we set the batch size to be $b = \alpha \lfloor n^{1/3} \rfloor$ where $\alpha$ is estimated using the methods of [@fleg:jone:2010; @liu2018batch]. For fast mixing Markov chains, we set $n = \lfloor n^{1/2} \rfloor$. For all examples, $h$ is the identity function[^2].

Bivariate normal Gibbs sampler
------------------------------

Consider sampling from a bivariate normal distribution using a Gibbs sampler. For $\omega_1, \omega_2 > 0$ and $\rho$ such that $\rho^2 < \omega_1 \omega_2$, the target distribution is $$\left(\begin{array}{c}
  X_1 \\ X_2
\end{array} \right) \sim 
 N \begin{pmatrix}
\begin{pmatrix}
\mu_{1}\\
\mu_{2}
\end{pmatrix}, 
\begin{pmatrix}
\omega_1 & \rho \\
\rho & \omega_2
\end{pmatrix}
\end{pmatrix}$$ The Gibbs sampler updates the Markov chain by using the following full conditional distributions: $$\label{eqn:gibbs1}
X_{1} \mid X_{2} \sim  N\left(\mu_{1} + \dfrac{\rho}{\omega_2}\left(X_{2} - \mu_{2}\right) , \omega_1 - \dfrac{\rho^{2}}{\omega_2}\right) \; .$$ $$\label{eqn:gibbs2}
X_{2} \mid X_{1} \sim  N\left(\mu_{2} + \dfrac{\rho}{\omega_1}\left(X_{1} - \mu_{1}\right) , \omega_2 - \dfrac{\rho^{2}}{\omega_1}\right) \; .$$ Although a seemingly simple example, the Gibbs sampler can exhibit arbitrarily fast or slow mixing based on the correlation in target distribution. In fact, [@rob:sahu:1996] make the case for approximating general Gibbs samplers with unimodal target distributions with a similar Gaussian Gibbs sampler. In the following theorem, we obtain the exact form of the asymptotic 