Hidden Markov Models, Theory and Applications
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2014-05-23
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1
History and Theoretical Basics
of Hidden Markov Models
Guy Leonard Kouemou
EADS Deutschland GmbH,
Germany
1. Introduction
The following chapter can be understood as one sort of brief introduction to the history and
basics of the Hidden Markov Models.
Hidden Markov Models (HMMs) are learnable finite stochastic automates. Nowadays, they
are considered as a specific form of dynamic Bayesian networks. Dynamic Bayesian
networks are based on the theory of Bayes (Bayes & Price, 1763).
A Hidden Markov Model consists of two stochastic processes. The first stochastic process is
a Markov chain that is characterized by states and transition probabilities. The states of the
chain are externally not visible, therefore “hidden”. The second stochastic process produces
emissions observable at each moment, depending on a state-dependent probability
distribution. It is important to notice that the denomination “hidden” while defining a
Hidden Markov Model is referred to the states of the Markov chain, not to the parameters of
the model.
The history of the HMMs consists of two parts. On the one hand there is the history of
Markov process and Markov chains, and on the other hand there is the history of algorithms
needed to develop Hidden Markov Models in order to solve problems in the modern
applied sciences by using for example a computer or similar electronic devices.
1.1. Brief history of Markov process and Markov chains
Andrey Andreyevich Markov (June 14, 1856 – July 20, 1922) was a Russian mathematician.
He is best known for his work on the theory of stochastic Markov processes. His research
area later became known as Markov process and Markov chains.
Andrey Andreyevich Markov introduced the Markov chains in 1906 when he produced the
first theoretical results for stochastic processes by using the term “chain” for the first time. In
1913 he calculated letter sequences of the Russian language.
A generalization to countable infinite state spaces was given by Kolmogorov (1931). Markov
chains are related to Brownian motion and the ergodic hypothesis, two topics in physics
which were important in the early years of the twentieth century. But Markov appears to
have pursued this out of a mathematical motivation, namely the extension of the law of
large numbers to dependent events.
Out of this approach grew a general statistical instrument, the so-called stochastic Markov
process.
In mathematics generally, probability theory and statistics particularly, a Markov process
can be considered as a time-varying random phenomenon for which Markov properties are
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