# Keeping track of the variance of a Metropolis-Hastings estimator

Let $$(E,\mathcal E,\lambda)$$ and $$(E',\mathcal E',\lambda')$$ be measure spaces, $$p,q$$ be probability densities on $$(E,\mathcal E,\lambda)$$, and $$\varphi:E'\to E$$ be bijective and $$(\mathcal E',\mathcal E)$$-measurable with $$\lambda'\circ\varphi^{-1}=q\lambda$$.

Let $$(X_n)_{n\in\mathbb N_0}$$ denote the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $$R$$, $$R(x,B)=\int_B\lambda({\rm d}y)q(y)r\left(\varphi^{-1}(x),\varphi^{-1}(y)\right)\;\;\;\text{for all }(x,B)\in E\times\mathcal E\tag1$$ for some $$\mathcal E'\otimes\mathcal E'$$-measurable $$r:E'\times E'\to[0,\infty)$$, and target distribution $$\mu:=p\lambda$$ and $$(Y_n)_{n\in\mathbb N}$$ denote the corresponding proposal sequence.

Consider the estimator $$A_ng:=\frac{\sum_{i=1}^n\rho(X_{i-1},Y_i)\frac gp(Y_i)}{\sum_{i=1}^n\rho(X_{i-1},Y_i)}$$ of $$\lambda g$$, where $$\rho(x,y):=\frac{p(y)}{q(y)r\left(\varphi^{-1}(x),\varphi^{-1}(y)\right)}\;\;\;\text{for }x,y\in E.$$ The asymptotic variance is given by $$\sigma^2(g):=\int\mu({\rm d}x)\underbrace{\int\lambda({\rm d}y)\frac{|g(y)|^2}{q(y)r\left(\varphi^{-1}(x),\varphi^{-1}(y)\right)}}_{=:\:a_g(x)}.$$

Are we able to estimate the quantity $$a_g(x)$$ (for fixed $$g$$ and $$x\in E$$) using the estimate $$A_ng$$?

Remark: What I'm trying to achieve is analogous to the way we estimate the variance in importance sampling; see, for example, equation 9.5 in https://statweb.stanford.edu/~owen/mc/Ch-var-is.pdf.