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.^[1]

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

θ k = ( θ k 1 , … , θ k N ) ∈ S 1 × ⋯ × S N {\displaystyle \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 { Λ n } n = 1 N {\displaystyle \{\Lambda ^{n}\}_{n=1}^{N}} such that

1. θ k 1 {\displaystyle \theta _{k}^{1}} is a Markov chain with transition probability matrix Λ 1 {\displaystyle \Lambda ^{1}}

   P ( θ k 1 = s ∣ θ k − 1 1 = r ) = Λ 1 ( s ∣ r ) {\displaystyle \mathbb {P} (\theta _{k}^{1}=s\mid \theta _{k-1}^{1}=r)=\Lambda ^{1}(s\mid r)}

2. there is a cascading dependence in every level of the hierarchy,

   P ( θ k n = s ∣ θ k − 1 n = r , θ k n − 1 = t ) = Λ n ( s ∣ r , t ) {\displaystyle \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 ≥ 2. {\displaystyle n\geq 2.}

3. θ k {\displaystyle \theta _{k}} satisfies a Markov property with a transition kernel that can be written in terms of the Λ {\displaystyle \Lambda } 's,

   P ( θ k + 1 = s → ∣ θ k = r → ) = Λ 1 ( s 1 ∣ r 1 ) ∏ ℓ = 2 N Λ ℓ ( s ℓ ∣ r ℓ , s ℓ − 1 ) {\displaystyle \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 s → = ( s 1 , … , s N ) ∈ S 1 × ⋯ × S N {\displaystyle {\vec {s}}=(s_{1},\ldots ,s_{N})\in {\mathcal {S}}^{1}\times \cdots \times {\mathcal {S}}^{N}} and r → = ( r 1 , … , r N ) ∈ S 1 × ⋯ × S N . {\displaystyle {\vec {r}}=(r_{1},\ldots ,r_{N})\in {\mathcal {S}}^{1}\times \cdots \times {\mathcal {S}}^{N}.}

References