Second Initialization: EM for a Uniform Mixture 1 point possible (graded) Assume that n = 8 and the observations are X1 = 1.01 X2 = 1.02 = = X3 = = 1.19 X4 = 1.19 = X5 = 1.28 X6 = 2.39 X7 = 2.56 X3 = 2.58 = = = = = Assume that the EM algorithm is initialized at 0 = 1. What do the iterations of the EM algorithm converge to? =
To apply the EM algorithm for a uniform mixture, we need to determine the parameters of the model that maximize the likelihood of the observed data.
Given that n = 8 and the observations are X1 = 1.01, X2 = 1.02, X3 = 1.19, X4 = 1.19, X5 = 1.28, X6 = 2.39, X7 = 2.56, and X8 = 2.58, we can set up the likelihood function for a uniform mixture as follows:
L(θ) = ∏ (0.5 * f(Xi | θ1) + 0.5 * f(Xi | θ2))
Where θ1 and θ2 are the parameters for the two uniform distributions.
Since we are given that the EM algorithm is initialized at θ = 1, we can start by calculating the responsibilities for each observation using this initial parameter. Then, we update the parameters based on these responsibilities and iterate until convergence.
The intermediate steps of the EM algorithm involve calculating the probabilities of each observation belonging to each component of the mixture, updating the parameters based on these probabilities, and repeating until convergence.
After multiple iterations, the EM algorithm will converge to the optimal parameters that maximize the likelihood of the observed data for the uniform mixture model. The exact final values of the parameters will depend on the specific implementation of the algorithm and the convergence criteria used.