hidden Markov model

(data structure)

Definition: A variant of a finite state machine having a set of states, Q, an output alphabet, O, transition probabilities, A, output probabilities, B, and initial state probabilities, Π. The current state is not observable. Instead, each state produces an output with a certain probability (B). Usually the states, Q, and outputs, O, are understood, so an HMM is said to be a triple, (A, B, Π).

Formal Definition: After Michael Cohen's lectures for CN760.

A = {a_ij = P(q_j at t+1 | q_i at t)}, where P(a | b) is the conditional probability of a given b, t ≥ 1 is time, and q_i ∈ Q.
Informally, A is the probability that the next state is q_j given that the current state is q_i.
B = {b_ik = P(o_k | q_i)}, where o_k ∈ O.
Informally, B is the probability that the output is o_k given that the current state is q_i.
Π = {p_i = P(q_i at t=1)}.

Also known as HMM.

Generalization (I am a kind of ...)
finite state machine.

Aggregate parent (I am a part of or used in ...)
Baum Welch algorithm, Viterbi algorithm.

See also Markov chain.

Note: Computing a model given sets of sequences of observed outputs is very difficult, since the states are not directly observable and transitions are probabilistic. One method is the Baum Welch algorithm.

Although the states cannot, by definition, be directly observed, the most likely sequence of sets for a given sequence of observed outputs can be computed in O(nt), where n is the number of states and t is the length of the sequence. One method is the Viterbi algorithm.

Thanks to Arvind <uk_arvind@mail.utexas.edu> May 2002.

Named after Andrei Andreyevich Markov (1856 - 1922), who studied poetry and other texts as stochastic sequences of characters.

Author: PEB

More information

L. E. Baum, An inequality and associated maximization technique in statistical estimation for probabilistic functions of Markov processes, Inequalities, 3:1-8, 1972.

Go to the Dictionary of Algorithms and Data Structures home page.

If you have suggestions, corrections, or comments, please get in touch with Paul Black.

Entry modified 21 December 2020.
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Cite this as:
Paul E. Black, "hidden Markov model", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed. 21 December 2020. (accessed TODAY) Available from: https://www.nist.gov/dads/HTML/hiddenMarkovModel.html