Login
Login

Telescoping Markov chain

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.https://ak2316.user.srcf.net/files/ib-markov-chains/markov-chains.pdf

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

: \theta_k = (\theta_k^1,\ldots,\theta_k^N)\in\mathcal S^1\times\cdots\times\mathcal S^N

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

  1. \theta_k^1 is a Markov chain with transition probability matrix \Lambda^1
  2. : \mathbb P(\theta_k^1=s\mid\theta_{k-1}^1=r)=\Lambda^1(s\mid r)
  3. there is a cascading dependence in every level of the hierarchy,
  4. : \mathbb P(\theta_k^n=s\mid\theta_{k-1}^n=r,\theta_k^{n-1}=t)=\Lambda^n(s\mid r,t)     for all n\geq 2.
  5. \theta_k satisfies a Markov property with a transition kernel that can be written in terms of the \Lambda's,
  6. :\mathbb P(\theta_{k+1}=\vec s\mid \theta_k=\vec r) = \Lambda^1(s_1\mid r_1) \prod_{\ell=2}^N \Lambda^\ell(s_\ell \mid r_\ell,s_{\ell-1})

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

References

{{reflist}}

{{DEFAULTSORT:Telescoping Markov Chain}}

{{Probability-stub}}

Category:Markov processes