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## Three fundamental problems of HMM

For the HMM to be useful in real-world applications in solving problems like sequence labeling, following three fundamental problems to be solved;

• Evaluation problem: Given an HMM λ = (A, B, π) and an observation sequence O = o1, o2, …, oT, how do we compute the probability that the observed sequence was produced by the model. In other words, it is about determining the likelihood of the observation sequence.
• Question answered by Evaluation problem: What is the probability that a particular sequence of symbols is produced by a particular model?

• Decoding problem: Given an HMM λ = (A, B, π) and an observation sequence O = o1, o2, …, oT, how do we choose the corresponding optimal hidden state sequence (most likely sequence) Q = q1, q2, …, qT that can best explain the observations.
• Question answered by Decoding problem: Given a sequence of symbols (your observations) and a model, what is the most likely sequence of states that produced the sequence.

• Learning problem: Given a sequence of observation O = o1, o2, …, oT, estimate the transition and emission probabilities that are most likely to give O. that is, using the observation sequence and HMM general structure, determine the HMM model λ = (A, B, π) that best fit training data.
• Question answered by Learning problem: Given a model structure and a set of sequences, find the model that best fits the data.

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