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| In [[probability theory]] and [[statistics]], the term '''Markov property''' refers to the memoryless property of a [[stochastic process]]. It is named after the [[Russia]]n [[mathematician]] [[Andrey Markov]].<ref>Markov, A. A. (1954). ''Theory of Algorithms''. [Translated by Jacques J. Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [Jerusalem, Israel Program for Scientific Translations, 1961; available from Office of Technical Services, [[United States Department of Commerce]]] Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: ''Teoriya algorifmov''. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS 60-51085.]</ref>
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| A stochastic process has the Markov property if the [[conditional probability distribution]] of future states of the process (conditional on both past and present values) depends only upon the present state, not on the sequence of events that preceded it. A process with this property is called a ''[[Markov process]]''. 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]]. Both the terms "Markov property" and "strong Markov property" have been used in connection with a particular "memoryless" property of the [[exponential distribution#Memorylessness|exponential distribution]].<ref>[[William Feller|Feller, W.]] (1971) ''Introduction to Probability Theory and Its Applications, Vol II'' (2nd edition),Wiley. ISBN 0-471-25709-5 (pages 9 and 20)</ref>
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| The term '''Markov assumption''' is used to describe a model where the Markov property is assumed to hold, such as a [[hidden Markov model]].
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| A [[Markov random field]],<ref>Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'' OUP. ISBN 0-19-850994-4</ref> extends this property to two or more dimensions or to random variables defined for an interconnected network of items. An example of a model for such a field is the [[Ising model]].
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| A discrete-time stochastic process satisfying the Markov property is known as a [[Markov chain]].
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| ==Introduction==
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| A stochastic process has the Markov property if the [[conditional probability distribution]] of future states of the process (conditional on both past and present values) depends only upon the present state; that is, given the present, the future does not depend on the past. A process with this property is said to be '''Markovian''' or a '''[[Markov process]]'''. The most famous Markov process is a [[Markov chain]]. [[Brownian motion]] is another well-known Markov process.
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| ==History==
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| {{Main|Markov chain#History}}
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| ==Definition==
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| Let <math>(\Omega,\mathcal{F},\mathbb{P})</math> be a [[probability space]] with a [[Filtration_(mathematics)#Measure_theory|filtration]] <math>(\mathcal{F}_s,\ s \in I)</math>, for some ([[totally ordered]]) index set <math>I</math>; and let <math>(S,\mathcal{S})</math> be a [[measurable space]]. A <math>(S,\mathcal{S})</math>-valued stochastic process <math>X=(X_t,\ t\in I)</math> adapted to the filtration is said to possess the ''Markov property'' if, for each <math>A \in \mathcal{S}</math> and each <math>s,t\in I</math> with <math>s<t</math>,
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| :<math>\mathbb{P}(X_t \in A |\mathcal{F}_s) = \mathbb{P}(X_t \in A| X_s).</math><ref>Durrett, Rick. ''Probability: Theory and Examples''. Fourth Edition. Cambridge: Cambridge University Press, 2010.</ref> | |
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| In the case where <math>S</math> is a discrete set with the [[Sigma-algebra#Examples|discrete sigma algebra]] and <math>I = \mathbb{N}</math>, this can be reformulated as follows:
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| :<math>\mathbb{P}(X_n=x_n|X_{n-1}=x_{n-1} \dots X_0=x_0)=\mathbb{P}(X_n=x_n|X_{n-1}=x_{n-1})</math>.
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| ==Alternative formulations==
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| Alternatively, the Markov property can be formulated as follows.
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| :<math>\mathbb{E}[f(X_t)|\mathcal{F}_s]=\mathbb{E}[f(X_t)|\sigma(X_s)]</math>
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| for all <math>t\geq s\geq 0</math> and <math>f:S\rightarrow \mathbb{R}</math> bounded and measurable.
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| ==Strong Markov property==
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| Let <math>\mathcal{F}_{\tau^+}=\{A \in \mathcal{F}: \{\tau=t\} \cap A \in \mathcal{F}_{t+} ,\, \forall t \geq 0\}</math>.
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| Suppose that <math>X=(X_t:t\geq 0)</math> is a [[stochastic process]] on a [[probability space]] <math>(\Omega,\mathcal{F},\mathbb{P})</math> with natural [[filtration (mathematics)|filtration]] <math>\{\mathcal{F}_t\}_{t\geq 0}</math>. Then <math>X</math> is said to have the strong Markov property if, for each [[stopping time]] <math>\tau</math>, conditioned on the event <math>\{\tau < \infty\}</math>, we have that for each <math>t\ge 0</math>, <math>X_{\tau + t}</math> is independent of <math>\mathcal{F}_{\tau^+}</math> given <math>X_\tau</math>.
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| The strong Markov property implies the ordinary Markov property, since by taking the stopping time <math>\tau=t</math>, the ordinary Markov property can be deduced.
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| <!-- Hide for now ... name '''Markov-type property''' seems doubtful
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| ==Markov Type Property==
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| A stochastic process has a '''Markov-type property''' if the process's [[random variable]]s determine a set of probabilities can be factored in a way that yields the '''Markov property'''. Useful in applied research, members of such classes{{Clarify|October 2009|date=October 2009}} defined by their mathematics or area of application{{Clarify|October 2009|date=October 2009}} are referred to as '''[[Markov random field]]s'''., and occur in many situations. The [[Ising model]] is a prototypical example. -->
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| ==Applications==
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| A very important{{Citation needed|date=March 2012}} application of the Markov property in a generalized form is in [[Markov chain Monte Carlo]] computations in the context of [[Bayesian statistics]].
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| ==See also==
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| *[[Markov chain]]
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| *[[Markov blanket]]
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| *[[Markov decision process]]
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| *[[Causal Markov condition]]
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| *[[Markov model]]
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| *[[Chapman–Kolmogorov equation]]
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| == References ==
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| {{Reflist}}
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| [[Category:Markov models]]
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| [[Category:Markov processes]]
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