In mathematics, a **Markov information source**, or simply, a **Markov source**, is an information source whose underlying dynamics are given by a stationary finite Markov chain.
[edit]Formal definition
An **information source** is a sequence of random variables ranging over a finite alphabet Γ, having a stationary distribution.
A Markov information source is then a (stationary) Markov chain *M*, together with a function
that maps states *S* in the Markov chain to letters in the alphabet Γ.
A **unifilar Markov source** is a Markov source for which the values *f*(*s*_{k}) are distinct whenever each of the states *s*_{k} are reachable, in one step, from a common prior state. Unifilar sources are notable in that many of their properties are far more easily analyzed, as compared to the general case.
[edit]Applications
Markov sources are commonly used in communication theory, as a model of a transmitter. Markov sources also occur in natural language processing, where they are used to represent hidden meaning in a text. Given the output of a Markov source, whose underlying Markov chain is unknown, the task of solving for the underlying chain is undertaken by the techniques of hidden Markov models, such as the Viterbi algorithm.
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A **Markov chain** is the characterization of a system that transits from one state to another, in a chainlike manner. It concerns any __random process__ endowed with the __Markov property__: *i.e.,* the property, simply said, that the next state depends only on the current state and not on the past. It is a __Markov model__, named for __Andrey Markov__, for a particular type of __Markov process__ in which the process can only be in a finite or countable number of states. Markov chains are useful as tools for __statistical modeling__ in almost all fields of modern applied mathematics.
in __probability theory__ and __statistics__, a **Markov process**, named after the Russian mathematician __Andrey Markov__, is a time-varying random phenomenon for which a specific property (the __Markov property__) holds.
the term **Markov property** refers to the memoryless property of a __stochastic process__.
in a stochastic or random process there is some indeterminacy in its future evolution described by probability distributions. This means that even if the initial condition (or starting point) is known, there are many possibilities the process might go to, but some paths may be more probable and others less so. |