I've been using the depmixS4 package in R to estimate a single hidden Markov Model from 30 different time series (i.e., 30 different people). Initially, I took the clumsy approach of fitting a model to each person's data and then averaging across parameters to end up with a single final model. But then I realized that, using the ntimes argument, I could fit a single model to all the time series at once.
My problem is that I don't specifically understand how one set of parameters is estimated from multiple time series. Is it simply the case that the maximum-likelihood estimator is maximizing the sum likelihood of all 30 people's data? That's my suspicion, but I haven't been able to confirm it.