Hidden Markov Models (HMMs) are commonly used in various fields, including natural language processing, speech recognition, and bioinformatics. They are sequence models that compute a sequence of outputs based on a given sequence of inputs. HMMs are represented as graphs, where nodes represent probability distributions over labels and edges represent the probability of transitioning from one node to another.

A notable tutorial by Rabiner introduced the concept of HMMs in the late 1980s, building upon earlier tutorials by Jack Ferguson in the 1960s. This tutorial emphasized the idea that HMMs should serve as hidden probabilistic models. In these models, the distribution of both observations and hidden states is considered, or equivalently, the prior distribution of these variables.

HMMs are generative models that aim to explicitly model the joint distribution of observations and hidden states. They can be used for tasks such as part-of-speech tagging, where the hidden states represent the different parts of speech, and the observations are the words in a sentence.

In the field of bioinformatics, HMMs are utilized for tasks like sequence alignment and gene finding. They can be used to model DNA or protein sequences, where the hidden states represent different structural elements, and the observations are the actual sequences.

One key characteristic of HMMs is that they operate under the Markov assumption, which assumes that the probability of transitioning to a particular state only depends on the previous state. This assumption simplifies the computation and allows for efficient algorithms, such as the Viterbi algorithm, which can efficiently find the most likely state sequence given the observations.

HMMs are also used in the field of speech recognition. In this application, the hidden states represent the phonemes and the observations correspond to the acoustic features of the speech signal.

HMMs can be trained using different algorithms, such as the Baum-Welch algorithm, which is a variation of the Expectation-Maximization (EM) algorithm. The Baum-Welch algorithm estimates the model parameters by iteratively updating them to maximize the likelihood of the training data.

Another important concept related to HMMs is the concept of the forward-backward algorithm, which can be used to compute the probability of being in a particular state at a given time, given the observed sequence.

In conclusion, Hidden Markov Models (HMMs) are probabilistic models widely used in various domains, including natural language processing, speech recognition, and bioinformatics. They are sequence models that compute a sequence of outputs based on a given sequence of inputs. HMMs model the joint distribution of observations and hidden states and operate under the Markov assumption. They can be trained using algorithms like the Baum-Welch algorithm and are utilized in tasks like sequence alignment, part-of-speech tagging, and speech recognition.