Recursive estimation of prior probabilities using a mixture

The problem of estimating the prior probabilitiesq = (q₁ ⋯ qₘ₋₁)ofmstatistical classes with known probability density functionsF₁(X) ⋯ Fₘ(x)on the basis ofnstatistically independent observations(Xₗ ⋯ xₙ)is considered. The mixture densityg(x|q) = ∑ᵐ⁻¹ⱼ₋₁qⱼFⱼ(x) + (1 - ∑ᵐ⁻¹τ = 1qτ)F_{m(x)is used to show that the maximum likelihood estimate ofqis asymptotically efficient and weakly consistent under very mild constraints on the set of density functions. A recursive estimate is proposed forq. By using stochastic approximation theory and optimizing the gain sequence, it is shown that the recursive estimate is asymptotically efficient for them = 2class case. Form > 2classes, the rate of convergence is computed and shown to be very close to asymptotic efficiency.