Classification of observation symbols in a HMM? I'm new to concept of HMMs. I have trained 2 HMMs separately. 
HMM1 is trained with symbols A, B, C. 
HMM2 is trained with symbols D, E, F. 
I have a set of observation symbols in the set V={A,B,C,D,E,F}.
In testing phase, I'm extracting a symbol by trying to associate a test vector to one of the symbols in V (using euclidean distance to cluster centers). 
How would I determine the log likelihood of a Observation Sequence  A, B, C,...  if that HMM was only trained with the symbols D, E, F....? 
Or is it that log likelihood isn't defined in this case?
Is it necessary that observation symbols have to be shared among various HMMs? i.e picked from a common observation Symbol set V?
Please help me figure out what I'm doing wrong?
 A: This is a classic Black Swan problem.  HMM1 will assign zero likelihood to symbols D, E, F and HMM2 will assign zero likelihood to symbols A, B, C.  Essentially from HMM1's perspective, D, E, F are impossible, while from HMM2s perspective D, E, F are.  They will never predict them.   (Note that there is nothing about HMMs in this answer -- you could replace "HMM" with "classifier" or "model" and the previous statement would still hold.)
If you knew something about the relationship between the symbols A, B, C and D, E, F you could  get creative with mapping them between each other.
In short, the loglikelihood of that sequence, i.e. a sequence A, B, C using a model trained on D, E, F is always -inf (= log 0).
A: Depending on how you define an observation, you can solve this problem by have a pseudo observation for rare training observations or unseen observations, e.g. number for all numbers. That way, when the HMM encounters an unseen observation, it looks for the closest pseudo observation. See 2.7.1 in this for more details.
On the other hand, if you can not have pseudo observation in you HMM model, the simplest way to handle unseen observation is just assign them zero probabilities!
