Hidden Markov Models (HMMs)

Let's consider a stochastic process X(t) that can assume N different states: s₁, s₂, ..., s_N with first-order Markov chain dynamics. Let's also suppose that we cannot observe the state of X(t), but we have access to another process O(t), connected to X(t), which produces observable outputs (often known as emissions). The resulting process is called a Hidden Markov Model (HMM), and a generic schema is shown in the following diagram:

Structure of a generic Hidden Markov Model

For each hidden state s_i, we need to define a transition probability P(i → j), normally represented as a matrix if the variable is discrete. For the Markov assumption, we have:

 

 

Moreover, given a sequence of observations o₁, o₂, ..., o_M, we also assume the following assumption about the independence of the emission probability:

In other words, the probability of the observation o_i (in this case, we mean the value at time i) is conditioned only by the state of the hidden variable at time i (x_i...