Kepler-4b - Revision history

en>Tom.Reding: WP:GenFixes (page/s, endash, , unicodify, and/or concising wikilinks, etc.), et al., ed., ref cleanup using AWB

2015-01-06T22:56:14Z

WP:GenFixes (page/s, endash, &nbsp, unicodify, and/or concising wikilinks, etc.), et al., ed., ref cleanup using AWB

← Older revision		Revision as of 00:56, 7 January 2015
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	~~In [[probability theory]], a '''telescoping Markov chain (TMC)''' is a vector-valued [[stochastic process]] that satisfies a [[Markov property]]~~ and ~~admits a hierarchical format through a network of transition matrices with cascading dependence.~~

	~~For any <math>N> 1</math> consider~~ the ~~set of spaces <math>\{\mathcal S^\ell\}_{\ell=1}^N<~~/~~math>. The hierarchical process <math>\theta_k<~~/~~math> defined in the product-space~~

	~~: <math>\theta_k = (\theta_k^1,.....,\theta_k^N)\in\mathcal S^1\times.~~..~~...\times\mathcal S^N</math>~~

	~~is said to be a TMC if there is a set of transition probability kernels <math>\{\Lambda^n\}_{n=1}^N<~~/~~math> such that~~

	~~'''(1)''' <math>\theta_k^1<~~/~~math> is a [[Markov chain]~~] with ~~transition probability matrix <math>\Lambda^1</math>~~

	: ~~<math>\mathbb P(\theta_k^1=s\|\theta_{k-1}^1=r)=\Lambda^1(s\|r)<~~/~~math>~~

	~~'''(2)''' there is a cascading dependence in every level of the hierarchy,~~

	~~: <math>\mathbb P(\theta_k^n=s\|\theta_{k-1}^n=r,\theta_k^{n-1}=t)=\Lambda^n(s\|r,t)<~~/~~math>     for all <math>n\geq 2~~.</~~math>~~

	~~'''(3)''' <math>\theta_k</math> satisfies a Markov property with a transition kernel that can be written in terms of the <math>\Lambda</math>'s,~~

	:~~<math>\mathbb P(\theta_{k+1}=\vec s\|\theta_k=\vec r)=\Lambda^1(s_1\|r_1)\prod_{\ell=2}^N\Lambda^\ell(s_\ell\|r_\ell,s_{\ell-1})<~~/~~math>~~

	~~where <math>\vec s = (s_1,\ldots,s_N)\in\mathcal S^1\times\cdots\times\mathcal S^N<~~/~~math> and <math>\vec r = (r_1,\ldots,r_N)\in\mathcal S^1\times\cdots\times\mathcal S^N~~.</~~math>~~

	~~{{Unreferenced\|date~~=~~September 2010}}~~

	~~{{DEFAULTSORT:Telescoping Markov Chain}}~~
	~~[[Category:Markov processes~~]]


	~~{{Probability-stub}}~~

en>Nickst: removed Category:Exoplanetology using HotCat

2013-11-23T01:32:19Z

removed Category:Exoplanetology using HotCat

← Older revision		Revision as of 03:32, 23 November 2013
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			In [[probability theory]], a '''telescoping Markov chain (TMC)''' is a vector-valued [[stochastic process]] that satisfies a [[Markov property]] and admits a hierarchical format through a network of transition matrices with cascading dependence.

			For any <math>N> 1</math> consider the set of spaces <math>\{\mathcal S^\ell\}_{\ell=1}^N</math>. The hierarchical process <math>\theta_k</math> defined in the product-space

			: <math>\theta_k = (\theta_k^1,.....,\theta_k^N)\in\mathcal S^1\times......\times\mathcal S^N</math>

			is said to be a TMC if there is a set of transition probability kernels <math>\{\Lambda^n\}_{n=1}^N</math> such that

			'''(1)''' <math>\theta_k^1</math> is a [[Markov chain]] with transition probability matrix <math>\Lambda^1</math>

			: <math>\mathbb P(\theta_k^1=s\|\theta_{k-1}^1=r)=\Lambda^1(s\|r)</math>

			'''(2)''' there is a cascading dependence in every level of the hierarchy,

			: <math>\mathbb P(\theta_k^n=s\|\theta_{k-1}^n=r,\theta_k^{n-1}=t)=\Lambda^n(s\|r,t)</math>     for all <math>n\geq 2.</math>

			'''(3)''' <math>\theta_k</math> satisfies a Markov property with a transition kernel that can be written in terms of the <math>\Lambda</math>'s,

			:<math>\mathbb P(\theta_{k+1}=\vec s\|\theta_k=\vec r)=\Lambda^1(s_1\|r_1)\prod_{\ell=2}^N\Lambda^\ell(s_\ell\|r_\ell,s_{\ell-1})</math>

			where <math>\vec s = (s_1,\ldots,s_N)\in\mathcal S^1\times\cdots\times\mathcal S^N</math> and <math>\vec r = (r_1,\ldots,r_N)\in\mathcal S^1\times\cdots\times\mathcal S^N.</math>

			{{Unreferenced\|date=September 2010}}

			{{DEFAULTSORT:Telescoping Markov Chain}}
			[[Category:Markov processes]]


			{{Probability-stub}}

en>Mike s: update eccentricity and ref

2012-08-05T19:10:33Z

update eccentricity and ref

New page

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