MATLAB Implementation of the Famous EM Algorithm

The EM (Expectation-Maximization) algorithm is a classical iterative method for parameter estimation, particularly suitable for scenarios involving latent variables or missing data. This algorithm progressively optimizes model parameters by alternating between E-step (Expectation step) and M-step (Maximization step) until convergence is achieved. MATLAB implementations typically leverage matrix operations for computational efficiency while providing visualization capabilities to demonstrate the parameter update dynamics. ### E-step (Expectation Step) In the E-step, the algorithm computes the conditional probability distribution of latent variables based on current parameter estimates, essentially calculating the expected values of hidden variables. This step constructs an expectation function concerning latent variables, establishing the foundation for parameter updates in the subsequent M-step. In MATLAB code, this typically involves probability calculations using functions like normpdf for Gaussian distributions or custom probability density functions, often vectorized for optimal performance. ### M-step (Maximization Step) During the M-step, the algorithm utilizes the expectation function obtained from the E-step to re-estimate model parameters, maximizing the current expected value. This step generally involves optimization of the log-likelihood function, which can be implemented through analytical methods or numerical optimization techniques. MATLAB implementations commonly use built-in optimization functions like fmincon or fminunc, or analytical solutions when available, to update parameters efficiently. ### Iterative Process The core of the EM algorithm lies in cyclically executing E-steps and M-steps until parameter changes fall below a specified threshold or the maximum number of iterations is reached. This iterative optimization approach ensures that the algorithm consistently moves toward improving the likelihood function, ultimately converging to a local optimum. MATLAB code typically implements convergence checks using while loops with tolerance conditions (e.g., norm(parameter_change) < tol) and includes iteration counters to monitor progress. For beginners, understanding the EM algorithm requires grasping the interaction between E-steps and M-steps, and how iterative methods approximate optimal parameters. MATLAB implementations often include debugging features such as parameter history tracking and likelihood progression plots, enabling users to visualize the algorithm's convergence behavior and intuitively comprehend the underlying principles.