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I found in many sources that Hidden Markov Models are linear-chain networks(e.g. in Predicting Structured Data book by MIT). However, as I understand it, HMMs can have any edges in its graph. Even simple example of HMM in wikipedia has non-linear graph: enter image description here .

So, the question is: what is the formal definition linear-chain structure and in which case forward-backward and Viterbi algorithms can give precise results.

I have also taken into consideration this picture, taken from CRF tutorial, which says, that linear-chain CRF is "generative-discriminative pair" to HMM.

enter image description here

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It is so called because it classifies the linear sequence, and not because the structure of the graph.

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