# What if transitions and emissions in a hidden Markov model are not independent?

A Hidden Markov Model is given by transition and emission matrices. The transition matrix determine probability of "next states" as a function of the current state. The emission matrix determine the probability of observations as a function of the current state.

In this formulation we assume that "transitions" and "emissions" are independent. But what if they are not? In other words, what if the probability of a particular observation (given the current state) and probability of a particular transitions are bound?

Can such a model be somehow transformed into a classical HMM (with a transition and emission matrices)? And, if not, how to approach such problem? For example, is there some replacement of the Viterbi and Baum-Welch algorithms for such an "extended" HMM?

• To be clear, say the hidden states are $x(t)$ and the outputs are $y(t)$. Are you saying that $y(t)$ and $x(t+1)$ are not independent? – Alex R. Nov 10 '17 at 20:46
• @AlexR. yes, this is what I meant. In other words we can say that probabilities of $y_t$ are (completely) determined by $x_t$ and $x_{t+1}$. – Roman Nov 11 '17 at 11:34