Let
- $(E,\mathcal E,\lambda)$ and $(E',\mathcal E',\lambda')$ be measure spaces
- $I$ be a finite nonempty set
- $\varphi_i:E'\to E$ be $(\mathcal E',\mathcal E)$-measurable with $$\lambda'\circ\varphi_i^{-1}=q_i\lambda\tag1$$ for $i\in I$
- $p,q_i:E\to[0,\infty)$ be $\mathcal E$-measurable with $$\int p\:{\rm d}\lambda=\int q_i\:{\rm d}\lambda=1$$ for $i\in I$
- $w_i:E\to[0,1]$ be $\mathcal E$-measurable, $w_i':=w_i\circ\varphi_i$ and $$p_i':=\begin{cases}\displaystyle\frac p{q_i}\circ\varphi_i&\text{on }\left\{q_i\circ\varphi_i>0\right\}\\0&\text{on }\left\{q_i\circ\varphi_i=0\right\}\end{cases}$$ for $i\in I$
- $\zeta$ denote the counting measure on $(I,2^I)$ and $$\nu':=w'p'(\zeta\otimes\lambda')$$
- $r:(I\times E')^2\to[0,\infty)$ be symmetric (i.e. $r((i,x'),(j,y'))=r((j,y'),(i,x'))$ for all $(i,x'),(j,y')\in I\times E'$) and $\left(2^I\otimes\mathcal E'\right)^{\otimes2}$-measurable and $$R((i,x'),\;\cdot\;):=r((i,x'),\;\cdot\;)(\zeta\otimes\lambda')\;\;\;\text{for }(i,x')\in I\times E'$$
Let $\kappa$ denote the transition kernel of the time-homogeneous Markov chain $\left((T_n,X'_n)\right)_{n\in\mathbb N}$ generated by the Metropolis-Hastings algorithm with proposal $R$ and target $\nu'$. Now let $$X_n:=\varphi_{T_n}(X'_n)\;\;\;\text{for }n\in\mathbb N_0.$$ Can we give a closed form expression for the transition kernel $\pi$ of $(X_n)_{n\in\mathbb N_0}$?
Note that, by definition, $$\kappa((i,x'),\{j\}\times C')=\int_{C'}\lambda'({\rm d}y')r((i,x'),(j,y'))\alpha((i,x'),(j,y'))+d(i,x')1_{\{i\}}(j)1_{C'}(x')\tag1$$ for all $(i,x')\in I\times E'$ and $(j,C')\in I\times\mathcal E'$, where $$\alpha((i,x'),(j,y'))=\begin{cases}\displaystyle\min\left(1,\frac{(w'p')(j,y')}{(w'p')(i,x')}\right)&\text{, if }(w'p')(i,x')\ne0\\1&\text{, otherwise}\end{cases}$$ for $(i,x'),(j,y')\in I\times E'$ and $$d(i,x')=1-\int R((i,x'),{\rm d}(j,y'))\alpha((i,x'),(j,y'))\;\;\;\text{for }(i,x')\in I\times E'.$$