## 3 Methodology
### Active Learning with Expectation Maximization  
Expectation maximization [65] is a widely used clustering algorithm that enables parameter estimation in probabilistic models with incomplete data. The EM algorithm involves two steps per iteration: (i) expectation, which computes the conditional expectation of the log likelihood (i.e., probability distribution) over completions of missing labeled data, given the current model; and (ii) maximization, which involves re-estimating the model parameters to maximize the expected log likelihood of the model, given the current expected value of the latent variables. In an unsupervised model, EM is initiated in cases in which the classes are predefined and parameters are random. However, in a semisupervised context, the initial model is trained using a subset of training data with ground truth labels. In the subsequent phase, which is unsupervised, the model is retrained on all the unlabeled data to infer the latent variables (i.e., labels). Here, the classification entropy provided by the neural network serves as the clustering metric, which adds on to the loss function. EM initially generates a classifier (\(\theta\)). The algorithm uses \(\theta\) to classify the data and generate a hypothesis based on the labels inferred in the previous step. EM attempts to  
Figure 1: An active, semisupervised expectation maximization (ASEM) framework for CAD tasks. determine latent variable \(Z\) (i.e., unknown labels), thereby maximizing the likelihood of observing the image \(X\) in the neural network model. The likelihood of the latent variable \(Z\) is given by the integral of the joint probability density of all the possible values of the latent variable \(Z\), as in Equation (2):  
\[L(\theta;X)=P(X|\theta)=\int p(X,Z|\theta)dZ \tag{2}\]  
EM alternates between the expectation and maximization phases to solve the integral in Equation (2). The expected value of the latent variable \(Z\), given the \(\mathrm{t}^{\mathrm{th}}\) iteration of the model \(\theta^{\mathrm{t}}\), is calculated in expectation. \(E_{Z|X,\theta^{\mathrm{t}}}\), the expected value of the latent variable (\(Z\)), can be computed by classifying the probabilities of the unlabeled image labels using the model coefficients (\(\theta^{\mathrm{t}}\)) in the \(\mathrm{t}^{\mathrm{th}}\) iteration,as shown below in Equation (3):  
\[Q(\theta|\theta^{\mathrm{t}})=E_{Z|X,\theta^{\mathrm{t}}}[logL(\theta;X,Z)] \tag{3}\]  
The maximum likelihood model (\(\theta^{\mathrm{t+1}}\)) is computed in the maximization phase, given the expected value of the latent variable (\(Z\)). The deep learning model is retrained using the expected values of the image labels in the \(\mathrm{t}^{\mathrm{th}}\) iteration to compute the maximum likelihood model as in Equation (4):  
\[\theta^{\mathrm{t+1}}=argmax_{\theta}Q(\theta|\theta^{\mathrm{t}}) \tag{4}\]  
Furthermore, active learning optimizes the expected posterior cross entropy of the model, given an alternate experimental design \(\xi\), along with the labeled sample (\(y_{i}\)). The expected posterior cross entropy is expressed as:  
\[U(\xi)=-\int log(p(X|\theta,y_{i},\xi))dy_{i} \tag{5}\]  
Applying the Bayes rule to Equation (5):  
\[U(\xi)=-\int log(p(y_{i}|\theta,X,\xi)\frac{p(y_{i}|\theta,\xi)}{p(X|\theta, \xi)}dy_{i} \tag{6}\]  
The integral in Equation (6) is computationally expensive since it requires retraining of the algorithm for each sample choice and sample label prior to choosing an appropriate sample. To simplify this process, we make an approximation since a single sample has a negligible impact on the model prediction for most samples, and the predicted sample \(y_{i}\) makes the greatest local contribution to posterior cross entropy.