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Let $\tilde\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $\tilde Q$ and target distribution $\tilde\mu$.

I want to choose $(w_i)_{i\in I}$ such that $\tilde\kappa$ is geometrically ergodic with a preferably small rate $\rho\in(0,1)$, i.e. $$\left\|\tilde\kappa^n(\tilde x,\;\cdot\;)-\tilde\mu\right\|\le M(x)\rho^n\;\;\;\text{for all }n\in\mathbb N,\tag1,$$ where $\left\|\;\cdot\;\right\|$ denotes the total variation norm and $M(x)<\infty$, for $\tilde\mu$-a.e. $\tilde x\in\tilde E$. What would be a sensible approach to find the right (possibly best) choice of $(w_i)_{i\in I}$ (nearly) minimizing $\rho$?

Definitions: Let

  • $(E,\mathcal E,\lambda)$, $(E',\mathcal E',\lambda')$ be measurable spaces;
  • $I$ be a finite nonempty set;
  • $p,q_i$ be probability densities on $(E,\mathcal E,\lambda)$ for $i\in I$;
  • $w_i:E\to[0,1]$ be $\mathcal E$-measurable with $\sum_{i\in I}w_i=1$;
  • $\varphi_i:E'\to E$ be bijective and $(\mathcal E',\mathcal E)$-measurable with $\lambda'\circ\varphi_i^{-1}=q_i\lambda$ for $i\in I$;
  • $w'_i:=w_i\circ\varphi_i$ and $p'_i:=\frac p{q_i}\circ\varphi_i$ for $i\in I$;
  • $\zeta$ denote the counting measure on $(I,2^I)$;
  • $(\tilde E,\tilde{\mathcal E},\tilde\lambda):=(I\times E',2^I\otimes\mathcal E',\zeta\otimes\lambda')$
  • $\tilde p:=w'p'$ and $\tilde\mu:=\tilde p\tilde\lambda$;
  • $\tilde q:\tilde E^2\to[0,\infty)$ be symmetric and $\tilde{\mathcal E}^{\otimes2}$-measurable with $\int\tilde\lambda({\rm d}\tilde y)\tilde q(\tilde x,\tilde y)=1$ for all $\tilde x\in\tilde E$ and $\tilde Q(\tilde x,\;\cdot\;):=\tilde q(\tilde x,\;\cdot\;)\tilde\lambda$;
  • $\tilde\alpha(\tilde x,\tilde y):=1\wedge\frac{\tilde p(\tilde y)}{\tilde p(\tilde x)}$ for $\tilde x,\tilde y\in\tilde E$;
  • $\tilde\kappa(\tilde x,\tilde B):=\int_{\tilde B}\tilde Q(\tilde x,{\rm d}\tilde y)\tilde\alpha(\tilde x,\tilde y)+\left(1-\int\tilde Q(\tilde x,{\rm d}\tilde y)\tilde\alpha(\tilde x,\tilde y)\right)1_{\tilde B}(\tilde x)$ for $(\tilde x,\tilde B)\in\tilde E\times\tilde{\mathcal E}$.
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