Estimation of prior and transition probabilities in multiclass finite Markov mixtures

Techniques for simultaneous estimation of prior probabilities of class labels of individual pattern samples and transition probabilities between class labels of successive samples from stationary unsupervised data are presented. The prior probability estimators derived by G. R. Dattatreya and L. N. Kanal (1990) are shown to be valid convergent estimators even when the class labels of successive pattern samples are Markov dependent. A simple class of convergent estimators for the joint probabilities of class labels of successive samples is derived by constructing M² linear equations involving 2M functions of observations, and their class conditional moments are derived. By using the properties of the tensor product of invertible matrices, it is shown that the same M functions required to estimate the prior probabilities are sufficient to ensure the uniqueness of the solution of the linear equations. Expressions for the variances and asymptotic variances of the estimates of joint class probabilities are worked out. Application areas are mentioned. Simulation results on a three class Markov problem are included