hidden markov model

a hidden markov model (HMM) is a probabilistic sequence model: given a sequence of units (words, letters, morphemes, sentences, whatever), it computes a probability distribution over possible sequences of labels and chooses the best label sequence.
a hidden markov model allows us to talk about both observed events (like words that we see in the input) and hidden events (like part-of-speech tags) that we think of as causal factors in our probabilistic model. an HMM is specified by the following components:

• <<>> : a set of states,
• <<>> : a transition probability matrix . each representing the probability of moving from state to state , \shortfor[such that]{s.t.} ,
• <<>> a sequence of observations, each one drawn from a vocabulary ,
• <<>> : a sequence of observation likelihoods, also called emission probabilities, each expressing the probability of an observation being generated from a state ,
• <<>> : an initial probability distribution over states. is the probability that the markov chain will start in state . some states may have , meaning that they cannot be initial states. also, .

a first-order hidden markov model instantiates two simplifying assumptions. first, as with a first-order markov chain, the probability of a particular state depends only on the previous state:
second, the probability of an output observation depends only on the state that produced the observation and not on any other states or any other observations:
[cite:;taken from @nlp_jurafsky_2020 chapter 8.4 part-of-speech tagging]