Markov chain - Revision history

Carl McBride at 11:56, 14 August 2007

2007-08-14T11:56:45Z

← Older revision		Revision as of 13:56, 14 August 2007
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	The concept of a '''Markov chain''' was developed by Andrey Andreyevich Markov.		The concept of a '''Markov chain''' was developed by Andrey Andreyevich Markov. A Markov chain is a sequence of random variables with the property that it is forgetful of all but its immediate past.
			For a process <math>{\mathbf \Phi}</math> evolving on a space <math>{\mathsf X}</math> and governed by an overall probability law <math>{\mathsf P}</math> to be a time-homogeneous Markov chain there must be a set of "transition probabilities" <math>\{P^n (x,A), x \in {\mathsf X}, A \subset {\mathsf X}\}</math> for appropriate sets <math>A</math> such that
			for times <math>n,m</math> in <math>{\mathbb Z}_+</math> (Ref. 1 Eq. 1.1)

			:<math>{\mathsf P} (\Phi_{n+m} \in A \vert \Phi_j,j \leq m; \Phi_m =x)= P^n(x,A);</math>

			that is <math>P^n(x,A)</math> denotes the probability that a chain at ''x'' will be in the set ''A'' after ''n'' steps, or transitions. The independence of <math>P^n</math> on the values of <math>\Phi_j,j \leq m</math> is the Markov property,
			and the independence of <math>P^n</math> and ''m'' is the time-homogeneity property.
	==References==		==References==
	#[http://probability.ca/MT/ S. P. Meyn and R. L. Tweedie "Markov Chains and Stochastic Stability", Springer-Verlag, London (1993)]		#[http://probability.ca/MT/ S. P. Meyn and R. L. Tweedie "Markov Chains and Stochastic Stability", Springer-Verlag, London (1993)]

Carl McBride at 11:29, 14 August 2007

2007-08-14T11:29:27Z

← Older revision		Revision as of 13:29, 14 August 2007
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			The concept of a '''Markov chain''' was developed by Andrey Andreyevich Markov.
	==References==		==References==
			#[http://probability.ca/MT/ S. P. Meyn and R. L. Tweedie "Markov Chains and Stochastic Stability", Springer-Verlag, London (1993)]
	#[http://dx.doi.org/10.1063/1.2743003 Ruichao Ren and G. Orkoulas "Parallel Markov chain Monte Carlo simulations", Journal of Chemical Physics '''126''' 211102 (2007)]		#[http://dx.doi.org/10.1063/1.2743003 Ruichao Ren and G. Orkoulas "Parallel Markov chain Monte Carlo simulations", Journal of Chemical Physics '''126''' 211102 (2007)]
	[[Category: Computer simulation techniques]]		[[Category: Computer simulation techniques]]
	[[category: Monte Carlo]]		[[category: Monte Carlo]]

Carl McBride: New page: ==References== #[http://dx.doi.org/10.1063/1.2743003 Ruichao Ren and G. Orkoulas "Parallel Markov chain Monte Carlo simulations", Journal of Chemical Physics '''126''' 211102 (2007)] [...

2007-06-12T15:43:25Z

New page: ==References== #[http://dx.doi.org/10.1063/1.2743003 Ruichao Ren and G. Orkoulas "Parallel Markov chain Monte Carlo simulations", Journal of Chemical Physics '''126''' 211102 (2007)] [...

New page

==References==
#[http://dx.doi.org/10.1063/1.2743003 Ruichao Ren and G. Orkoulas "Parallel Markov chain Monte Carlo simulations", Journal of Chemical Physics '''126''' 211102 (2007)]
[[Category: Computer simulation techniques]]
[[category: Monte Carlo]]