Gaussian Mixtures and the E-M Algorithm

The EM algorithm is an effecive way to perform maximum likelihood (ML) estimation when the data PDF can be easily maximized if a certain set of unknown parameters are known. These “unknown" parameters, or missing data, are the mode assignments. The mode assignments can be understood if we assume that each data sample from the Gaussian mixture had been produced by exactly one of the modes. The mode assignment for sample $n$ is denoted $k_n$ and ${\bf k}$ denotes a particular set of assignments ${\bf k}= \{k_1, k_2 \ldots k_N\}$.