Is it possible to model a Non-Markov process using Hidden Markov Models?
In other words, can we look at the hidden states as the memory of a Non-Markovian system?
Thanks.
$\begingroup$
$\endgroup$
$\begingroup$
$\endgroup$
Yes, it is possible. And it's usually what happens.
When one uses a HMM to model some observed time series $\{y_t\}$ using some hidden Markov chain $\{x_t\}$, one assumes that $x_t$ is Markov (i.e. that $p(x_t|x_{1:t-1}) = f(x_t|x_{t-1})$, but this is never assumed about $\{y_t\}$.
Looking at the standard filtering recursions, \begin{align*} p(y_t|y_{1:t-1}) &= \sum_{x_t}g(y_t|x_t)p(x_t|y_{1:t-1})\\ &= \sum_{x_t}g(y_t|x_t)f(x_t|x_{t-1})p(x_{t-1}|y_{1:t-1}) \end{align*} you can see that $y_t$ is Markov iff $p(x_{t-1}|y_{1:t-1}) = p(x_{t-1}|y_{t-1})$. I actually can't think of a situation where this is true at the moment.
