Let's consider a stochastic process X(t) that can assume N different states: s1, s2, ..., sN 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 si, 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, ..., oM, we also assume the following assumption about the independence of the emission probability:
In other words, the probability of the observation oi (in this case, we mean the value at time i) is conditioned only by the state of the hidden variable at time i (xi)....