\[fig:hmm\]. HMM models a (time-dependent) sequence of data $\{Y_t\}_{t=0}^T$ as indirect observations of an underlying Markov chain $\{X_t\}_{t=0}^T$ which is not available to us. Homogeneous HMMs are parsimonious models, in the sense that they are fully characterized by the transition probability $\Pr[X_{t+1}|X_{t}]$ and the emission probability $\Pr[Y_t|X_t]$ even though the size of the given data $\{Y_t\}_{t=0}^T$ can be very large.

Consider a homogeneous HMM such that:

-   a latent variable $X_t$ can take $K$ possible outcomes $x_1,...,x_K$;

-   an ambient variable $Y_t$ can take $N$ possible outcomes $y_1,...,y_N$.

Recall that [@rabiner1986introduction; @Ghahramani2001]:

-   Given both $\{X_t\}_{t=0}^T$ and $\{Y_t\}_{t=0}^T$, the complete joint probability factors, and we can easily estimate the transition probability $\Pr[X_{t+1}|X_{t}]$ and the emission probability $\Pr[Y_t|X_t]$.

-   Given only $\{Y_t\}_{t=0}^T$, but assuming we know the underlying transition and emission probabilities, we can calculate the observation likelihood using the forward algorithm, estimate the most likely hidden sequence using the Viterbi algorithm, and compute the posterior probability of the hidden states using the forward-backward algorithm.

The most natural problem setting, however, is when neither the hidden state sequence nor the underlying probabilities are known to us—we only have access to a sequence of observations, and our job is to reveal the 