Mastering Maximum Likelihood Estimation for Multivariate Normal Distribution

Maximum Likelihood Estimation (MLE) for multivariate normal distribution is a crucial concept in statistics and machine learning. Mastering this technique through practical experiments not only provides deep understanding of multivariate normal distribution properties but also establishes a solid theoretical foundation for subsequent classification problems. ### Maximum Likelihood Estimation for Multivariate Normal Distribution The Multivariate Normal Distribution represents an extension of univariate normal distribution to higher-dimensional spaces. The core objective of its Maximum Likelihood Estimation is to estimate the mean vector and covariance matrix using sample data. Specifically, given a set of independent and identically distributed sample data, we can find optimal parameter estimates by optimizing the likelihood function. Mean Vector Estimation: The sample mean vector serves as the maximum likelihood estimator for the multivariate normal distribution's mean vector. Its calculation parallels the univariate case, involving arithmetic averaging across each dimension. In code implementation, this can be achieved using numpy's mean() function along specific axes. Covariance Matrix Estimation: The sample covariance matrix (either unbiased or maximum likelihood estimator) is used to fit the data structure of multivariate normal distribution. Under MLE, covariance matrix calculation requires centering the sample data first, then computing the average of outer products. Programming implementation typically involves subtracting the mean vector and using matrix multiplication operations. ### Bayesian Classification with Minimum Error Rate Under the multivariate normal distribution assumption, Bayesian classifiers can minimize classification error rates. The core concept utilizes class-conditional probability density functions (multivariate normal distributions) and prior probabilities to compute posterior probabilities, ultimately assigning samples to classes with highest posterior probability. Discriminant Function: For multivariate normal distributions, discriminant functions typically adopt logarithmic likelihood ratio forms. Leveraging covariance matrix properties, these can be further simplified to linear or quadratic discriminant functions. Code implementation often involves calculating Mahalanobis distances and determinant computations. Classification Boundaries: Depending on whether covariance matrices are identical across classes, classification boundaries may be linear (Linear Discriminant Analysis, LDA) or nonlinear (Quadratic Discriminant Analysis, QDA). Algorithm implementation requires comparing covariance matrices and constructing appropriate decision boundaries. ### Extensions to Other Parameter Estimation Methods This experimental approach enhances understanding of other parameter estimation methods, such as: Method of Moments: Matching sample moments with theoretical moments, suitable for certain non-normal distribution scenarios. Implementation involves solving equations derived from moment conditions. Bayesian Estimation: Incorporating prior distributions and updating posterior distributions with data, particularly useful for small-sample situations. Code implementation typically involves conjugate prior distributions and Markov Chain Monte Carlo (MCMC) sampling. Ultimately, this learning process not only improves parameter estimation capabilities but also strengthens practical skills in high-dimensional data processing and classification tasks.