|
|
| Line 1: |
Line 1: |
| | | Bolts will certainly fit on the size of shed and saving money at the side If you [http://www.twitpic.com/tag/beloved beloved] this informative article along with you would want to obtain more details relating to [http://steelbuildingsguide.net/ Metal Garages Prices Nc] [https://www.gov.uk/search?q=kindly+check kindly check] out the site. . |
| 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}}
| |
Bolts will certainly fit on the size of shed and saving money at the side If you beloved this informative article along with you would want to obtain more details relating to Metal Garages Prices Nc kindly check out the site. .