Hands-On Markov Models with Python

Parameterization of HMM

In the previous section, we saw an example of an HMM to get an idea of how the model works. Let's now formally parameterize an HMM.

As the latent variables of an HMM are discrete multinomial variables, we can use the 1-of-K encoding scheme to represent it, where the z_n variable is represented by a K-dimensional vector of binary variables, z_nk ∈ {0,1}, such that z_nk = 1 and z_nj = 0 for j ≠ k if the z_n variable is in the k state.

With this in mind, we can create a matrix with the transition probability matrix A, where A_ij = Pr(Z_nj = 1| z_n-1, i = 1). As the A_ij represent the probability of moving from state i to state j, it holds the property of  and can be expressed using the K(K-1) parameters. Thus we can represent ...