6.4.2. Hidden Markov Models

Hidden Markov Model

A Hidden Markov Model (HMM) is a probabilistic model that can be viewed as a Markov Model whose states are not directly observable. The underlying chain of hidden variables $X_t$ evolves according to the same Markov property,
$P(X_t|X_0,X_1,\dots ,X_{t-1})=P(X_t\mid X_{t-1}),$
but instead of observing each state $X_t$ itself, we only observe an evidence variable $E_t$ that depends probabilistically on that hidden state.

The hidden layer forms a Markov Chain, while the observation layer adds an emission process that provides the update step
$P(E_t|X_{0:t},E_{0:t-1})=P(E_t\mid X_t).$

Each timestep thus contains two components:

1. A transition model $P(X_{t+1}\mid X_t)$, governing how hidden states evolve, just like a Markov Chain.
2. An emission model $P(E_t\mid X_t)$, governing how each hidden state generates observable evidence.

Both models are assumed stationary, so their conditional tables do not change with $t$.

The full process is a Markov Chain “wrapped” by an observation model. The resulting joint distribution over both hidden states and evidence variables factorizes as
$P(X_{0:n},E_{0:n})=P(X_0)P(E_0\mid X_0){\prod }_{i=0}^{n-1}P(X_{i+1}\mid X_i)P(E_{i+1}\mid X_{i+1}).$

This structure allows inference over hidden states using algorithms such as forward-backward, Viterbi, or particle filtering, all of which operate on the same transition dynamics as a Markov Chain but must also account for the uncertainty introduced by the hidden layer.