I have a mixture Markov model (containing K clusters, or components) that I am trying to train, e.g perform clustering over a set of varying length sequences. Each component of the model is a first order Markov chain. The model is defined by the following:
is the model describing behaviour of sequences in the k-th cluster.
Each kth model component(cluster) has initial probability distribution vector (theta_KI) and a transition matrix (theta_tK) - see equation 2. Each sequence seq_i in the set of sequences has a different length L (so the L in the 2nd equation would be different for each sequence in the set).
There is n possible visible states in the model and m sequences in each cluster, and the total amount of sequences is M.
In this case, how would I define e- and m-steps for my EM algorithm? My understanding is that I have to maximize log likelihood function of the model, which would be a sum of likelihoods across all vectors v_i ( synonym with sequence seq_i)- something like:
However, I get stuck not knowing if this is the right definition for what needs to be maximized and which arguments need to be found for that. I have looked for literature but it is very sparse, almost nonexistent.
Another question: is there actually an R library that has functions that can be extended to include such a model for EM algorithm?