HMM: Number of states > emission? My question is: is it possible to use HMM when the number of latent states ($S_i \mid i = 1..N$) is more than the number of emission symbols ($Y_j \mid j = 1..K$), i.e., $N > K$.
 A: Yes this is allowed. There are $N$ emission distributions, each emission distribution can take on $K$ values. 
If you look at one of the seminal papers on HMM "A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition" by LAWRENCE R. RABINER pg 261 the only constraint is the must be at least two states and emission symbols. 

4) The observation symbol probability distribution in
state $i$, $B = {b_{j}(k)}$, where
$b_{j}(k)=p[v_k\ at \ t|q_{t}=S_j] \ \ \ \ \ \ \ \ \ \ \ \ 1<j<N\ \ \ \ \   1<k<M$

Where $M$ is the number of distinct observation symbols and $N$ the number of states in the model. For Continuous Observation Densities there is a non-countable number of observation values. 
A: It is possible. 
Number of hidden state in HMM is similar to number of Gaussian in mixture of Gaussian model.
One important thing is increasing number of hidden state may cause over-fitting. Therefore how to choose number of hidden statement is one important problem and cross validation on likelihood can be used. 
