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The term strong Markov property is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a stopping time. The term Markov assumption is used to describe a model where the Markov property is assumed to hold, such as a hidden Markov model .
In this context, the Markov property indicates that the distribution for this variable depends only on the distribution of a previous state. An example use of a Markov chain is Markov chain Monte Carlo, which uses the Markov property to prove that a particular method for performing a random walk will sample from the joint distribution.
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The Hammersley–Clifford theorem is a result in probability theory, mathematical statistics and statistical mechanics that gives necessary and sufficient conditions under which a strictly positive probability distribution (of events in a probability space) [clarification needed] can be represented as events generated by a Markov network (also known as a Markov random field).
In probability theory and ergodic theory, a Markov operator is an operator on a certain function space that conserves the mass (the so-called Markov property). If the underlying measurable space is topologically sufficiently rich enough, then the Markov operator admits a kernel representation. Markov operators can be linear or non-linear.
The prototypical Markov random field is the Ising model; indeed, the Markov random field was introduced as the general setting for the Ising model. [2] In the domain of artificial intelligence, a Markov random field is used to model various low- to mid-level tasks in image processing and computer vision. [3]
Markov random fields, also known as undirected graphical models are common representations for this problem.Given an undirected graph G = (V, E), a set of random variables X = (X v) v ∈ V indexed by V, form a Markov random field with respect to G if they satisfy the pairwise Markov property:
Markov chain; Markov chain central limit theorem; Markov chain geostatistics; Markov chain Monte Carlo; Markov partition; Markov property; Markov switching multifractal; Markovian discrimination; Maximum-entropy Markov model; MegaHAL; Models of DNA evolution; MRF optimization via dual decomposition; Multiple sequence alignment