Since we have the model ready, we now need to estimate the parameters of the model. We need to estimate the mean vectors μm; the covariance matrices ∑m; and the transition probabilities am,n,l, where m,n,l = 1,..., M, and M is the total number of states. We will use the expectation maximization (EM) algorithm.
As we have seen in previous chapters, EM is an iterative algorithm that can learn the maximum likelihood estimates in the case of missing data; that is, when we have unobserved variables in our data. Let's say that our unobserved variable x is in the sample space x, and the observed variable y is in the sample space y. If we postulate a family of distribution f(x|Φ), with parameters Φ ∈ Ω, then the distribution ...
