cluster - - - - - - - mean vector - - - - covariance matrix - - figure cluster the data points (leftinto three clusterswithout making any assumptions about the probability distribution of the data in factthe data were generated from three bivariate normal distributionswhose parameters are listed on the right be bin( / distributed howeverin the actual simulationthe number of points in each cluster is exactly neverthelessthe mixture model would be an accurate (although not exactmodel for these data figure displays the "targetgaussian mixture density for the data in figure that isthe mixture with equal weights and with the exact parameters as specified in figure figure the target mixture density for the data in figure in the next section we will carry out the clustering by using the em algorithm em algorithm for mixture models as we saw in section instead of maximizing the log-likelihood function ( directly from the data { xn }the em algorithm first augments the data with the vector of latent variables -in this case the hidden cluster labels { zn the idea is that is data augmentation