## 3 Expectation Maximization Algorithm
M-step (maximization step): In the M-step, the algorithm updates the parameter estimates to maximize the expected log likelihood computed in the E-step. It treats the expected values of the missing data as if they were observed and finds the parameter values that maximize the log likelihood of the complete data. To maximize the expected log likelihood, the algorithm employs standard optimization techniques, such as gradient descent or closed-form solutions tailored to specific models. The M-step involves solving the optimal values of the parameters by maximizing the Q-function for the parameters. This can be achieved through numerical optimization methods, which iteratively update the parameter estimates until convergence is reached. Gradient-based methods estimate the log-likelihood gradient for the parameters and adjust the parameter values in the direction of the steepest ascent. On the other hand, closed-form solutions exploit the specific structure of the model to derive explicit expressions for the optimal parameter estimates (Equation (5)). \[Q^{k+1}=\text{arg max }Q\Big{(}\theta,\theta^{k}\Big{)}\] (5) The choice of the optimization technique depends on the complexity of the model and the computational efficiency required. Gradient-based methods are widely used due to their versatility and ability to handle many models. However, for simpler models, closed-form solutions may offer faster and more efficient estimation. The M-step has a crucial role in refining the parameter estimates by iteratively improving their values based on the available data. By maximizing the log likelihood of the complete data, the algorithm finds parameter values that optimize the fit between the model and the observed and expected data. This step is essential for obtaining accurate and reliable parameter estimates in the presence of incomplete data. 4. Iteration: After completing the E-step and M-step, the algorithm checks for convergence. If the change in the log-likelihood or the parameter estimates falls below a certain threshold, the algorithm terminates. Otherwise, it continues to iterate by returning to the E-step and repeating the process until convergence is achieved. The convergence guarantees of the EM algorithm ensure that the likelihood of the model increases or remains constant with each iteration. However, it is essential to note that the algorithm may converge to a local maximum of the likelihood function rather than the global maximum. This behavior arises due to the inherent non-convex nature of the likelihood function. Consequently, running the algorithm multiple times with different initializations is often recommended [31]. This strategy helps mitigate the risk of getting trapped in suboptimal solutions and increases the chances of finding the global maximum. Figure 1 illustrates the interactive process of the EM algorithm, demonstrating how the algorithm iteratively updates the parameter estimates and improves the likelihood of the model. The stepwise nature of the algorithm is evident, with the E-step estimating the missing data and the M-step refining the parameter estimates based on the complete data. The iteration continues until convergence, resulting in optimized parameter estimates, which maximize the likelihood of the complete data. The iterative nature of the EM algorithm allows it to handle complex models and effectively accommodate incomplete or censored data. The algorithm refines the parameter estimates by leveraging the E-step and M-step iteratively, improving the overall fit between the model and the observed data. Convergence serves as a criterion for determining when the algorithm has reached a stable solution, ensuring the reliability and accuracy of the estimated parameters. Reliability studies often involve analyzing data where the failure times of equipment or systems are subject to right censoring. Right censoring occurs when the study's duration limits the observed failure times, and failures beyond that duration are not observed.