Recursive estimation of prior probabilities using a mixture

The problem of estimating the prior probabilities q = (q_{1} \cdots q_{m-1}) of m statistical classes with known probability density functions F_{1}(X) \cdots F_{m}(x) on the basis of n statistically independent observations (X_{l} \cdots x_{n}) is considered. The mixture density g(x|q) = \sum^{m-1}_{j-1}q_{j}F_{j}(x) + (1 - \sum^{m-1}_{\tau = 1}q_{\tau})F_{m(x) is used to show that the maximum likelihood estimate of q is asymptotically efficient and weakly consistent under very mild constraints on the set of density functions. A recursive estimate is proposed for q . By using stochastic approximation theory and optimizing the gain sequence, it is shown that the recursive estimate is asymptotically efficient for the m = 2 class case. For m > 2 classes, the rate of convergence is computed and shown to be very close to asymptotic efficiency.