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Concept
Hidden Markov model
Summary
A hidden Markov model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process — call it — with unobservable ("hidden") states. As part of the definition, HMM requires that there be an observable process whose outcomes are "influenced" by the outcomes of in a known way. Since cannot be observed directly, the goal is to learn about by observing HMM has an additional requirement that the outcome of at time must be "influenced" exclusively by the outcome of at and that the outcomes of and at must be conditionally independent of at given at time
Hidden Markov models are known for their applications to thermodynamics, statistical mechanics, physics, chemistry, economics, finance, signal processing, information theory, pattern recognition—such as speech, handwriting, gesture recognition, part-of-speech tagging, musical score following, partial discharges and bioinformatics.
Let and be discrete-time stochastic processes and . The pair is a hidden Markov model if
is a Markov process whose behavior is not directly observable ("hidden");
for every and every Borel set .
Let and be continuous-time stochastic processes. The pair is a hidden Markov model if
is a Markov process whose behavior is not directly observable ("hidden");
for every every Borel set and every family of Borel sets
The states of the process (resp. are called hidden states, and (resp. is called emission probability or output probability.
In its discrete form, a hidden Markov process can be visualized as a generalization of the urn problem with replacement (where each item from the urn is returned to the original urn before the next step). Consider this example: in a room that is not visible to an observer there is a genie. The room contains urns X1, X2, X3, ... each of which contains a known mix of balls, each ball labeled y1, y2, y3, ... . The genie chooses an urn in that room and randomly draws a ball from that urn. It then puts the ball onto a conveyor belt, where the observer can observe the sequence of the balls but not the sequence of urns from which they were drawn.
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