First and last state/observation – not truly 'hidden'?

[Joseph M]

Given the allowed pathways of our HMM’s (in the speech rec assignment), is it correct to say that each emitting state must generate a minimum of one observation (which represents one feature vector, which represents one 25ms frame of the utterance we are attempting to generate/classify/recognize)? If so, in a 5-state HMM (3 emitting states) would it not follow that the very first observation in the sequence MUST be generated by the gaussian in state 1, and the very last observation MUST be emitted by the gaussian in state 3? And therefore that we then know at least these 2 facts about the state sequence, which in some sense violates the ‘hidden’ quality of our Markov Model?

[Simon]

In the specific case of a left-to-right HMM topology, you are right that the first and last observations in any observation sequence must indeed be emitted from the first and last states, respectively.

But, we still only know about some part of the state sequence, and the complete state sequence remains unknown to us: it is still a hidden random variable. It’s just that the distribution of this hidden random variable is ever so slightly restricted.

In the general case of an arbitrary model topology and an observation sequence that is longer than the shortest path through that model, this is not the case. But, even in this general case, we still know something about possible values of the hidden state sequence. Any state sequences that are shorter or longer than the observation sequence have zero probability, and non-zero-probability values of the state sequence are restricted to those of exactly the right length.

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