Let f(x; θ) be a mixture of M univariate distributions; θ = (θ1, . . . , θM);
Assume no prior, M = 2 and f(x; θm) = N(x, µm, σm), m = 1, 2.For M = 2, assume you take the first observation X1 and define the following estimates µ1 = X1, with π1 =1/N, and µ2 =Sum(n=2 to N) of Xn, with π2 =N-1/N.
For M = 2 assume σ1 = σ2.
Write out the MLE equations for the parameters when the Yn's are observed and then derive the iterations of an EM algorithm for the situation in (2d) when the Yn's are not observed.
Can you propose a different solution to this degeneracy without imposing σ1 = σ2? How would the EM iterations change?