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Question

Problem 4.6 Mixture of exponentials​

Consider a mixture of exponential densities,

p(x)=∑j=1Kπjp(x∣j),p(x∣j)=λje−λjx.p(x) = \sum_{j=1}^K \pi_j p(x|j), \quad p(x|j) = \lambda_j e^{-\lambda_j x}.

where λj>0\lambda_j > 0 is the parameter of component jj. Given a set of samples {x1,…,xn}\{x_1, \dots, x_n\}, derive the EM algorithm to estimate the parameters θ={πj,λj}j=1K\theta = \{\pi_j, \lambda_j\}_{j=1}^K.