## 3 Methodology
### Bayesian Approximation  
The main factors impacting the performance of the EM meta-algorithm (Algorithm 1) are (i) accuracy of the trained initial model; (ii) the label quality, which depends on Equation (2); and (iii) the acquisition policy for additional ground truth [76]. Monte Carlo (MC) dropout has been used as a Bayesian approximation method to generate labels in a semisupervised approach [77] and to evaluate the prediction uncertainty with active learning [76].  
The posterior cross entropy in Equation (5) can be optimized to decrease the model loss, given the model weights \(\theta\), experimental design \(\zeta\), and the labeled sample (\(y_{i}\)). MC dropout is applied to infer the model output (\(\hat{y}\)) for a fixed number of iterations to calculate the average probability of outputs as:  
\[\hat{y}=\hat{f}_{T}(X,\theta)=\frac{1}{T}\sum_{t=1}^{T}f^{(t)}(X,\theta)\forall X \tag{9}\]  
where \(f^{(t)}\) indicates the neural network function with dropout. Classification entropy, as defined in (7), is the metric used to determine how close the output of neural network (\(\hat{y}\)) is from the predicted labels (pseudolabels) of the samples (as defined in line 8 of Algorithm 1). Other approaches for label acquisition are discussed in [76].  
```
Data: Input Data \(X\), labels \(G\), unlabeled data \(X^{\prime}\), ground truth \(g^{\prime}\) Result: weights \(w\)
1 Initialize the data (\(X\),\(G\)) with minimal samples and the weights \(w\) train \(w\) on (\(X\),\(G\)) to minimize the loss \(\mathcal{L}\) repeat
2 Expectation Maximization step while\(\forall x^{\prime}\in X^{\prime}\)do
3 infer the model output \(y^{\prime}\) using \(f\), \(w\) and \(x^{\prime}\) with Bayesian Approximation estimate the classification uncertainty via entropy ifclassification entropy is below threshold \(\theta\)then
4\((X,G)\leftarrow\) (\(X\),\(G\)) \(\cup\) (\(x^{\prime},\psi(y^{\prime})\))
5\(X^{\prime}\gets X^{\prime}\setminus x^{\prime}\)
6 end if
7 ifclassification entropy higher than threshold \(\theta\)then
8 Active Learning: ask the oracle for ground truth g' \((X,G)\leftarrow\) (\(X,G\)) \(\cup\) (\(x^{\prime},g^{\prime}\))
9\(X^{\prime}\gets X^{\prime}\setminus x^{\prime}\)
10 end if
11
12 end if
13 train \(w\) for \(N\) epochs with the updated (\(X\), \(G\)) to minimize the loss \(\mathcal{L}\) untilconvergence;
```  
**Algorithm 1**Active Learning with Bayesian Approximation  
MC dropout inference is applied to all data points for which the ground truth is acquired from the oracle during training for threshold estimation. The average classification entropy is calculated as in Equation (8). In every iteration of the EM algorithm, all the data samples with entropy values below the chosen threshold are added. The entropy of the average classification results under MC dropout is used for for both semi-supervised and active learning (lines 8 and 12 in Algorithm 1) for all \(x^{\prime}\in X^{\prime}\), where \(X^{\prime}\) represents unlabeled data points. For the samples with highest entropy, the oracle provides the ground truth (\(g^{\prime}\)). The newly labeled data point (\(x^{\prime},g^{\prime}\)) is added to the set of original data points (X) and labels (G), thereby removing \(x^{\prime}\) from \(X^{\prime}\).