Maximum Likelihood Estimation - Rapid Parameter Estimation with High Accuracy

Maximum Likelihood Estimation (MLE) is a widely used parameter estimation method that optimally estimates model parameters by leveraging probability distribution functions, resulting in highly accurate estimation outcomes. This method's key advantage lies in its ability to estimate model parameters using minimal data points while maintaining applicability across various distribution functions. Through MLE implementation, we can significantly enhance parameter estimation accuracy with optimal computational efficiency, providing better solutions for practical problems. Implementation typically involves defining a likelihood function based on the observed data and the assumed probability distribution. The core algorithm maximizes this function using optimization techniques like gradient descent or Newton-Raphson methods. Key computational steps include: 1. Formulating the joint probability density function for observed data 2. Converting to log-likelihood for numerical stability 3. Applying optimization algorithms to find parameter values that maximize likelihood For Gaussian distributions, MLE directly calculates mean and variance from sample data. More complex distributions may require iterative optimization methods where the algorithm computes partial derivatives and updates parameters until convergence criteria are met. This approach ensures both statistical efficiency and computational practicality for real-world applications.