Estimation of mixing probabilities in multiclass finite mixtures

The problem of estimating prior probabilities in a mixture of M classes with known class conditional distributions is studied. The observation is a sequence of n independent, identically distributed mixture random variables. The first moments of appropriately formulated functions of observations are used to facilitate estimation. The complexity of these functions may vary from linear functions of the observations (in some cases) to complex functions of class conditional density functions of observations, depending on the desired balance between computational simplicity and theoretical properties. A closed-form, recursive, unbiased, convergent estimator using the density function is presented: the result is valid for any problem in which prior probabilities are identifiable. Discrete and mixed densities require a minor modification. Three application examples are described. The class conditional expectations of density functions, required for the initialization of the estimator algorithm, are analytically evaluated for Gaussian and exponential densities