I am reading up on Hidden Markov Models (HMMs) for my research and would like to know if it is applicable to the problem I wish to tackle.
My problem is to detect/estimate the next value of a sequence of observations that comes from a finite alphabet. I think I can model the observations as coming from an HMM, as in I believe it is possible to devise a Markov chain where the probability distribution of the observation is only dependent on the current state and at each state a new observation is derived.
Now, supposing I formulate such an HMM, is it now possible to make use of the tools available for HMMs to predict the next observation or is it only useful to predict the next (hidden) state?
Best Answer
A HMM is usually comprised of a transition model $p(z_i|z_{i-1})$ and an observation model $p(x_i|z_i)$. It should be fairly easy to take the predictive distribution for hidden state $z_i$ and feed it through the observation model to get a distribution for the observation $x_i$.
For example, if $z_i$ were discrete and you'd found yourself a distribution $p(z_i)$ over the $i$th hidden variable, the $i$th observation would be distributed as
$$p(x_i) = \sum_k p(x_i|z_i = k )p(z_i = k)$$
This is called marginalization.