whose meaning is as above, be the new label for each training sample (the calculation process for \(y_{i}^{k}\) is the same as above). This forms a new training dataset \(\mathbf{D}_{k}=\left\{\mathbf{x}_{i},\mathbf{y}_{i}^{k}\right\}\), and each individual classifier trains a regression model \(p_{k}\) on \(\mathbf{D}_{k}\).  
For each testing sample \((\mathbf{x}_{i},\mathbf{y}_{i})\in\mathbf{T}\), we calculate the prior and posterior probabilities for the sample using each classifier's regression model.  
\[P(X_{i}):=\frac{\sum_{k}p_{k}(x_{i})}{K} \tag{15}\]  
\[Q(X_{i}):=\frac{\sum_{k}C_{i}^{k}\left(O_{i}^{k}=1\right)}{\sum_{k=1}^{N}C_{i} ^{k}} \tag{16}\]  
The decision rule of this new method differs from CTS because this approach does not incorporate the concept of a predetermined answer. If we still apply \(P(X_{i})<Q(X_{i})\), then whether the answer is 1 or 0 holds no practical significance.  
In fact, the prior probability \(P\) in this paper describes the consistency of the classifier ensemble's decision for a new sample based on past learning experience. If \(P>Q\), this indicates a high level of agreement among the classifiers, in which case we follow the majority voting result. However, when \(P<Q\), it suggests that the decision on this event differs from previous instances, indicating a higher level of surprise. To obtain the result from knowledgeable individuals who hold the correct answer, we still apply the \"surprisingly popular\" concept. This concept implies that in a survey on an obscure question, those who know the true answer often predict a lower consensus rate. We denote this value as \(R_{i}^{k}\), In other words, these individuals truly know the answer and believe that most of the group does not. We therefore select these individuals' judgments as the final answer.  
As the machine cannot determine whether its answer is correct, we use confidence to express this. We assume that classifiers with high confidence in their classification result and low predicted consensus are more likely to hold the correct answer. Therefore, the decision rule is expressed as follows:  
\[O_{i}=\left\{\begin{matrix}\text{MajorityVote}\big{(}O_{i}^{1},O_{i}^{2}, \ldots,O_{i}^{N}\big{)}&,\text{if }P>Q\\ O_{i}^{k}\mid\text{argmax}_{K}\big{(}C_{i}^{k}-R_{i}^{k}\big{)}&,\text{if }P<Q \end{matrix}\right. \tag{17}\]