In this question we consider clustering 1D data with a mixture of 2 Gaussians using the EM algorithm
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In this question we consider clustering 1D data with a mixture of 2 Gaussians using the EM algorithm

Manual calculation of the M step for a GMM (Source: de Freitas.) In this question we consider clustering 1D data with a mixture of 2 Gaussians using the EM algorithm. You are given the 1-D data points x = [1 10 20]. Suppose the output of the E step is the following matrix:


where entry ri,c is the probability of obervation xi belonging to cluster c (the responsibility of cluster c for data point i). You just have to compute the M step. You may state the equations for maximum likelihood estimates of these quantities (which you should know) without proof; you just have to apply the equations to this data set. You may leave your answer in fractional form. Show your work.

a. Write down the likelihood function you are trying to optimize.

b. After performing the M step for the mixing weights π1, π2, what are the new values?

c. After performing the M step for the means μ1 and μ2, what are the new values?

Hint
MathematicsThe EM i.e. expectation–maximization algorithm is an iterative method to find minimum or maximum likelihood a posteriori (MAP) estimates of parameters in statistical models, the model where depends on the unobserved latent variables. It alternates between performing an expectation (E) step, that helps in creating a function for the expectation of the log-likelihood which for the paramet...

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