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 A Hidden Markov Model (HMM) is a statistical model used to represent systems that undergo a sequence of transitions between hidden states, where the actual state sequence is unknown but can be inferred from observable data. HMMs are widely used in speech recognition, natural language processing, bioinformatics, and other sequence prediction tasks.

Structure of HMM

An HMM consists of:

1. States ($S$): A set of hidden states that the system transitions through.

2. Observations ($O$): A sequence of observable events dependent on the hidden states.

3. Transition Probabilities ($A$): The probabilities of moving from one state to another.

4. Emission Probabilities ($B$): The likelihood of an observation being generated from a given state.

5. Initial Probabilities ($\pi$): The probability distribution of starting in each state.

HMM for Sequence Prediction

HMMs predict sequences by leveraging three fundamental problems:

1. Evaluation (Forward Algorithm): Computes the likelihood of an observed sequence given the model.

2. Decoding (Viterbi Algorithm): Finds the most probable sequence of hidden states for a given observation sequence.

3. Learning (Baum-Welch Algorithm): Adjusts model parameters to best fit observed data.

In sequence prediction tasks, HMMs model temporal dependencies and infer future events based on past observations, making them crucial for applications like speech recognition, handwriting recognition, and biological sequence analysis.