Ptmclust - a Generative Model for Finding Ptm Groups

In this section we provide a brief summary of the ExpectationMaximization (EM) algorithm. For details and theoretical understanding of the EM algorithm we recommend readers to refer to the following references (Dempster et al. (1977); McLachlan and Krishnan (1997)). The EM algorithm is an elegant and powerful method for finding the maximum likelihood of models with hidden variables. The key concept in the EM algorithm is that it iterates between the Estep and M-step until convergence. In the E-step, the algorithm estimates the posterior distribution of the hidden variables Q (also refer to as the responsibility) given the observed data and the current parameter settings; and in the M-step the algorithm calculates the ML parameter settings withQ fixed. At the end of each iteration the lower bound on the likelihood is optimized for the given parameter setting (M-step) and the likelihood is set to that bound (E-step), which guarantees an increase in the likelihood and convergence to a local maximum, or global maximum if the likelihood function is unimodal. Formally, for N input data points at iteration t, given the current parameter settings θt−1 calculated in the previous iteration, the Estep estimates the responsibility Q for the nth input data point as