Recursive estimation of prior probabilities using a mixture

The problem of estimating the prior probabilities of statistical classes with known probability density functions on the basis of statistically independent observations is considered. The mixture density is used to show that the maximum likelihood estimate of is asymptotically efficient and weakly consistent under very mild constraints on the set of density functions. A recursive estimate is proposed for . By using stochastic approximation theory and optimizing the gain sequence, it is shown that the recursive estimate is asymptotically efficient for the class case. For classes, the rate of convergence is computed and shown to be very close to asymptotic efficiency.