HMM (hidden Markov model)

Hidden Markov models (HMMs) are statistical models used to describe the probability distribution of a sequence of observations based on an underlying sequence of hidden states. HMMs are widely used in a range of applications including speech recognition, natural language processing, image recognition, bioinformatics, and finance. The basic idea behind HMMs is to model the probability distribution of a sequence of observations given a sequence of underlying hidden states. The model is based on the assumption that the current state depends only on the previous state and not on any previous observation.

HMMs are made up of two main components: the hidden state sequence and the observation sequence. The hidden state sequence is a sequence of states that are not directly observable, but can be inferred from the observations. The observation sequence is a sequence of observable events that are generated by the hidden states. The probability of each observation is dependent on the hidden state that generated it.

The basic structure of an HMM consists of a set of states, a set of observations, a transition matrix, an emission matrix, and an initial state distribution. The set of states represents the hidden states of the model. The set of observations represents the observed events that are generated by the hidden states. The transition matrix specifies the probability of transitioning from one state to another. The emission matrix specifies the probability of observing a particular event given a hidden state. The initial state distribution specifies the probability of starting in each state.

In an HMM, the states are not directly observable, but the observations are. The goal of the HMM is to infer the underlying sequence of states that generated the observed sequence of events. This is done by computing the probability of each possible sequence of states given the observed sequence of events. This computation is done using the forward-backward algorithm, which is a dynamic programming algorithm that efficiently computes the probability of a sequence of states given a sequence of observations.

The forward-backward algorithm works by computing two sets of probabilities: the forward probabilities and the backward probabilities. The forward probabilities represent the probability of being in a particular state at a particular time given the previous observations. The backward probabilities represent the probability of observing the rest of the sequence given the current state. These probabilities are computed recursively and can be used to compute the probability of a particular sequence of states given the observations.

Once the probability of each possible sequence of states has been computed, the most likely sequence of states can be determined using the Viterbi algorithm. The Viterbi algorithm works by finding the most probable path through the HMM that generates the observed sequence of events. This path represents the most likely sequence of hidden states that generated the observed sequence of events.

In addition to the Viterbi algorithm, HMMs can also be used for a range of other tasks, including sequence alignment, parameter estimation, and model selection. Sequence alignment involves finding the best alignment between two sequences, such as DNA sequences or protein sequences. Parameter estimation involves estimating the parameters of the HMM, such as the transition matrix and the emission matrix, given a set of observed sequences. Model selection involves selecting the best HMM to use for a particular task, based on the observed data.

In summary, HMMs are statistical models used to describe the probability distribution of a sequence of observations based on an underlying sequence of hidden states. HMMs are widely used in a range of applications including speech recognition, natural language processing, image recognition, bioinformatics, and finance. The basic idea behind HMMs is to model the probability distribution of a sequence of observations given a sequence of underlying hidden states. The model is based on the assumption that the current state depends only on the previous state and not on any previous observation. HMMs consist of a set of states, a set of observations, a transition matrix, an emission matrix,